LMFDB - Embedded newform 338.3.d.f.239.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,3,Mod(99,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.99"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	338.d (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.20983293538$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.612074651904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 $ x^8 - 74x^6 + 2067x^4 - 25778*x^2 + 121801
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			239.2
Root			$-4.71318 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	338.239
Dual form			338.3.d.f.99.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} -3.84716 q^{3} -2.00000i q^{4} +(-3.77418 + 3.77418i) q^{5} +(3.84716 - 3.84716i) q^{6} +(-7.25532 - 7.25532i) q^{7} +(2.00000 + 2.00000i) q^{8} +5.80063 q^{9} -7.54837i q^{10} +(-7.40816 - 7.40816i) q^{11} +7.69432i q^{12} +14.5106 q^{14} +(14.5199 - 14.5199i) q^{15} -4.00000 q^{16} -4.88811i q^{17} +(-5.80063 + 5.80063i) q^{18} +(-18.6785 + 18.6785i) q^{19} +(7.54837 + 7.54837i) q^{20} +(27.9124 + 27.9124i) q^{21} +14.8163 q^{22} -19.9590i q^{23} +(-7.69432 - 7.69432i) q^{24} -3.48892i q^{25} +12.3085 q^{27} +(-14.5106 + 14.5106i) q^{28} +14.3044 q^{29} +29.0398i q^{30} +(-19.0056 + 19.0056i) q^{31} +(4.00000 - 4.00000i) q^{32} +(28.5004 + 28.5004i) q^{33} +(4.88811 + 4.88811i) q^{34} +54.7658 q^{35} -11.6013i q^{36} +(42.9072 + 42.9072i) q^{37} -37.3569i q^{38} -15.0967 q^{40} +(-3.54099 + 3.54099i) q^{41} -55.8247 q^{42} +11.9728i q^{43} +(-14.8163 + 14.8163i) q^{44} +(-21.8926 + 21.8926i) q^{45} +(19.9590 + 19.9590i) q^{46} +(7.59168 + 7.59168i) q^{47} +15.3886 q^{48} +56.2792i q^{49} +(3.48892 + 3.48892i) q^{50} +18.8053i q^{51} +77.0450 q^{53} +(-12.3085 + 12.3085i) q^{54} +55.9195 q^{55} -29.0213i q^{56} +(71.8590 - 71.8590i) q^{57} +(-14.3044 + 14.3044i) q^{58} +(-44.4819 - 44.4819i) q^{59} +(-29.0398 - 29.0398i) q^{60} -56.2764 q^{61} -38.0112i q^{62} +(-42.0854 - 42.0854i) q^{63} +8.00000i q^{64} -57.0007 q^{66} +(4.32207 - 4.32207i) q^{67} -9.77622 q^{68} +76.7856i q^{69} +(-54.7658 + 54.7658i) q^{70} +(40.4291 - 40.4291i) q^{71} +(11.6013 + 11.6013i) q^{72} +(-12.7990 - 12.7990i) q^{73} -85.8143 q^{74} +13.4224i q^{75} +(37.3569 + 37.3569i) q^{76} +107.497i q^{77} +7.98532 q^{79} +(15.0967 - 15.0967i) q^{80} -99.5584 q^{81} -7.08199i q^{82} +(35.8343 - 35.8343i) q^{83} +(55.8247 - 55.8247i) q^{84} +(18.4486 + 18.4486i) q^{85} +(-11.9728 - 11.9728i) q^{86} -55.0312 q^{87} -29.6326i q^{88} +(-57.2867 - 57.2867i) q^{89} -43.7853i q^{90} -39.9181 q^{92} +(73.1175 - 73.1175i) q^{93} -15.1834 q^{94} -140.992i q^{95} +(-15.3886 + 15.3886i) q^{96} +(38.7928 - 38.7928i) q^{97} +(-56.2792 - 56.2792i) q^{98} +(-42.9720 - 42.9720i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 6 q^{5} - 10 q^{7} + 16 q^{8} + 84 q^{9} - 42 q^{11} + 20 q^{14} - 60 q^{15} - 32 q^{16} - 84 q^{18} - 22 q^{19} + 12 q^{20} + 102 q^{21} + 84 q^{22} + 72 q^{27} - 20 q^{28} + 12 q^{29}+ \cdots - 252 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - 6 * q^5 - 10 * q^7 + 16 * q^8 + 84 * q^9 - 42 * q^11 + 20 * q^14 - 60 * q^15 - 32 * q^16 - 84 * q^18 - 22 * q^19 + 12 * q^20 + 102 * q^21 + 84 * q^22 + 72 * q^27 - 20 * q^28 + 12 * q^29 - 32 * q^31 + 32 * q^32 - 54 * q^33 + 60 * q^34 + 156 * q^35 - 32 * q^37 - 24 * q^40 + 12 * q^41 - 204 * q^42 - 84 * q^44 + 102 * q^45 + 108 * q^46 - 60 * q^47 + 88 * q^50 - 132 * q^53 - 72 * q^54 + 324 * q^55 + 294 * q^57 - 12 * q^58 - 234 * q^59 + 120 * q^60 - 72 * q^61 + 156 * q^63 + 108 * q^66 + 14 * q^67 - 120 * q^68 - 156 * q^70 - 162 * q^71 + 168 * q^72 - 166 * q^73 + 64 * q^74 + 44 * q^76 - 96 * q^79 + 24 * q^80 + 24 * q^81 + 240 * q^83 + 204 * q^84 + 234 * q^85 - 132 * q^86 - 720 * q^87 - 210 * q^89 - 216 * q^92 + 444 * q^93 + 120 * q^94 - 146 * q^97 + 16 * q^98 - 252 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$3$	−3.84716			−1.28239			−0.641193	−	0.767380i	$-0.721560\pi$
$3$	−3.84716			−1.28239			−0.641193	+	0.767380i	$0.721560\pi$
$4$		−	2.00000i		−	0.500000i
$5$	−3.77418	+	3.77418i	−0.754837	+	0.754837i	−0.975378	−	0.220541i	$-0.929218\pi$
$5$	−3.77418	+	3.77418i	−0.754837	+	0.754837i	0.220541	+	0.975378i	$0.429218\pi$
$6$	3.84716	−	3.84716i	0.641193	−	0.641193i
$7$	−7.25532	−	7.25532i	−1.03647	−	1.03647i	−0.999309	−	0.0371646i	$-0.988167\pi$
$7$	−7.25532	−	7.25532i	−1.03647	−	1.03647i	−0.0371646	−	0.999309i	$-0.511833\pi$
$8$	2.00000	+	2.00000i	0.250000	+	0.250000i
$9$	5.80063			0.644514
$10$		−	7.54837i		−	0.754837i
$11$	−7.40816	−	7.40816i	−0.673469	−	0.673469i	0.285045	−	0.958514i	$-0.407991\pi$
$11$	−7.40816	−	7.40816i	−0.673469	−	0.673469i	−0.958514	+	0.285045i	$0.907991\pi$
$12$			7.69432i			0.641193i
$13$		0			0
$13$		0			0
$14$	14.5106			1.03647
$15$	14.5199	−	14.5199i	0.967992	−	0.967992i
$16$	−4.00000			−0.250000
$17$		−	4.88811i		−	0.287536i	−0.989611	−	0.143768i	$-0.954078\pi$
$17$		−	4.88811i		−	0.287536i	0.989611	−	0.143768i	$-0.0459219\pi$
$18$	−5.80063	+	5.80063i	−0.322257	+	0.322257i
$19$	−18.6785	+	18.6785i	−0.983077	+	0.983077i	−0.999859	−	0.0167821i	$-0.994658\pi$
$19$	−18.6785	+	18.6785i	−0.983077	+	0.983077i	0.0167821	+	0.999859i	$0.494658\pi$
$20$	7.54837	+	7.54837i	0.377418	+	0.377418i
$21$	27.9124	+	27.9124i	1.32916	+	1.32916i
$22$	14.8163			0.673469
$23$		−	19.9590i		−	0.867785i	−0.900965	−	0.433892i	$-0.857140\pi$
$23$		−	19.9590i		−	0.867785i	0.900965	−	0.433892i	$-0.142860\pi$
$24$	−7.69432	−	7.69432i	−0.320597	−	0.320597i
$25$		−	3.48892i		−	0.139557i
$26$		0			0
$27$	12.3085			0.455870
$28$	−14.5106	+	14.5106i	−0.518237	+	0.518237i
$29$	14.3044			0.493254			0.246627	−	0.969110i	$-0.420678\pi$
$29$	14.3044			0.493254			0.246627	+	0.969110i	$0.420678\pi$
$30$			29.0398i			0.967992i
$31$	−19.0056	+	19.0056i	−0.613083	+	0.613083i	−0.943748	−	0.330665i	$-0.892727\pi$
$31$	−19.0056	+	19.0056i	−0.613083	+	0.613083i	0.330665	+	0.943748i	$0.392727\pi$
$32$	4.00000	−	4.00000i	0.125000	−	0.125000i
$33$	28.5004	+	28.5004i	0.863647	+	0.863647i
$34$	4.88811	+	4.88811i	0.143768	+	0.143768i
$35$	54.7658			1.56474
$36$		−	11.6013i		−	0.322257i
$37$	42.9072	+	42.9072i	1.15965	+	1.15965i	0.984550	+	0.175103i	$0.0560259\pi$
$37$	42.9072	+	42.9072i	1.15965	+	1.15965i	0.175103	+	0.984550i	$0.443974\pi$
$38$		−	37.3569i		−	0.983077i
$39$		0			0
$40$	−15.0967			−0.377418
$41$	−3.54099	+	3.54099i	−0.0863657	+	0.0863657i	−0.748970	−	0.662604i	$-0.769451\pi$
$41$	−3.54099	+	3.54099i	−0.0863657	+	0.0863657i	0.662604	+	0.748970i	$0.269451\pi$
$42$	−55.8247			−1.32916
$43$			11.9728i			0.278438i	0.990262	+	0.139219i	$0.0444592\pi$
$43$			11.9728i			0.278438i	−0.990262	+	0.139219i	$0.955541\pi$
$44$	−14.8163	+	14.8163i	−0.336734	+	0.336734i
$45$	−21.8926	+	21.8926i	−0.486503	+	0.486503i
$46$	19.9590	+	19.9590i	0.433892	+	0.433892i
$47$	7.59168	+	7.59168i	0.161525	+	0.161525i	0.783242	−	0.621717i	$-0.213565\pi$
$47$	7.59168	+	7.59168i	0.161525	+	0.161525i	−0.621717	+	0.783242i	$0.713565\pi$
$48$	15.3886			0.320597
$49$			56.2792i			1.14856i
$50$	3.48892	+	3.48892i	0.0697783	+	0.0697783i
$51$			18.8053i			0.368732i
$52$		0			0
$53$	77.0450			1.45368			0.726840	−	0.686807i	$-0.240988\pi$
$53$	77.0450			1.45368			0.726840	+	0.686807i	$0.240988\pi$
$54$	−12.3085	+	12.3085i	−0.227935	+	0.227935i
$55$	55.9195			1.01672
$56$		−	29.0213i		−	0.518237i
$57$	71.8590	−	71.8590i	1.26068	−	1.26068i
$58$	−14.3044	+	14.3044i	−0.246627	+	0.246627i
$59$	−44.4819	−	44.4819i	−0.753930	−	0.753930i	0.221280	−	0.975210i	$-0.428976\pi$
$59$	−44.4819	−	44.4819i	−0.753930	−	0.753930i	−0.975210	+	0.221280i	$0.928976\pi$
$60$	−29.0398	−	29.0398i	−0.483996	−	0.483996i
$61$	−56.2764			−0.922564			−0.461282	−	0.887254i	$-0.652610\pi$
$61$	−56.2764			−0.922564			−0.461282	+	0.887254i	$0.652610\pi$
$62$		−	38.0112i		−	0.613083i
$63$	−42.0854	−	42.0854i	−0.668022	−	0.668022i
