LMFDB - Embedded newform 338.5.d.c.239.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,5,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, 5, 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, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	338.d (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,-12,0,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$34.9390475223$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 522x^{4} + 68121x^{2} + 54756 $ x^6 + 522x^4 + 68121x^2 + 54756
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			239.1
Root			$16.5864i$ of defining polynomial
Character	$\chi$	$=$	338.239
Dual form			338.5.d.c.99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 + 2.00000i) q^{2} -16.5864 q^{3} -8.00000i q^{4} +(13.5581 - 13.5581i) q^{5} +(33.1728 - 33.1728i) q^{6} +(40.1445 + 40.1445i) q^{7} +(16.0000 + 16.0000i) q^{8} +194.108 q^{9} +54.2326i q^{10} +(12.0598 + 12.0598i) q^{11} +132.691i q^{12} -160.578 q^{14} +(-224.880 + 224.880i) q^{15} -64.0000 q^{16} +223.703i q^{17} +(-388.216 + 388.216i) q^{18} +(267.563 - 267.563i) q^{19} +(-108.465 - 108.465i) q^{20} +(-665.852 - 665.852i) q^{21} -48.2392 q^{22} -983.993i q^{23} +(-265.382 - 265.382i) q^{24} +257.354i q^{25} -1876.05 q^{27} +(321.156 - 321.156i) q^{28} -560.708 q^{29