$64$			8.00000i			0.125000i
$65$		0			0
$66$	−57.0007			−0.863647
$67$	4.32207	−	4.32207i	0.0645085	−	0.0645085i	−0.674117	−	0.738625i	$-0.735475\pi$
$67$	4.32207	−	4.32207i	0.0645085	−	0.0645085i	0.738625	+	0.674117i	$0.235475\pi$
$68$	−9.77622			−0.143768
$69$			76.7856i			1.11283i
$70$	−54.7658	+	54.7658i	−0.782368	+	0.782368i
$71$	40.4291	−	40.4291i	0.569424	−	0.569424i	−0.362543	−	0.931967i	$-0.618091\pi$
$71$	40.4291	−	40.4291i	0.569424	−	0.569424i	0.931967	+	0.362543i	$0.118091\pi$
$72$	11.6013	+	11.6013i	0.161129	+	0.161129i
$73$	−12.7990	−	12.7990i	−0.175329	−	0.175329i	0.613987	−	0.789316i	$-0.289565\pi$
$73$	−12.7990	−	12.7990i	−0.175329	−	0.175329i	−0.789316	+	0.613987i	$0.789565\pi$
$74$	−85.8143			−1.15965
$75$			13.4224i			0.178966i
$76$	37.3569	+	37.3569i	0.491539	+	0.491539i
$77$			107.497i			1.39607i
$78$		0			0
$79$	7.98532			0.101080			0.0505400	−	0.998722i	$-0.483906\pi$
$79$	7.98532			0.101080			0.0505400	+	0.998722i	$0.483906\pi$
$80$	15.0967	−	15.0967i	0.188709	−	0.188709i
$81$	−99.5584			−1.22912
$82$		−	7.08199i		−	0.0863657i
$83$	35.8343	−	35.8343i	0.431738	−	0.431738i	−0.457481	−	0.889219i	$-0.651248\pi$
$83$	35.8343	−	35.8343i	0.431738	−	0.431738i	0.889219	+	0.457481i	$0.151248\pi$
$84$	55.8247	−	55.8247i	0.664580	−	0.664580i
$85$	18.4486	+	18.4486i	0.217043	+	0.217043i
$86$	−11.9728	−	11.9728i	−0.139219	−	0.139219i
$87$	−55.0312			−0.632542
$88$		−	29.6326i		−	0.336734i
$89$	−57.2867	−	57.2867i	−0.643671	−	0.643671i	0.307785	−	0.951456i	$-0.400412\pi$
$89$	−57.2867	−	57.2867i	−0.643671	−	0.643671i	−0.951456	+	0.307785i	$0.900412\pi$
$90$		−	43.7853i		−	0.486503i
$91$		0			0
$92$	−39.9181			−0.433892
$93$	73.1175	−	73.1175i	0.786209	−	0.786209i
$94$	−15.1834			−0.161525
$95$		−	140.992i		−	1.48413i
$96$	−15.3886	+	15.3886i	−0.160298	+	0.160298i
$97$	38.7928	−	38.7928i	0.399926	−	0.399926i	−0.478281	−	0.878207i	$-0.658740\pi$
$97$	38.7928	−	38.7928i	0.399926	−	0.399926i	0.878207	+	0.478281i	$0.158740\pi$
$98$	−56.2792	−	56.2792i	−0.574278	−	0.574278i
$99$	−42.9720	−	42.9720i	−0.434060	−	0.434060i
$100$	−6.97783			−0.0697783
$101$			14.0635i			0.139242i	0.997573	+	0.0696212i	$0.0221791\pi$
$101$			14.0635i			0.139242i	−0.997573	+	0.0696212i	$0.977821\pi$
$102$	−18.8053	−	18.8053i	−0.184366	−	0.184366i
$103$		−	80.8399i		−	0.784853i	−0.919783	−	0.392426i	$-0.871636\pi$
$103$		−	80.8399i		−	0.784853i	0.919783	−	0.392426i	$-0.128364\pi$
$104$		0			0
$105$	−210.693			−2.00660
$106$	−77.0450	+	77.0450i	−0.726840	+	0.726840i
$107$	162.705			1.52061			0.760304	−	0.649568i	$-0.225050\pi$
$107$	162.705			1.52061			0.760304	+	0.649568i	$0.225050\pi$
$108$		−	24.6170i		−	0.227935i
$109$	62.6738	−	62.6738i	0.574989	−	0.574989i	−0.358530	−	0.933518i	$-0.616722\pi$
$109$	62.6738	−	62.6738i	0.574989	−	0.574989i	0.933518	+	0.358530i	$0.116722\pi$
$110$	−55.9195	+	55.9195i	−0.508359	+	0.508359i
$111$	−165.071	−	165.071i	−1.48712	−	1.48712i
$112$	29.0213	+	29.0213i	0.259118	+	0.259118i
$113$	−69.8809			−0.618415			−0.309208	−	0.950995i	$-0.600064\pi$
$113$	−69.8809			−0.618415			−0.309208	+	0.950995i	$0.600064\pi$
$114$			143.718i			1.26068i
$115$	75.3291	+	75.3291i	0.655036	+	0.655036i
$116$		−	28.6087i		−	0.246627i
$117$		0			0
$118$	88.9637			0.753930
$119$	−35.4648	+	35.4648i	−0.298024	+	0.298024i
$120$	58.0795			0.483996
$121$		−	11.2384i		−	0.0928793i
$122$	56.2764	−	56.2764i	0.461282	−	0.461282i
$123$	13.6228	−	13.6228i	0.110754	−	0.110754i
$124$	38.0112	+	38.0112i	0.306542	+	0.306542i
$125$	−81.1868	−	81.1868i	−0.649494	−	0.649494i
$126$	84.1708			0.668022
$127$			82.7908i			0.651896i	0.945388	+	0.325948i	$0.105683\pi$
$127$			82.7908i			0.651896i	−0.945388	+	0.325948i	$0.894317\pi$
$128$	−8.00000	−	8.00000i	−0.0625000	−	0.0625000i
$129$		−	46.0614i		−	0.357065i
$130$		0			0
$131$	47.8506			0.365272			0.182636	−	0.983181i	$-0.441537\pi$
$131$	47.8506			0.365272			0.182636	+	0.983181i	$0.441537\pi$
$132$	57.0007	−	57.0007i	0.431824	−	0.431824i
$133$	271.036			2.03787
$134$			8.64414i			0.0645085i
$135$	−46.4545	+	46.4545i	−0.344107	+	0.344107i
$136$	9.77622	−	9.77622i	0.0718840	−	0.0718840i
$137$	136.082	+	136.082i	0.993298	+	0.993298i	0.999978	−	0.00667939i	$-0.00212613\pi$
$137$	136.082	+	136.082i	0.993298	+	0.993298i	−0.00667939	+	0.999978i	$0.502126\pi$
$138$	−76.7856	−	76.7856i	−0.556417	−	0.556417i
$139$	193.093			1.38916			0.694578	−	0.719417i	$-0.255591\pi$
$139$	193.093			1.38916			0.694578	+	0.719417i	$0.255591\pi$
$140$		−	109.532i		−	0.782368i
$141$	−29.2064	−	29.2064i	−0.207137	−	0.207137i
$142$			80.8582i			0.569424i
$143$		0			0
$144$	−23.2025			−0.161129
$145$	−53.9873	+	53.9873i	−0.372326	+	0.372326i
$146$	25.5980			0.175329
$147$		−	216.515i		−	1.47289i
$148$	85.8143	−	85.8143i	0.579827	−	0.579827i
$149$	−95.0866	+	95.0866i	−0.638165	+	0.638165i	−0.950103	−	0.311938i	$-0.899022\pi$
$149$	−95.0866	+	95.0866i	−0.638165	+	0.638165i	0.311938	+	0.950103i	$0.399022\pi$
$150$	−13.4224	−	13.4224i	−0.0894828	−	0.0894828i
$151$	−26.5821	−	26.5821i	−0.176041	−	0.176041i	0.613587	−	0.789627i	$-0.289726\pi$
$151$	−26.5821	−	26.5821i	−0.176041	−	0.176041i	−0.789627	+	0.613587i	$0.789726\pi$
$152$	−74.7139			−0.491539
$153$		−	28.3541i		−	0.185321i
$154$	−107.497	−	107.497i	−0.698033	−	0.698033i
$155$		−	143.461i		−	0.925555i
$156$		0			0
$157$	−251.005			−1.59876			−0.799380	−	0.600826i	$-0.794838\pi$
$157$	−251.005			−1.59876			−0.799380	+	0.600826i	$0.794838\pi$
$158$	−7.98532	+	7.98532i	−0.0505400	+	0.0505400i
$159$	−296.404			−1.86418
$160$			30.1935i			0.188709i
$161$	−144.809	+	144.809i	−0.899436	+	0.899436i
$162$	99.5584	−	99.5584i	0.614558	−	0.614558i
$163$	85.9701	+	85.9701i	0.527424	+	0.527424i	0.919803	−	0.392380i	$-0.128348\pi$
$163$	85.9701	+	85.9701i	0.527424	+	0.527424i	−0.392380	+	0.919803i	$0.628348\pi$
$164$	7.08199	+	7.08199i	0.0431829	+	0.0431829i
$165$	−215.131			−1.30383
$166$			71.6686i			0.431738i
$167$	43.7710	+	43.7710i	0.262102	+	0.262102i	0.825907	−	0.563806i	$-0.190663\pi$
$167$	43.7710	+	43.7710i	0.262102	+	0.262102i	−0.563806	+	0.825907i	$0.690663\pi$
$168$			111.649i			0.664580i
$169$		0			0
$170$	−36.8973			−0.217043
$171$	−108.347	+	108.347i	−0.633607	+	0.633607i
$172$	23.9457			0.139219
$173$			310.835i			1.79674i	0.439244	+	0.898368i	$0.355246\pi$
$173$			310.835i			1.79674i	−0.439244	+	0.898368i	$0.644754\pi$
$174$	55.0312	−	55.0312i	0.316271	−	0.316271i
$175$	−25.3132	+	25.3132i	−0.144647	+	0.144647i
$176$	29.6326	+	29.6326i	0.168367	+	0.168367i
$177$	171.129	+	171.129i	0.966829	+	0.966829i
$178$	114.573			0.643671
$179$		−	63.3403i		−	0.353856i	−0.984224	−	0.176928i	$-0.943384\pi$
$179$		−	63.3403i		−	0.353856i	0.984224	−	0.176928i	$-0.0566160\pi$
$180$	43.7853	+	43.7853i	0.243251	+	0.243251i
$181$			186.504i			1.03041i	0.857067	+	0.515205i	$0.172284\pi$
$181$			186.504i			1.03041i	−0.857067	+	0.515205i	$0.827716\pi$
$182$		0			0
$183$	216.504			1.18308
$184$	39.9181	−	39.9181i	0.216946	−	0.216946i
$185$	−323.879			−1.75070
$186$			146.235i			0.786209i
$187$	−36.2119	+	36.2119i	−0.193647	+	0.193647i
$188$	15.1834	−	15.1834i	0.0807625	−	0.0807625i
$189$	−89.3020	−	89.3020i	−0.472497	−	0.472497i
$190$	140.992	+	140.992i	0.742063	+	0.742063i
$191$	−182.691			−0.956500			−0.478250	−	0.878224i	$-0.658729\pi$
$191$	−182.691			−0.956500			−0.478250	+	0.878224i	$0.658729\pi$
$192$		−	30.7773i		−	0.160298i
$193$	64.4548	+	64.4548i	0.333963	+	0.333963i	0.854089	−	0.520126i	$-0.174115\pi$
$193$	64.4548	+	64.4548i	0.333963	+	0.333963i	−0.520126	+	0.854089i	$0.674115\pi$
$194$			77.5856i			0.399926i
$195$		0			0
$196$	112.558			0.574278
$197$	152.596	−	152.596i	0.774600	−	0.774600i	−0.204307	−	0.978907i	$-0.565494\pi$
$197$	152.596	−	152.596i	0.774600	−	0.774600i	0.978907	+	0.204307i	$0.0654940\pi$
$198$	85.9439			0.434060
$199$			78.4893i			0.394418i	0.980361	+	0.197209i	$0.0631878\pi$
$199$			78.4893i			0.394418i	−0.980361	+	0.197209i	$0.936812\pi$
$200$	6.97783	−	6.97783i	0.0348892	−	0.0348892i
$201$	−16.6277	+	16.6277i	−0.0827248	+	0.0827248i
$202$	−14.0635	−	14.0635i	−0.0696212	−	0.0696212i
$203$	−103.783	−	103.783i	−0.511245	−	0.511245i
$204$	37.6107			0.184366
$205$		−	26.7287i		−	0.130384i
$206$	80.8399	+	80.8399i	0.392426	+	0.392426i
$207$		−	115.775i		−	0.559300i
$208$		0			0
$209$	276.746			1.32414
$210$	210.693	−	210.693i	1.00330	−	1.00330i
$211$	−132.398			−0.627478			−0.313739	−	0.949509i	$-0.601582\pi$
$211$	−132.398			−0.627478			−0.313739	+	0.949509i	$0.601582\pi$
$212$		−	154.090i		−	0.726840i
$213$	−155.537	+	155.537i	−0.730221	+	0.730221i
$214$	−162.705	+	162.705i	−0.760304	+	0.760304i
$215$	−45.1876	−	45.1876i	−0.210175	−	0.210175i
$216$	24.6170	+	24.6170i	0.113968	+	0.113968i
$217$	275.783			1.27089
$218$			125.348i			0.574989i
$219$	49.2398	+	49.2398i	0.224839	+	0.224839i
$220$		−	111.839i		−	0.508359i
$221$		0			0
$222$	330.141			1.48712
$223$	−84.4985	+	84.4985i	−0.378917	+	0.378917i	−0.870711	−	0.491794i	$-0.836341\pi$
$223$	−84.4985	+	84.4985i	−0.378917	+	0.378917i	0.491794	+	0.870711i	$0.336341\pi$
$224$	−58.0425			−0.259118
$225$		−	20.2379i		−	0.0899462i
$226$	69.8809	−	69.8809i	0.309208	−	0.309208i
$227$	299.942	−	299.942i	1.32133	−	1.32133i	0.408632	−	0.912699i	$-0.366006\pi$
$227$	299.942	−	299.942i	1.32133	−	1.32133i	0.912699	−	0.408632i	$-0.133994\pi$
$228$	−143.718	−	143.718i	−0.630342	−	0.630342i
$229$	158.206	+	158.206i	0.690856	+	0.690856i	0.962420	−	0.271564i	$-0.0875409\pi$
$229$	158.206	+	158.206i	0.690856	+	0.690856i	−0.271564	+	0.962420i	$0.587541\pi$
$230$	−150.658			−0.655036
$231$		−	413.558i		−	1.79030i
$232$	28.6087	+	28.6087i	0.123314	+	0.123314i
$233$			163.030i			0.699701i	0.936806	+	0.349850i	$0.113768\pi$
$233$			163.030i			0.699701i	−0.936806	+	0.349850i	$0.886232\pi$
$234$		0			0
$235$	−57.3048			−0.243850
$236$	−88.9637	+	88.9637i	−0.376965	+	0.376965i
$237$	−30.7208			−0.129624
$238$		−	70.9296i		−	0.298024i
$239$	185.455	−	185.455i	0.775962	−	0.775962i	−0.203179	−	0.979142i	$-0.565127\pi$
$239$	185.455	−	185.455i	0.775962	−	0.775962i	0.979142	+	0.203179i	$0.0651275\pi$
$240$	−58.0795	+	58.0795i	−0.241998	+	0.241998i
$241$	321.239	+	321.239i	1.33294	+	1.33294i	0.902728	+	0.430212i	$0.141561\pi$
$241$	321.239	+	321.239i	1.33294	+	1.33294i	0.430212	+	0.902728i	$0.358439\pi$
$242$	11.2384	+	11.2384i	0.0464397	+	0.0464397i
$243$	272.240			1.12033
$244$			112.553i			0.461282i
$245$	−212.408	−	212.408i	−0.866972	−	0.866972i
$246$			27.2455i			0.110754i
$247$		0			0
$248$	−76.0223			−0.306542
$249$	−137.860	+	137.860i	−0.553655	+	0.553655i
$250$	162.374			0.649494
$251$			438.996i			1.74899i	0.485035	+	0.874495i	$0.338807\pi$
$251$			438.996i			1.74899i	−0.485035	+	0.874495i	$0.661193\pi$
$252$	−84.1708	+	84.1708i	−0.334011	+	0.334011i
$253$	−147.860	+	147.860i	−0.584426	+	0.584426i
$254$	−82.7908	−	82.7908i	−0.325948	−	0.325948i
$255$	−70.9748	−	70.9748i	−0.278333	−	0.278333i
$256$	16.0000			0.0625000
$257$		−	437.208i		−	1.70120i	−0.525815	−	0.850599i	$-0.676240\pi$
$257$		−	437.208i		−	1.70120i	0.525815	−	0.850599i	$-0.323760\pi$
$258$	46.0614	+	46.0614i	0.178532	+	0.178532i
$259$		−	622.610i		−	2.40390i
$260$		0			0
$261$	82.9743			0.317909
$262$	−47.8506	+	47.8506i	−0.182636	+	0.182636i
$263$	−287.959			−1.09490			−0.547451	−	0.836838i	$-0.684402\pi$
$263$	−287.959			−1.09490			−0.547451	+	0.836838i	$0.684402\pi$
$264$			114.001i			0.431824i
$265$	−290.782	+	290.782i	−1.09729	+	1.09729i
$266$	−271.036	+	271.036i	−1.01893	+	1.01893i
$267$	220.391	+	220.391i	0.825435	+	0.825435i
$268$	−8.64414	−	8.64414i	−0.0322542	−	0.0322542i
$269$	343.400			1.27658			0.638290	−	0.769796i	$-0.279642\pi$
$269$	343.400			1.27658			0.638290	+	0.769796i	$0.279642\pi$
$270$		−	92.9090i		−	0.344107i
$271$	−5.87429	−	5.87429i	−0.0216764	−	0.0216764i	0.696186	−	0.717862i	$-0.254879\pi$
$271$	−5.87429	−	5.87429i	−0.0216764	−	0.0216764i	−0.717862	+	0.696186i	$0.754879\pi$
$272$			19.5524i			0.0718840i
$273$		0			0
$274$	−272.164			−0.993298
$275$	−25.8464	+	25.8464i	−0.0939871	+	0.0939871i
$276$	153.571			0.556417
$277$		−	444.247i		−	1.60378i	−0.597471	−	0.801891i	$-0.703828\pi$
$277$		−	444.247i		−	1.60378i	0.597471	−	0.801891i	$-0.296172\pi$
$278$	−193.093	+	193.093i	−0.694578	+	0.694578i
$279$	−110.244	+	110.244i	−0.395141	+	0.395141i
$280$	109.532	+	109.532i	0.391184	+	0.391184i
$281$	163.678	+	163.678i	0.582486	+	0.582486i	0.935586	−	0.353100i	$-0.114872\pi$
$281$	163.678	+	163.678i	0.582486	+	0.582486i	−0.353100	+	0.935586i	$0.614872\pi$
$282$	58.4128			0.207137
$283$			464.139i			1.64007i	0.572315	+	0.820034i	$0.306045\pi$
$283$			464.139i			1.64007i	−0.572315	+	0.820034i	$0.693955\pi$
$284$	−80.8582	−	80.8582i	−0.284712	−	0.284712i
$285$			542.418i			1.90322i
$286$		0			0
$287$	51.3821			0.179032
$288$	23.2025	−	23.2025i	0.0805643	−	0.0805643i
$289$	265.106			0.917323
$290$		−	107.975i		−	0.372326i
$291$	−149.242	+	149.242i	−0.512859	+	0.512859i
$292$	−25.5980	+	25.5980i	−0.0876644	+	0.0876644i
$293$	−44.9476	−	44.9476i	−0.153405	−	0.153405i	0.626232	−	0.779637i	$-0.284596\pi$
$293$	−44.9476	−	44.9476i	−0.153405	−	0.153405i	−0.779637	+	0.626232i	$0.784596\pi$
$294$	216.515	+	216.515i	0.736446	+	0.736446i
$295$	335.765			1.13819
$296$			171.629i			0.579827i
$297$	−91.1832	−	91.1832i	−0.307014	−	0.307014i
$298$		−	190.173i		−	0.638165i
$299$		0			0
$300$	26.8448			0.0894828
$301$	86.8667	−	86.8667i	0.288594	−	0.288594i
$302$	53.1643			0.176041
$303$		−	54.1045i		−	0.178563i
$304$	74.7139	−	74.7139i	0.245769	−	0.245769i
$305$	212.397	−	212.397i	0.696385	−	0.696385i
$306$	28.3541	+	28.3541i	0.0926605	+	0.0926605i
$307$	−67.1395	−	67.1395i	−0.218695	−	0.218695i	0.589253	−	0.807948i	$-0.299422\pi$
$307$	−67.1395	−	67.1395i	−0.218695	−	0.218695i	−0.807948	+	0.589253i	$0.799422\pi$
$308$	214.994			0.698033
$309$			311.004i			1.00648i
$310$	143.461	+	143.461i	0.462778	+	0.462778i
$311$			331.141i			1.06476i	0.846505	+	0.532381i	$0.178703\pi$
$311$			331.141i			1.06476i	−0.846505	+	0.532381i	$0.821297\pi$
$312$		0			0
$313$	−391.075			−1.24944			−0.624721	−	0.780848i	$-0.714787\pi$
$313$	−391.075			−1.24944			−0.624721	+	0.780848i	$0.714787\pi$
$314$	251.005	−	251.005i	0.799380	−	0.799380i
$315$	317.676			1.00849
$316$		−	15.9706i		−	0.0505400i
$317$	322.964	−	322.964i	1.01881	−	1.01881i	0.0189927	−	0.999820i	$-0.493954\pi$
$317$	322.964	−	322.964i	1.01881	−	1.01881i	0.999820	−	0.0189927i	$-0.00604594\pi$
$318$	296.404	−	296.404i	0.932089	−	0.932089i
$319$	−105.969	−	105.969i	−0.332191	−	0.332191i
$320$	−30.1935	−	30.1935i	−0.0943546	−	0.0943546i
$321$	−625.952			−1.95001
$322$		−	289.618i		−	0.899436i
$323$	91.3024	+	91.3024i	0.282670	+	0.282670i
$324$			199.117i			0.614558i
$325$		0			0
$326$	−171.940			−0.527424
$327$	−241.116	+	241.116i	−0.737357	+	0.737357i
$328$	−14.1640			−0.0431829
$329$		−	110.160i		−	0.334833i
$330$	215.131	−	215.131i	0.651913	−	0.651913i
$331$	−346.516	+	346.516i	−1.04688	+	1.04688i	−0.0480317	+	0.998846i	$0.515295\pi$
$331$	−346.516	+	346.516i	−1.04688	+	1.04688i	−0.998846	+	0.0480317i	$0.984705\pi$
$332$	−71.6686	−	71.6686i	−0.215869	−	0.215869i
$333$	248.889	+	248.889i	0.747413	+	0.747413i
$334$	−87.5419			−0.262102
$335$			32.6246i			0.0973867i
$336$	−111.649	−	111.649i	−0.332290	−	0.332290i
$337$		−	498.500i		−	1.47923i	−0.673031	−	0.739614i	$-0.735008\pi$
$337$		−	498.500i		−	1.47923i	0.673031	−	0.739614i	$-0.264992\pi$
$338$		0			0
$339$	268.843			0.793047
$340$	36.8973	−	36.8973i	0.108521	−	0.108521i
$341$	281.593			0.825785
$342$		−	216.694i		−	0.633607i
$343$	52.8131	−	52.8131i	0.153974	−	0.153974i
$344$	−23.9457	+	23.9457i	−0.0696095	+	0.0696095i
$345$	−289.803	−	289.803i	−0.840009	−	0.840009i
$346$	−310.835	−	310.835i	−0.898368	−	0.898368i
$347$	−234.625			−0.676153			−0.338076	−	0.941119i	$-0.609776\pi$
$347$	−234.625			−0.676153			−0.338076	+	0.941119i	$0.609776\pi$
$348$			110.062i			0.316271i
$349$	−125.061	−	125.061i	−0.358341	−	0.358341i	0.504860	−	0.863201i	$-0.331544\pi$
$349$	−125.061	−	125.061i	−0.358341	−	0.358341i	−0.863201	+	0.504860i	$0.831544\pi$
$350$		−	50.6264i		−	0.144647i
$351$		0			0
$352$	−59.2653			−0.168367
$353$	159.033	−	159.033i	0.450518	−	0.450518i	−0.445009	−	0.895526i	$-0.646799\pi$
$353$	159.033	−	159.033i	0.450518	−	0.450518i	0.895526	+	0.445009i	$0.146799\pi$
$354$	−342.258			−0.966829
$355$			305.174i			0.859644i
$356$	−114.573	+	114.573i	−0.321836	+	0.321836i
$357$	136.439	−	136.439i	0.382181	−	0.382181i
$358$	63.3403	+	63.3403i	0.176928	+	0.176928i
$359$	−290.278	−	290.278i	−0.808574	−	0.808574i	0.175844	−	0.984418i	$-0.443735\pi$
$359$	−290.278	−	290.278i	−0.808574	−	0.808574i	−0.984418	+	0.175844i	$0.943735\pi$
$360$	−87.5705			−0.243251
$361$		−	336.770i		−	0.932881i
$362$	−186.504	−	186.504i	−0.515205	−	0.515205i
$363$			43.2359i			0.119107i
$364$		0			0
$365$	96.6115			0.264689
$366$	−216.504	+	216.504i	−0.591542	+	0.591542i
$367$	−220.744			−0.601481			−0.300741	−	0.953706i	$-0.597234\pi$
$367$	−220.744			−0.601481			−0.300741	+	0.953706i	$0.597234\pi$
$368$			79.8362i			0.216946i
$369$	−20.5400	+	20.5400i	−0.0556639	+	0.0556639i
$370$	323.879	−	323.879i	0.875349	−	0.875349i
$371$	−558.986	−	558.986i	−1.50670	−	1.50670i
$372$	−146.235	−	146.235i	−0.393105	−	0.393105i
$373$	−62.1593			−0.166647			−0.0833235	−	0.996523i	$-0.526553\pi$
$373$	−62.1593			−0.166647			−0.0833235	+	0.996523i	$0.526553\pi$
$374$		−	72.4238i		−	0.193647i
$375$	312.338	+	312.338i	0.832902	+	0.832902i
$376$			30.3667i			0.0807625i
$377$		0			0
$378$	178.604			0.472497
$379$	102.724	−	102.724i	0.271038	−	0.271038i	−0.558480	−	0.829518i	$-0.688615\pi$
$379$	102.724	−	102.724i	0.271038	−	0.271038i	0.829518	+	0.558480i	$0.188615\pi$
$380$	−281.984			−0.742063
$381$		−	318.509i		−	0.835982i
$382$	182.691	−	182.691i	0.478250	−	0.478250i
$383$	355.526	−	355.526i	0.928265	−	0.928265i	−0.0693288	−	0.997594i	$-0.522086\pi$
$383$	355.526	−	355.526i	0.928265	−	0.928265i	0.997594	+	0.0693288i	$0.0220858\pi$
$384$	30.7773	+	30.7773i	0.0801491	+	0.0801491i
$385$	−405.714	−	405.714i	−1.05380	−	1.05380i
$386$	−128.910			−0.333963
$387$			69.4499i			0.179457i
$388$	−77.5856	−	77.5856i	−0.199963	−	0.199963i
$389$		−	284.973i		−	0.732579i	−0.930501	−	0.366289i	$-0.880628\pi$
$389$		−	284.973i		−	0.732579i	0.930501	−	0.366289i	$-0.119372\pi$
$390$		0			0
$391$	−97.5621			−0.249519
$392$	−112.558	+	112.558i	−0.287139	+	0.287139i
$393$	−184.089			−0.468420
$394$			305.193i			0.774600i
$395$	−30.1380	+	30.1380i	−0.0762988	+	0.0762988i
$396$	−85.9439	+	85.9439i	−0.217030	+	0.217030i
$397$	209.336	+	209.336i	0.527294	+	0.527294i	0.919764	−	0.392471i	$-0.128380\pi$
$397$	209.336	+	209.336i	0.527294	+	0.527294i	−0.392471	+	0.919764i	$0.628380\pi$
$398$	−78.4893	−	78.4893i	−0.197209	−	0.197209i
$399$	−1042.72			−2.61333
$400$			13.9557i			0.0348892i
$401$	−171.024	−	171.024i	−0.426494	−	0.426494i	0.460938	−	0.887432i	$-0.347513\pi$
$401$	−171.024	−	171.024i	−0.426494	−	0.426494i	−0.887432	+	0.460938i	$0.847513\pi$
$402$		−	33.2554i		−	0.0827248i
$403$		0			0
$404$	28.1270			0.0696212
$405$	375.752	−	375.752i	0.927782	−	0.927782i
$406$	207.565			0.511245
$407$		−	635.726i		−	1.56198i
$408$	−37.6107	+	37.6107i	−0.0921830	+	0.0921830i
$409$	−137.980	+	137.980i	−0.337361	+	0.337361i	−0.855373	−	0.518013i	$-0.826672\pi$
$409$	−137.980	+	137.980i	−0.337361	+	0.337361i	0.518013	+	0.855373i	$0.326672\pi$
$410$	26.7287	+	26.7287i	0.0651920	+	0.0651920i
$411$	−523.529	−	523.529i	−1.27379	−	1.27379i
$412$	−161.680			−0.392426
$413$			645.460i			1.56286i
$414$	115.775	+	115.775i	0.279650	+	0.279650i
$415$			270.490i			0.651784i
$416$		0			0
$417$	−742.859			−1.78144
$418$	−276.746	+	276.746i	−0.662072	+	0.662072i
$419$	47.9110			0.114346			0.0571730	−	0.998364i	$-0.481791\pi$
$419$	47.9110			0.114346			0.0571730	+	0.998364i	$0.481791\pi$
$420$			421.385i			1.00330i
$421$	477.018	−	477.018i	1.13306	−	1.13306i	0.143394	−	0.989666i	$-0.454198\pi$
$421$	477.018	−	477.018i	1.13306	−	1.13306i	0.989666	−	0.143394i	$-0.0458016\pi$
$422$	132.398	−	132.398i	0.313739	−	0.313739i
$423$	44.0365	+	44.0365i	0.104105	+	0.104105i
$424$	154.090	+	154.090i	0.363420	+	0.363420i
$425$	−17.0542			−0.0401276
$426$		−	311.074i		−	0.730221i
$427$	408.303	+	408.303i	0.956214	+	0.956214i
$428$		−	325.410i		−	0.760304i
$429$		0			0
$430$	90.3753			0.210175
$431$	−55.0595	+	55.0595i	−0.127748	+	0.127748i	−0.768090	−	0.640342i	$-0.778793\pi$
$431$	−55.0595	+	55.0595i	−0.127748	+	0.127748i	0.640342	+	0.768090i	$0.278793\pi$
$432$	−49.2340			−0.113968
$433$			497.603i			1.14920i	0.818435	+	0.574599i	$0.194842\pi$
$433$			497.603i			1.14920i	−0.818435	+	0.574599i	$0.805158\pi$
$434$	−275.783	+	275.783i	−0.635445	+	0.635445i
$435$	207.698	−	207.698i	0.477466	−	0.477466i
$436$	−125.348	−	125.348i	−0.287494	−	0.287494i
$437$	372.804	+	372.804i	0.853099	+	0.853099i
$438$	−98.4795			−0.224839
$439$			167.510i			0.381572i	0.981632	+	0.190786i	$0.0611037\pi$
$439$			167.510i			0.381572i	−0.981632	+	0.190786i	$0.938896\pi$
$440$	111.839	+	111.839i	0.254179	+	0.254179i
$441$			326.455i			0.740260i
$442$		0			0
$443$	−443.835			−1.00188			−0.500942	−	0.865481i	$-0.667013\pi$
$443$	−443.835			−1.00188			−0.500942	+	0.865481i	$0.667013\pi$
$444$	−330.141	+	330.141i	−0.743562	+	0.743562i
$445$	432.421			0.971733
$446$		−	168.997i		−	0.378917i
$447$	365.813	−	365.813i	0.818374	−	0.818374i
$448$	58.0425	−	58.0425i	0.129559	−	0.129559i
$449$	181.585	+	181.585i	0.404421	+	0.404421i	0.879788	−	0.475366i	$-0.157685\pi$
$449$	181.585	+	181.585i	0.404421	+	0.404421i	−0.475366	+	0.879788i	$0.657685\pi$
$450$	20.2379	+	20.2379i	0.0449731	+	0.0449731i
$451$	52.4645			0.116329
$452$			139.762i			0.309208i
$453$	102.266	+	102.266i	0.225752	+	0.225752i
$454$			599.884i			1.32133i
$455$		0			0
$456$	287.436			0.630342
$457$	−61.5957	+	61.5957i	−0.134783	+	0.134783i	−0.771279	−	0.636497i	$-0.780383\pi$
$457$	−61.5957	+	61.5957i	−0.134783	+	0.134783i	0.636497	+	0.771279i	$0.280383\pi$
$458$	−316.412			−0.690856
$459$		−	60.1653i		−	0.131079i
$460$	150.658	−	150.658i	0.327518	−	0.327518i
$461$	−349.897	+	349.897i	−0.758996	+	0.758996i	−0.976140	−	0.217144i	$-0.930326\pi$
$461$	−349.897	+	349.897i	−0.758996	+	0.758996i	0.217144	+	0.976140i	$0.430326\pi$
$462$	413.558	+	413.558i	0.895148	+	0.895148i
$463$	399.472	+	399.472i	0.862789	+	0.862789i	0.991661	−	0.128872i	$-0.0411356\pi$
$463$	399.472	+	399.472i	0.862789	+	0.862789i	−0.128872	+	0.991661i	$0.541136\pi$
$464$	−57.2175			−0.123314
$465$			551.917i			1.18692i
$466$	−163.030	−	163.030i	−0.349850	−	0.349850i
$467$			409.816i			0.877550i	0.898597	+	0.438775i	$0.144587\pi$
$467$			409.816i			0.877550i	−0.898597	+	0.438775i	$0.855413\pi$
$468$		0			0
$469$	−62.7159			−0.133723
$470$	57.3048	−	57.3048i	0.121925	−	0.121925i
$471$	965.657			2.05023
$472$		−	177.927i		−	0.376965i
$473$	88.6966	−	88.6966i	0.187519	−	0.187519i
$474$	30.7208	−	30.7208i	0.0648118	−	0.0648118i
$475$	65.1676	+	65.1676i	0.137195	+	0.137195i
$476$	70.9296	+	70.9296i	0.149012	+	0.149012i
$477$	446.909			0.936917
$478$			370.910i			0.775962i
$479$	−376.568	−	376.568i	−0.786153	−	0.786153i	0.194708	−	0.980861i	$-0.437624\pi$
$479$	−376.568	−	376.568i	−0.786153	−	0.786153i	−0.980861	+	0.194708i	$0.937624\pi$
$480$		−	116.159i		−	0.241998i
$481$		0			0
$482$	−642.477			−1.33294
$483$	557.104	−	557.104i	1.15342	−	1.15342i
$484$	−22.4768			−0.0464397
$485$			292.822i			0.603757i
$486$	−272.240	+	272.240i	−0.560165	+	0.560165i
$487$	−476.048	+	476.048i	−0.977512	+	0.977512i	−0.999753	−	0.0222407i	$-0.992920\pi$
$487$	−476.048	+	476.048i	−0.977512	+	0.977512i	0.0222407	+	0.999753i	$0.492920\pi$
$488$	−112.553	−	112.553i	−0.230641	−	0.230641i
$489$	−330.740	−	330.740i	−0.676361	−	0.676361i
$490$	424.816			0.866972
$491$			117.456i			0.239219i	0.992821	+	0.119609i	$0.0381642\pi$
$491$			117.456i			0.239219i	−0.992821	+	0.119609i	$0.961836\pi$
$492$	−27.2455	−	27.2455i	−0.0553771	−	0.0553771i
$493$		−	69.9214i		−	0.141828i
$494$		0			0
$495$	324.368			0.655289
$496$	76.0223	−	76.0223i	0.153271	−	0.153271i
$497$	−586.652			−1.18039
$498$		−	275.720i		−	0.553655i
$499$	−155.387	+	155.387i	−0.311396	+	0.311396i	−0.845450	−	0.534054i	$-0.820668\pi$
$499$	−155.387	+	155.387i	−0.311396	+	0.311396i	0.534054	+	0.845450i	$0.320668\pi$
$500$	−162.374	+	162.374i	−0.324747	+	0.324747i
$501$	−168.394	−	168.394i	−0.336116	−	0.336116i
$502$	−438.996	−	438.996i	−0.874495	−	0.874495i
$503$	325.036			0.646196			0.323098	−	0.946366i	$-0.395276\pi$
$503$	325.036			0.646196			0.323098	+	0.946366i	$0.395276\pi$
$504$		−	168.342i		−	0.334011i
$505$	−53.0782	−	53.0782i	−0.105105	−	0.105105i
$506$		−	295.720i		−	0.584426i
$507$		0			0
$508$	165.582			0.325948
$509$	−410.842	+	410.842i	−0.807154	+	0.807154i	−0.984202	−	0.177048i	$-0.943345\pi$
$509$	−410.842	+	410.842i	−0.807154	+	0.807154i	0.177048	+	0.984202i	$0.443345\pi$
$510$	141.950			0.278333
$511$			185.722i			0.363447i
$512$	−16.0000	+	16.0000i	−0.0312500	+	0.0312500i
$513$	−229.904	+	229.904i	−0.448155	+	0.448155i
$514$	437.208	+	437.208i	0.850599	+	0.850599i
$515$	305.104	+	305.104i	0.592436	+	0.592436i
$516$	−92.1227			−0.178532
$517$		−	112.481i		−	0.217564i
$518$	622.610	+	622.610i	1.20195	+	1.20195i
$519$		−	1195.83i		−	2.30411i
$520$		0			0
$521$	400.067			0.767883			0.383942	−	0.923357i	$-0.374566\pi$
$521$	400.067			0.767883			0.383942	+	0.923357i	$0.374566\pi$
$522$	−82.9743	+	82.9743i	−0.158955	+	0.158955i
$523$	468.671			0.896120			0.448060	−	0.894004i	$-0.352115\pi$
$523$	468.671			0.896120			0.448060	+	0.894004i	$0.352115\pi$
$524$		−	95.7013i		−	0.182636i
$525$	97.3839	−	97.3839i	0.185493	−	0.185493i
$526$	287.959	−	287.959i	0.547451	−	0.547451i
$527$	92.9014	+	92.9014i	0.176283	+	0.176283i
$528$	−114.001	−	114.001i	−0.215912	−	0.215912i
$529$	130.636			0.246950
$530$		−	581.564i		−	1.09729i
$531$	−258.023	−	258.023i	−0.485918	−	0.485918i
$532$		−	542.073i		−	1.01893i
$533$		0			0
$534$	−440.782			−0.825435
$535$	−614.079	+	614.079i	−1.14781	+	1.14781i
$536$	17.2883			0.0322542
$537$			243.680i			0.453781i
$538$	−343.400	+	343.400i	−0.638290	+	0.638290i
$539$	416.925	−	416.925i	0.773516	−	0.773516i
$540$	92.9090	+	92.9090i	0.172054	+	0.172054i
$541$	−155.599	−	155.599i	−0.287614	−	0.287614i	0.548522	−	0.836136i	$-0.315191\pi$
$541$	−155.599	−	155.599i	−0.287614	−	0.287614i	−0.836136	+	0.548522i	$0.815191\pi$
$542$	11.7486			0.0216764
$543$		−	717.511i		−	1.32138i
$544$	−19.5524	−	19.5524i	−0.0359420	−	0.0359420i
$545$			473.085i			0.868045i
$546$		0			0
$547$	858.888			1.57018			0.785090	−	0.619382i	$-0.212617\pi$
$547$	858.888			1.57018			0.785090	+	0.619382i	$0.212617\pi$
$548$	272.164	−	272.164i	0.496649	−	0.496649i
$549$	−326.439			−0.594606
$550$		−	51.6929i		−	0.0939871i
$551$	−267.184	+	267.184i	−0.484907	+	0.484907i
$552$	−153.571	+	153.571i	−0.278209	+	0.278209i
$553$	−57.9360	−	57.9360i	−0.104767	−	0.104767i
$554$	444.247	+	444.247i	0.801891	+	0.801891i
$555$	1246.01			2.24507
$556$		−	386.186i		−	0.694578i
$557$	377.023	+	377.023i	0.676881	+	0.676881i	0.959293	−	0.282412i	$-0.0911345\pi$
$557$	377.023	+	377.023i	0.676881	+	0.676881i	−0.282412	+	0.959293i	$0.591135\pi$
$558$		−	220.489i		−	0.395141i
$559$		0			0
$560$	−219.063			−0.391184
$561$	139.313	−	139.313i	0.248330	−	0.248330i
$562$	−327.357			−0.582486
$563$		−	434.624i		−	0.771979i	−0.922503	−	0.385990i	$-0.873860\pi$
$563$		−	434.624i		−	0.771979i	0.922503	−	0.385990i	$-0.126140\pi$
$564$	−58.4128	+	58.4128i	−0.103569	+	0.103569i
$565$	263.743	−	263.743i	0.466803	−	0.466803i
$566$	−464.139	−	464.139i	−0.820034	−	0.820034i
$567$	722.327	+	722.327i	1.27395	+	1.27395i
$568$	161.716			0.284712
$569$		−	332.271i		−	0.583955i	−0.956425	−	0.291978i	$-0.905687\pi$
$569$		−	332.271i		−	0.583955i	0.956425	−	0.291978i	$-0.0943133\pi$
$570$	−542.418	−	542.418i	−0.951611	−	0.951611i
$571$			555.806i			0.973391i	0.873572	+	0.486696i	$0.161798\pi$
$571$			555.806i			0.973391i	−0.873572	+	0.486696i	$0.838202\pi$
$572$		0			0
$573$	702.843			1.22660
$574$	−51.3821	+	51.3821i	−0.0895158	+	0.0895158i
$575$	−69.6354			−0.121105
$576$			46.4050i			0.0805643i
$577$	−287.753	+	287.753i	−0.498705	+	0.498705i	−0.911035	−	0.412330i	$-0.864715\pi$
$577$	−287.753	+	287.753i	−0.498705	+	0.498705i	0.412330	+	0.911035i	$0.364715\pi$
$578$	−265.106	+	265.106i	−0.458662	+	0.458662i
$579$	−247.968	−	247.968i	−0.428269	−	0.428269i
$580$	107.975	+	107.975i	0.186163	+	0.186163i
$581$	−519.978			−0.894971
$582$		−	298.484i		−	0.512859i
$583$	−570.762	−	570.762i	−0.979008	−	0.979008i
$584$		−	51.1960i		−	0.0876644i
$585$		0			0
$586$	89.8951			0.153405
$587$	473.776	−	473.776i	0.807114	−	0.807114i	−0.177082	−	0.984196i	$-0.556666\pi$
$587$	473.776	−	473.776i	0.807114	−	0.807114i	0.984196	+	0.177082i	$0.0566657\pi$
$588$	−433.030			−0.736446
$589$		−	709.990i		−	1.20542i
$590$	−335.765	+	335.765i	−0.569094	+	0.569094i
$591$	−587.062	+	587.062i	−0.993337	+	0.993337i
$592$	−171.629	−	171.629i	−0.289913	−	0.289913i
$593$	69.0570	+	69.0570i	0.116454	+	0.116454i	0.762932	−	0.646479i	$-0.223759\pi$
$593$	69.0570	+	69.0570i	0.116454	+	0.116454i	−0.646479	+	0.762932i	$0.723759\pi$
$594$	182.366			0.307014
$595$		−	267.701i		−	0.449918i
$596$	190.173	+	190.173i	0.319082	+	0.319082i
$597$		−	301.961i		−	0.505797i
$598$		0			0
$599$	461.140			0.769850			0.384925	−	0.922948i	$-0.374227\pi$
$599$	461.140			0.769850			0.384925	+	0.922948i	$0.374227\pi$
$600$	−26.8448	+	26.8448i	−0.0447414	+	0.0447414i
$601$	176.162			0.293115			0.146558	−	0.989202i	$-0.453181\pi$
$601$	176.162			0.293115			0.146558	+	0.989202i	$0.453181\pi$
$602$			173.733i			0.288594i
$603$	25.0707	−	25.0707i	0.0415766	−	0.0415766i
$604$	−53.1643	+	53.1643i	−0.0880203	+	0.0880203i
$605$	42.4158	+	42.4158i	0.0701087	+	0.0701087i
$606$	54.1045	+	54.1045i	0.0892813	+	0.0892813i
$607$	−315.414			−0.519628			−0.259814	−	0.965659i	$-0.583661\pi$
$607$	−315.414			−0.519628			−0.259814	+	0.965659i	$0.583661\pi$
$608$			149.428i			0.245769i
$609$	399.269	+	399.269i	0.655613	+	0.655613i
$610$			424.795i			0.696385i
$611$		0			0
$612$	−56.7082			−0.0926605
$613$	638.354	−	638.354i	1.04136	−	1.04136i	0.0422542	−	0.999107i	$-0.486546\pi$
$613$	638.354	−	638.354i	1.04136	−	1.04136i	0.999107	−	0.0422542i	$-0.0134539\pi$
$614$	134.279			0.218695
$615$			102.830i			0.167203i
$616$	−214.994	+	214.994i	−0.349016	+	0.349016i
$617$	483.018	−	483.018i	0.782849	−	0.782849i	−0.197461	−	0.980311i	$-0.563270\pi$
$617$	483.018	−	483.018i	0.782849	−	0.782849i	0.980311	+	0.197461i	$0.0632697\pi$
$618$	−311.004	−	311.004i	−0.503242	−	0.503242i
$619$	413.378	+	413.378i	0.667816	+	0.667816i	0.957210	−	0.289394i	$-0.0934538\pi$
$619$	413.378	+	413.378i	0.667816	+	0.667816i	−0.289394	+	0.957210i	$0.593454\pi$
$620$	−286.922			−0.462778
$621$		−	245.666i		−	0.395597i
$622$	−331.141	−	331.141i	−0.532381	−	0.532381i
$623$			831.267i			1.33430i
$624$		0			0
$625$	700.050			1.12008
$626$	391.075	−	391.075i	0.624721	−	0.624721i
$627$	−1064.69			−1.69806
$628$			502.010i			0.799380i
$629$	209.735	−	209.735i	0.333442	−	0.333442i
$630$	−317.676	+	317.676i	−0.504247	+	0.504247i
$631$	−1.18650	−	1.18650i	−0.00188035	−	0.00188035i	0.706166	−	0.708046i	$-0.250423\pi$
$631$	−1.18650	−	1.18650i	−0.00188035	−	0.00188035i	−0.708046	+	0.706166i	$0.750423\pi$
$632$	15.9706	+	15.9706i	0.0252700	+	0.0252700i
$633$	509.356			0.804670
$634$			645.927i			1.01881i
$635$	−312.468	−	312.468i	−0.492075	−	0.492075i
$636$			592.809i			0.932089i
$637$		0			0
$638$	211.938			0.332191
$639$	234.514	−	234.514i	0.367002	−	0.367002i
$640$	60.3869			0.0943546
$641$			1227.53i			1.91502i	0.288406	+	0.957508i	$0.406875\pi$
$641$			1227.53i			1.91502i	−0.288406	+	0.957508i	$0.593125\pi$
$642$	625.952	−	625.952i	0.975003	−	0.975003i
$643$	426.567	−	426.567i	0.663401	−	0.663401i	−0.292779	−	0.956180i	$-0.594580\pi$
$643$	426.567	−	426.567i	0.663401	−	0.663401i	0.956180	+	0.292779i	$0.0945801\pi$
$644$	289.618	+	289.618i	0.449718	+	0.449718i
$645$	173.844	+	173.844i	0.269526	+	0.269526i
$646$	−182.605			−0.282670
$647$			876.797i			1.35517i	0.735443	+	0.677587i	$0.236974\pi$
$647$			876.797i			1.35517i	−0.735443	+	0.677587i	$0.763026\pi$
$648$	−199.117	−	199.117i	−0.307279	−	0.307279i
$649$			659.057i			1.01550i
$650$		0			0
$651$	−1060.98			−1.62977
$652$	171.940	−	171.940i	0.263712	−	0.263712i
$653$	150.361			0.230261			0.115131	−	0.993350i	$-0.463271\pi$
$653$	150.361			0.230261			0.115131	+	0.993350i	$0.463271\pi$
$654$		−	482.232i		−	0.737357i
$655$	−180.597	+	180.597i	−0.275721	+	0.275721i
$656$	14.1640	−	14.1640i	0.0215914	−	0.0215914i
$657$	−74.2422	−	74.2422i	−0.113002	−	0.113002i
$658$	110.160	+	110.160i	0.167416	+	0.167416i
$659$	873.541			1.32556			0.662778	−	0.748816i	$-0.269377\pi$
$659$	873.541			1.32556			0.662778	+	0.748816i	$0.269377\pi$
$660$			430.262i			0.651913i
$661$	336.481	+	336.481i	0.509048	+	0.509048i	0.914234	−	0.405186i	$-0.132793\pi$
$661$	336.481	+	336.481i	0.509048	+	0.509048i	−0.405186	+	0.914234i	$0.632793\pi$
$662$		−	693.033i		−	1.04688i
$663$		0			0
$664$	143.337			0.215869
$665$	−1022.94	+	1022.94i	−1.53826	+	1.53826i
$666$	−497.777			−0.747413
$667$		−	285.502i		−	0.428038i
$668$	87.5419	−	87.5419i	0.131051	−	0.131051i
$669$	325.079	−	325.079i	0.485918	−	0.485918i
$670$	−32.6246	−	32.6246i	−0.0486934	−	0.0486934i
$671$	416.905	+	416.905i	0.621318	+	0.621318i
$672$	223.299			0.332290
$673$		−	1105.76i		−	1.64302i	−0.570191	−	0.821512i	$-0.693131\pi$
$673$		−	1105.76i		−	1.64302i	0.570191	−	0.821512i	$-0.306869\pi$
$674$	498.500	+	498.500i	0.739614	+	0.739614i
$675$		−	42.9433i		−	0.0636197i
$676$		0			0
$677$	474.341			0.700651			0.350326	−	0.936628i	$-0.386071\pi$
$677$	474.341			0.700651			0.350326	+	0.936628i	$0.386071\pi$
$678$	−268.843	+	268.843i	−0.396524	+	0.396524i
$679$	−562.908			−0.829025
$680$			73.7945i			0.108521i
$681$	−1153.93	+	1153.93i	−1.69446	+	1.69446i
$682$	−281.593	+	281.593i	−0.412892	+	0.412892i
$683$	741.412	+	741.412i	1.08552	+	1.08552i	0.995983	+	0.0895398i	$0.0285396\pi$
$683$	741.412	+	741.412i	1.08552	+	1.08552i	0.0895398	+	0.995983i	$0.471460\pi$
$684$	216.694	+	216.694i	0.316804	+	0.316804i
$685$	−1027.20			−1.49956
$686$			105.626i			0.153974i
$687$	−608.644	−	608.644i	−0.885945	−	0.885945i
$688$		−	47.8913i		−	0.0696095i
$689$		0			0
$690$	579.606			0.840009
$691$	286.793	−	286.793i	0.415040	−	0.415040i	−0.468450	−	0.883490i	$-0.655187\pi$
$691$	286.793	−	286.793i	0.415040	−	0.415040i	0.883490	+	0.468450i	$0.155187\pi$
$692$	621.671			0.898368
$693$			623.550i			0.899784i
$694$	234.625	−	234.625i	0.338076	−	0.338076i
$695$	−728.768	+	728.768i	−1.04859	+	1.04859i
$696$	−110.062	−	110.062i	−0.158136	−	0.158136i
$697$	17.3088	+	17.3088i	0.0248333	+	0.0248333i
$698$	250.122			0.358341
$699$		−	627.203i		−	0.897287i
$700$	50.6264	+	50.6264i	0.0723234	+	0.0723234i
$701$		−	322.739i		−	0.460398i	−0.973144	−	0.230199i	$-0.926062\pi$
$701$		−	322.739i		−	0.460398i	0.973144	−	0.230199i	$-0.0739378\pi$
$702$		0			0
$703$	−1602.88			−2.28006
$704$	59.2653	−	59.2653i	0.0841836	−	0.0841836i
$705$	220.460			0.312710
$706$			318.065i			0.450518i
$707$	102.035	−	102.035i	0.144321	−	0.144321i
$708$	342.258	−	342.258i	0.483415	−	0.483415i
$709$	−68.9061	−	68.9061i	−0.0971877	−	0.0971877i	0.656841	−	0.754029i	$-0.271892\pi$
$709$	−68.9061	−	68.9061i	−0.0971877	−	0.0971877i	−0.754029	+	0.656841i	$0.771892\pi$
$710$	−305.174	−	305.174i	−0.429822	−	0.429822i
$711$	46.3198			0.0651475
$712$		−	229.147i		−	0.321836i
$713$	379.333	+	379.333i	0.532024	+	0.532024i
$714$			272.877i			0.382181i
$715$		0			0
$716$	−126.681			−0.176928
$717$	−713.475	+	713.475i	−0.995083	+	0.995083i
$718$	580.556			0.808574
$719$		−	892.454i		−	1.24124i	−0.784110	−	0.620622i	$-0.786880\pi$
$719$		−	892.454i		−	1.24124i	0.784110	−	0.620622i	$-0.213120\pi$
$720$	87.5705	−	87.5705i	0.121626	−	0.121626i
$721$	−586.519	+	586.519i	−0.813479	+	0.813479i
$722$	336.770	+	336.770i	0.466440	+	0.466440i
$723$	−1235.86	−	1235.86i	−1.70934	−	1.70934i
$724$	373.008			0.515205
$725$		−	49.9067i		−	0.0688369i
$726$	−43.2359	−	43.2359i	−0.0595536	−	0.0595536i
$727$		−	1093.66i		−	1.50435i	−0.658964	−	0.752175i	$-0.729005\pi$
$727$		−	1093.66i		−	1.50435i	0.658964	−	0.752175i	$-0.270995\pi$
$728$		0			0
$729$	−151.327			−0.207581
$730$	−96.6115	+	96.6115i	−0.132345	+	0.132345i
$731$	58.5245			0.0800609
$732$		−	433.009i		−	0.591542i
$733$	−136.804	+	136.804i	−0.186636	+	0.186636i	−0.794240	−	0.607604i	$-0.792131\pi$
$733$	−136.804	+	136.804i	−0.186636	+	0.186636i	0.607604	+	0.794240i	$0.292131\pi$
$734$	220.744	−	220.744i	0.300741	−	0.300741i
$735$	817.168	+	817.168i	1.11179	+	1.11179i
$736$	−79.8362	−	79.8362i	−0.108473	−	0.108473i
$737$	−64.0371			−0.0868889
$738$		−	41.0800i		−	0.0556639i
$739$	568.031	+	568.031i	0.768649	+	0.768649i	0.977869	−	0.209220i	$-0.0670924\pi$
$739$	568.031	+	568.031i	0.768649	+	0.768649i	−0.209220	+	0.977869i	$0.567092\pi$
$740$			647.758i			0.875349i
$741$		0			0
$742$	1117.97			1.50670
$743$	−505.313	+	505.313i	−0.680098	+	0.680098i	−0.960022	−	0.279924i	$-0.909691\pi$
$743$	−505.313	+	505.313i	−0.680098	+	0.680098i	0.279924	+	0.960022i	$0.409691\pi$
$744$	292.470			0.393105
$745$		−	717.748i		−	0.963421i
$746$	62.1593	−	62.1593i	0.0833235	−	0.0833235i
$747$	207.861	−	207.861i	0.278261	−	0.278261i
$748$	72.4238	+	72.4238i	0.0968233	+	0.0968233i
$749$	−1180.48	−	1180.48i	−1.57607	−	1.57607i
$750$	−624.677			−0.832902
$751$		−	24.3942i		−	0.0324823i	−0.999868	−	0.0162412i	$-0.994830\pi$
$751$		−	24.3942i		−	0.0324823i	0.999868	−	0.0162412i	$-0.00516995\pi$
$752$	−30.3667	−	30.3667i	−0.0403813	−	0.0403813i
$753$		−	1688.89i		−	2.24288i
$754$		0			0
$755$	200.652			0.265764
$756$	−178.604	+	178.604i	−0.236249	+	0.236249i
$757$	−1025.22			−1.35433			−0.677163	−	0.735833i	$-0.736791\pi$
$757$	−1025.22			−1.35433			−0.677163	+	0.735833i	$0.736791\pi$
$758$			205.447i			0.271038i
$759$	568.840	−	568.840i	0.749460	−	0.749460i
$760$	281.984	−	281.984i	0.371031	−	0.371031i
$761$	272.647	+	272.647i	0.358275	+	0.358275i	0.863177	−	0.504902i	$-0.168471\pi$
$761$	272.647	+	272.647i	0.358275	+	0.358275i	−0.504902	+	0.863177i	$0.668471\pi$
$762$	318.509	+	318.509i	0.417991	+	0.417991i
$763$	−909.436			−1.19192
$764$			365.383i			0.478250i
$765$	107.014	+	107.014i	0.139887	+	0.139887i
$766$			711.051i			0.928265i
$767$		0			0
$768$	−61.5545			−0.0801491
$769$	−171.616	+	171.616i	−0.223168	+	0.223168i	−0.809831	−	0.586663i	$-0.800441\pi$
$769$	−171.616	+	171.616i	−0.223168	+	0.223168i	0.586663	+	0.809831i	$0.300441\pi$
$770$	811.427			1.05380
$771$			1682.01i			2.18159i
$772$	128.910	−	128.910i	0.166981	−	0.166981i
$773$	−367.857	+	367.857i	−0.475883	+	0.475883i	−0.903812	−	0.427929i	$-0.859243\pi$
$773$	−367.857	+	367.857i	−0.475883	+	0.475883i	0.427929	+	0.903812i	$0.359243\pi$
$774$	−69.4499	−	69.4499i	−0.0897286	−	0.0897286i
$775$	66.3089	+	66.3089i	0.0855598	+	0.0855598i
$776$	155.171			0.199963
$777$			2395.28i			3.08273i
$778$	284.973	+	284.973i	0.366289	+	0.366289i
$779$		−	132.281i		−	0.169808i
$780$		0			0
$781$	−599.010			−0.766978
$782$	97.5621	−	97.5621i	0.124760	−	0.124760i
$783$	176.065			0.224860
$784$		−	225.117i		−	0.287139i
$785$	947.340	−	947.340i	1.20680	−	1.20680i
$786$	184.089	−	184.089i	0.234210	−	0.234210i
$787$	−1028.17	−	1028.17i	−1.30644	−	1.30644i	−0.923965	−	0.382478i	$-0.875071\pi$
$787$	−1028.17	−	1028.17i	−1.30644	−	1.30644i	−0.382478	−	0.923965i	$-0.624929\pi$
$788$	−305.193	−	305.193i	−0.387300	−	0.387300i
$789$	1107.83			1.40409
$790$		−	60.2761i		−	0.0762988i
$791$	507.008	+	507.008i	0.640971	+	0.640971i
$792$		−	171.888i		−	0.217030i
$793$		0			0
$794$	−418.671			−0.527294
$795$	1118.68	−	1118.68i	1.40715	−	1.40715i
$796$	156.979			0.197209
$797$			472.216i			0.592491i	0.955112	+	0.296246i	$0.0957347\pi$
$797$			472.216i			0.592491i	−0.955112	+	0.296246i	$0.904265\pi$
$798$	1042.72	−	1042.72i	1.30667	−	1.30667i
$799$	37.1090	−	37.1090i	0.0464443	−	0.0464443i
$800$	−13.9557	−	13.9557i	−0.0174446	−	0.0174446i
$801$	−332.299	−	332.299i	−0.414855	−	0.414855i
$802$	342.048			0.426494
$803$			189.634i			0.236157i
$804$	33.2554	+	33.2554i	0.0413624	+	0.0413624i
$805$		−	1093.07i		−	1.35785i
$806$		0			0
$807$	−1321.11			−1.63707
$808$	−28.1270	+	28.1270i	−0.0348106	+	0.0348106i
$809$	954.431			1.17977			0.589883	−	0.807489i	$-0.299174\pi$
$809$	954.431			1.17977			0.589883	+	0.807489i	$0.299174\pi$
$810$			751.503i			0.927782i
$811$	−643.215	+	643.215i	−0.793114	+	0.793114i	−0.981999	−	0.188885i	$-0.939513\pi$
$811$	−643.215	+	643.215i	−0.793114	+	0.793114i	0.188885	+	0.981999i	$0.439513\pi$
$812$	−207.565	+	207.565i	−0.255622	+	0.255622i
$813$	22.5993	+	22.5993i	0.0277975	+	0.0277975i
$814$	635.726	+	635.726i	0.780990	+	0.780990i
$815$	−648.934			−0.796238
$816$		−	75.2214i		−	0.0921830i
$817$	−223.634	−	223.634i	−0.273726	−	0.273726i
$818$		−	275.961i		−	0.337361i
$819$		0			0
$820$	−53.4574			−0.0651920
$821$	−609.239	+	609.239i	−0.742069	+	0.742069i	−0.972976	−	0.230907i	$-0.925831\pi$
$821$	−609.239	+	609.239i	−0.742069	+	0.742069i	0.230907	+	0.972976i	$0.425831\pi$
$822$	1047.06			1.27379
$823$		−	747.037i		−	0.907700i	−0.891078	−	0.453850i	$-0.850050\pi$
$823$		−	747.037i		−	0.907700i	0.891078	−	0.453850i	$-0.149950\pi$
$824$	161.680	−	161.680i	0.196213	−	0.196213i
$825$	99.4354	−	99.4354i	0.120528	−	0.120528i
$826$	−645.460	−	645.460i	−0.781428	−	0.781428i
$827$	1064.80	+	1064.80i	1.28755	+	1.28755i	0.936273	+	0.351272i	$0.114251\pi$
$827$	1064.80	+	1064.80i	1.28755	+	1.28755i	0.351272	+	0.936273i	$0.385749\pi$
$828$	−231.550			−0.279650
$829$		−	127.571i		−	0.153886i	−0.997036	−	0.0769429i	$-0.975484\pi$
$829$		−	127.571i		−	0.153886i	0.997036	−	0.0769429i	$-0.0245159\pi$
$830$	−270.490	−	270.490i	−0.325892	−	0.325892i
$831$			1709.09i			2.05667i
$832$		0			0
$833$	275.099			0.330251
$834$	742.859	−	742.859i	0.890718	−	0.890718i
$835$	−330.399			−0.395688
$836$		−	553.492i		−	0.662072i
$837$	−233.930	+	233.930i	−0.279486	+	0.279486i
$838$	−47.9110	+	47.9110i	−0.0571730	+	0.0571730i
$839$	335.660	+	335.660i	0.400071	+	0.400071i	0.878258	−	0.478187i	$-0.158706\pi$
$839$	335.660	+	335.660i	0.400071	+	0.400071i	−0.478187	+	0.878258i	$0.658706\pi$
$840$	−421.385	−	421.385i	−0.501649	−	0.501649i
$841$	−636.385			−0.756700
$842$			954.036i			1.13306i
$843$	−629.697	−	629.697i	−0.746972	−	0.746972i
$844$			264.796i			0.313739i
$845$		0			0
$846$	−88.0730			−0.104105
$847$	−81.5381	+	81.5381i	−0.0962670	+	0.0962670i
$848$	−308.180			−0.363420
$849$		−	1785.62i		−	2.10320i
$850$	17.0542	−	17.0542i	0.0200638	−	0.0200638i
$851$	856.386	−	856.386i	1.00633	−	1.00633i
$852$	311.074	+	311.074i	0.365111	+	0.365111i
$853$	−425.319	−	425.319i	−0.498616	−	0.498616i	0.412391	−	0.911007i	$-0.364694\pi$
$853$	−425.319	−	425.319i	−0.498616	−	0.498616i	−0.911007	+	0.412391i	$0.864694\pi$
$854$	−816.606			−0.956214
$855$		−	817.841i		−	0.956540i
$856$	325.410	+	325.410i	0.380152	+	0.380152i
$857$		−	48.5237i		−	0.0566204i	−0.999599	−	0.0283102i	$-0.990987\pi$
$857$		−	48.5237i		−	0.0566204i	0.999599	−	0.0283102i	$-0.00901262\pi$
$858$		0			0
$859$	165.922			0.193157			0.0965787	−	0.995325i	$-0.469210\pi$
$859$	165.922			0.193157			0.0965787	+	0.995325i	$0.469210\pi$
$860$	−90.3753	+	90.3753i	−0.105088	+	0.105088i
$861$	−197.675			−0.229588
$862$		−	110.119i		−	0.127748i
$863$	−376.277	+	376.277i	−0.436010	+	0.436010i	−0.890667	−	0.454657i	$-0.849762\pi$
$863$	−376.277	+	376.277i	−0.436010	+	0.436010i	0.454657	+	0.890667i	$0.349762\pi$
$864$	49.2340	−	49.2340i	0.0569838	−	0.0569838i
$865$	−1173.15	−	1173.15i	−1.35624	−	1.35624i
$866$	−497.603	−	497.603i	−0.574599	−	0.574599i
$867$	−1019.91			−1.17636
$868$		−	551.566i		−	0.635445i
$869$	−59.1565	−	59.1565i	−0.0680742	−	0.0680742i
$870$			415.395i			0.477466i
$871$		0			0
$872$	250.695			0.287494
$873$	225.023	−	225.023i	0.257758	−	0.257758i
$874$	−745.609			−0.853099
$875$			1178.07i			1.34637i
$876$	98.4795	−	98.4795i	0.112420	−	0.112420i
$877$	903.857	−	903.857i	1.03062	−	1.03062i	0.0311071	−	0.999516i	$-0.490097\pi$
$877$	903.857	−	903.857i	1.03062	−	1.03062i	0.999516	−	0.0311071i	$-0.00990331\pi$
$878$	−167.510	−	167.510i	−0.190786	−	0.190786i
$879$	172.920	+	172.920i	0.196724	+	0.196724i
$880$	−223.678			−0.254179
$881$		−	1248.35i		−	1.41697i	−0.705727	−	0.708484i	$-0.749379\pi$
$881$		−	1248.35i		−	1.41697i	0.705727	−	0.708484i	$-0.250621\pi$
$882$	−326.455	−	326.455i	−0.370130	−	0.370130i
$883$			1217.43i			1.37874i	0.724408	+	0.689371i	$0.242113\pi$
$883$			1217.43i			1.37874i	−0.724408	+	0.689371i	$0.757887\pi$
$884$		0			0
$885$	−1291.74			−1.45960
$886$	443.835	−	443.835i	0.500942	−	0.500942i
$887$	−31.0970			−0.0350587			−0.0175293	−	0.999846i	$-0.505580\pi$
$887$	−31.0970			−0.0350587			−0.0175293	+	0.999846i	$0.505580\pi$
$888$		−	660.283i		−	0.743562i
$889$	600.673	−	600.673i	0.675673	−	0.675673i
$890$	−432.421	+	432.421i	−0.485867	+	0.485867i
$891$	737.544	+	737.544i	0.827771	+	0.827771i
$892$	168.997	+	168.997i	0.189459	+	0.189459i
$893$	−283.602			−0.317583
$894$			731.626i			0.818374i
$895$	239.058	+	239.058i	0.267104	+	0.267104i
$896$			116.085i			0.129559i
$897$		0			0
$898$	−363.170			−0.404421
$899$	−271.863	+	271.863i	−0.302406	+	0.302406i
$900$	−40.4758			−0.0449731
$901$		−	376.605i		−	0.417985i
$902$	−52.4645	+	52.4645i	−0.0581646	+	0.0581646i
$903$	−334.190	+	334.190i	−0.370088	+	0.370088i
$904$	−139.762	−	139.762i	−0.154604	−	0.154604i
$905$	−703.901	−	703.901i	−0.777791	−	0.777791i
$906$	−204.531			−0.225752
$907$			1539.44i			1.69728i	0.528968	+	0.848642i	$0.322579\pi$
$907$			1539.44i			1.69728i	−0.528968	+	0.848642i	$0.677421\pi$
$908$	−599.884	−	599.884i	−0.660666	−	0.660666i
$909$			81.5771i			0.0897438i
$910$		0			0
$911$	−148.518			−0.163027			−0.0815136	−	0.996672i	$-0.525975\pi$
$911$	−148.518			−0.163027			−0.0815136	+	0.996672i	$0.525975\pi$
$912$	−287.436	+	287.436i	−0.315171	+	0.315171i
$913$	−530.932			−0.581525
$914$		−	123.191i		−	0.134783i
$915$	−817.127	+	817.127i	−0.893035	+	0.893035i
$916$	316.412	−	316.412i	0.345428	−	0.345428i
$917$	−347.171	−	347.171i	−0.378595	−	0.378595i
$918$	60.1653	+	60.1653i	0.0655395	+	0.0655395i
$919$	−1750.17			−1.90442			−0.952212	−	0.305438i	$-0.901197\pi$
$919$	−1750.17			−1.90442			−0.952212	+	0.305438i	$0.901197\pi$
$920$			301.316i			0.327518i
$921$	258.296	+	258.296i	0.280452	+	0.280452i
$922$		−	699.794i		−	0.758996i
$923$		0			0
$924$	−827.116			−0.895148
$925$	149.700	−	149.700i	0.161837	−	0.161837i
$926$	−798.943			−0.862789
$927$		−	468.922i		−	0.505849i
$928$	57.2175	−	57.2175i	0.0616568	−	0.0616568i
$929$	696.265	−	696.265i	0.749478	−	0.749478i	−0.224903	−	0.974381i	$-0.572207\pi$
$929$	696.265	−	696.265i	0.749478	−	0.749478i	0.974381	+	0.224903i	$0.0722066\pi$
$930$	−551.917	−	551.917i	−0.593460	−	0.593460i
$931$	−1051.21	−	1051.21i	−1.12912	−	1.12912i
$932$	326.061			0.349850
$933$		−	1273.95i		−	1.36544i
$934$	−409.816	−	409.816i	−0.438775	−	0.438775i
$935$		−	273.341i		−	0.292343i
$936$		0			0
$937$	835.853			0.892052			0.446026	−	0.895020i	$-0.352839\pi$
$937$	835.853			0.892052			0.446026	+	0.895020i	$0.352839\pi$
$938$	62.7159	−	62.7159i	0.0668613	−	0.0668613i
$939$	1504.53			1.60227
$940$			114.610i			0.121925i
$941$	−759.883	+	759.883i	−0.807527	+	0.807527i	−0.984259	−	0.176732i	$-0.943447\pi$
$941$	−759.883	+	759.883i	−0.807527	+	0.807527i	0.176732	+	0.984259i	$0.443447\pi$
$942$	−965.657	+	965.657i	−1.02511	+	1.02511i
$943$	70.6749	+	70.6749i	0.0749468	+	0.0749468i
$944$	177.927	+	177.927i	0.188482	+	0.188482i
$945$	674.084			0.713317
$946$			177.393i			0.187519i
$947$	−816.039	−	816.039i	−0.861709	−	0.861709i	0.129827	−	0.991537i	$-0.458558\pi$
$947$	−816.039	−	816.039i	−0.861709	−	0.861709i	−0.991537	+	0.129827i	$0.958558\pi$
$948$			61.4415i			0.0648118i
$949$		0			0
$950$	−130.335			−0.137195
$951$	−1242.49	+	1242.49i	−1.30651	+	1.30651i
$952$	−141.859			−0.149012
$953$		−	1628.14i		−	1.70844i	−0.519914	−	0.854219i	$-0.674036\pi$
$953$		−	1628.14i		−	1.70844i	0.519914	−	0.854219i	$-0.325964\pi$
$954$	−446.909	+	446.909i	−0.468458	+	0.468458i
$955$	689.511	−	689.511i	0.722001	−	0.722001i
$956$	−370.910	−	370.910i	−0.387981	−	0.387981i
$957$	407.680	+	407.680i	0.425998	+	0.425998i
$958$	753.135			0.786153
$959$		−	1974.63i		−	2.05906i
$960$	116.159	+	116.159i	0.120999	+	0.120999i
$961$			238.576i			0.248258i
$962$		0			0
$963$	943.791			0.980053
$964$	642.477	−	642.477i	0.666470	−	0.666470i
$965$	−486.529			−0.504175
$966$			1114.21i			1.15342i
$967$	1097.29	−	1097.29i	1.13473	−	1.13473i	0.145354	−	0.989380i	$-0.453568\pi$
$967$	1097.29	−	1097.29i	1.13473	−	1.13473i	0.989380	−	0.145354i	$-0.0464320\pi$
$968$	22.4768	−	22.4768i	0.0232198	−	0.0232198i
$969$	−351.255	−	351.255i	−0.362492	−	0.362492i
$970$	−292.822	−	292.822i	−0.301879	−	0.301879i
$971$	1289.79			1.32831			0.664157	−	0.747593i	$-0.268791\pi$
$971$	1289.79			1.32831			0.664157	+	0.747593i	$0.268791\pi$
$972$		−	544.481i		−	0.560165i
$973$	−1400.95	−	1400.95i	−1.43982	−	1.43982i
$974$		−	952.097i		−	0.977512i
$975$		0			0
$976$	225.106			0.230641
$977$	1105.01	−	1105.01i	1.13102	−	1.13102i	0.141012	−	0.990008i	$-0.454964\pi$
$977$	1105.01	−	1105.01i	1.13102	−	1.13102i	0.990008	−	0.141012i	$-0.0450357\pi$
$978$	661.481			0.676361
$979$			848.778i			0.866985i
$980$	−424.816	+	424.816i	−0.433486	+	0.433486i
$981$	363.547	−	363.547i	0.370588	−	0.370588i
$982$	−117.456	−	117.456i	−0.119609	−	0.119609i
$983$	−79.2765	−	79.2765i	−0.0806475	−	0.0806475i	0.665632	−	0.746280i	$-0.268162\pi$
$983$	−79.2765	−	79.2765i	−0.0806475	−	0.0806475i	−0.746280	+	0.665632i	$0.768162\pi$
$984$	54.4911			0.0553771
$985$			1151.85i			1.16939i
$986$	69.9214	+	69.9214i	0.0709142	+	0.0709142i
$987$			423.803i			0.429385i
$988$		0			0
$989$	238.966			0.241624
$990$	−324.368	+	324.368i	−0.327645	+	0.327645i
$991$	−203.185			−0.205030			−0.102515	−	0.994731i	$-0.532689\pi$
$991$	−203.185			−0.205030			−0.102515	+	0.994731i	$0.532689\pi$
$992$			152.045i			0.153271i
$993$	1333.10	−	1333.10i	1.34250	−	1.34250i
$994$	586.652	−	586.652i	0.590193	−	0.590193i
$995$	−296.233	−	296.233i	−0.297721	−	0.297721i
$996$	275.720	+	275.720i	0.276828	+	0.276828i
$997$	1972.04			1.97798			0.988989	−	0.147987i	$-0.0472792\pi$
$997$	1972.04			1.97798			0.988989	+	0.147987i	$0.0472792\pi$
$998$		−	310.774i		−	0.311396i
$999$	528.123	+	528.123i	0.528651	+	0.528651i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.3.d.f.239.2		8
13.2	odd	12		26.3.f.b.7.2	✓	8
13.3	even	3		338.3.f.j.19.2		8
13.4	even	6		26.3.f.b.15.2	yes	8
13.5	odd	4		338.3.d.g.99.2		8
13.6	odd	12		338.3.f.h.89.2		8
13.7	odd	12		338.3.f.j.89.2		8
13.8	odd	4	inner	338.3.d.f.99.2		8
13.9	even	3		338.3.f.i.249.2		8
13.10	even	6		338.3.f.h.19.2		8
13.11	odd	12		338.3.f.i.319.2		8
13.12	even	2		338.3.d.g.239.2		8
39.2	even	12		234.3.bb.f.163.1		8
39.17	odd	6		234.3.bb.f.145.1		8
52.15	even	12		208.3.bd.f.33.1		8
52.43	odd	6		208.3.bd.f.145.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.3.f.b.7.2	✓	8	13.2	odd	12
26.3.f.b.15.2	yes	8	13.4	even	6
208.3.bd.f.33.1		8	52.15	even	12
208.3.bd.f.145.1		8	52.43	odd	6
234.3.bb.f.145.1		8	39.17	odd	6
234.3.bb.f.163.1		8	39.2	even	12
338.3.d.f.99.2		8	13.8	odd	4	inner
338.3.d.f.239.2		8	1.1	even	1	trivial
338.3.d.g.99.2		8	13.5	odd	4
338.3.d.g.239.2		8	13.12	even	2
338.3.f.h.19.2		8	13.10	even	6
338.3.f.h.89.2		8	13.6	odd	12
338.3.f.i.249.2		8	13.9	even	3
338.3.f.i.319.2		8	13.11	odd	12
338.3.f.j.19.2		8	13.3	even	3
338.3.f.j.89.2		8	13.7	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	338.d (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -3.84716 q^{3} -2.00000i q^{4} +(-3.77418 + 3.77418i) q^{5} +(3.84716 - 3.84716i) q^{6} +(-7.25532 - 7.25532i) q^{7} +(2.00000 + 2.00000i) q^{8} +5.80063 q^{9} -7.54837i q^{10} +(-7.40816 - 7.40816i) q^{11} +7.69432i q^{12} +14.5106 q^{14} +(14.5199 - 14.5199i) q^{15} -4.00000 q^{16} -4.88811i q^{17} +(-5.80063 + 5.80063i) q^{18} +(-18.6785 + 18.6785i) q^{19} +(7.54837 + 7.54837i) q^{20} +(27.9124 + 27.9124i) q^{21} +14.8163 q^{22} -19.9590i q^{23} +(-7.69432 - 7.69432i) q^{24} -3.48892i q^{25} +12.3085 q^{27} +(-14.5106 + 14.5106i) q^{28} +14.3044 q^{29} +29.0398i q^{30} +(-19.0056 + 19.0056i) q^{31} +(4.00000 - 4.00000i) q^{32} +(28.5004 + 28.5004i) q^{33} +(4.88811 + 4.88811i) q^{34} +54.7658 q^{35} -11.6013i q^{36} +(42.9072 + 42.9072i) q^{37} -37.3569i q^{38} -15.0967 q^{40} +(-3.54099 + 3.54099i) q^{41} -55.8247 q^{42} +11.9728i q^{43} +(-14.8163 + 14.8163i) q^{44} +(-21.8926 + 21.8926i) q^{45} +(19.9590 + 19.9590i) q^{46} +(7.59168 + 7.59168i) q^{47} +15.3886 q^{48} +56.2792i q^{49} +(3.48892 + 3.48892i) q^{50} +18.8053i q^{51} +77.0450 q^{53} +(-12.3085 + 12.3085i) q^{54} +55.9195 q^{55} -29.0213i q^{56} +(71.8590 - 71.8590i) q^{57} +(-14.3044 + 14.3044i) q^{58} +(-44.4819 - 44.4819i) q^{59} +(-29.0398 - 29.0398i) q^{60} -56.2764 q^{61} -38.0112i q^{62} +(-42.0854 - 42.0854i) q^{63} +8.00000i q^{64} -57.0007 q^{66} +(4.32207 - 4.32207i) q^{67} -9.77622 q^{68} +76.7856i q^{69} +(-54.7658 + 54.7658i) q^{70} +(40.4291 - 40.4291i) q^{71} +(11.6013 + 11.6013i) q^{72} +(-12.7990 - 12.7990i) q^{73} -85.8143 q^{74} +13.4224i q^{75} +(37.3569 + 37.3569i) q^{76} +107.497i q^{77} +7.98532 q^{79} +(15.0967 - 15.0967i) q^{80} -99.5584 q^{81} -7.08199i q^{82} +(35.8343 - 35.8343i) q^{83} +(55.8247 - 55.8247i) q^{84} +(18.4486 + 18.4486i) q^{85} +(-11.9728 - 11.9728i) q^{86} -55.0312 q^{87} -29.6326i q^{88} +(-57.2867 - 57.2867i) q^{89} -43.7853i q^{90} -39.9181 q^{92} +(73.1175 - 73.1175i) q^{93} -15.1834 q^{94} -140.992i q^{95} +(-15.3886 + 15.3886i) q^{96} +(38.7928 - 38.7928i) q^{97} +(-56.2792 - 56.2792i) q^{98} +(-42.9720 - 42.9720i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 6 q^{5} - 10 q^{7} + 16 q^{8} + 84 q^{9} - 42 q^{11} + 20 q^{14} - 60 q^{15} - 32 q^{16} - 84 q^{18} - 22 q^{19} + 12 q^{20} + 102 q^{21} + 84 q^{22} + 72 q^{27} - 20 q^{28} + 12 q^{29}+ \cdots - 252 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 6 * q^5 - 10 * q^7 + 16 * q^8 + 84 * q^9 - 42 * q^11 + 20 * q^14 - 60 * q^15 - 32 * q^16 - 84 * q^18 - 22 * q^19 + 12 * q^20 + 102 * q^21 + 84 * q^22 + 72 * q^27 - 20 * q^28 + 12 * q^29 - 32 * q^31 + 32 * q^32 - 54 * q^33 + 60 * q^34 + 156 * q^35 - 32 * q^37 - 24 * q^40 + 12 * q^41 - 204 * q^42 - 84 * q^44 + 102 * q^45 + 108 * q^46 - 60 * q^47 + 88 * q^50 - 132 * q^53 - 72 * q^54 + 324 * q^55 + 294 * q^57 - 12 * q^58 - 234 * q^59 + 120 * q^60 - 72 * q^61 + 156 * q^63 + 108 * q^66 + 14 * q^67 - 120 * q^68 - 156 * q^70 - 162 * q^71 + 168 * q^72 - 166 * q^73 + 64 * q^74 + 44 * q^76 - 96 * q^79 + 24 * q^80 + 24 * q^81 + 240 * q^83 + 204 * q^84 + 234 * q^85 - 132 * q^86 - 720 * q^87 - 210 * q^89 - 216 * q^92 + 444 * q^93 + 120 * q^94 - 146 * q^97 + 16 * q^98 - 252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more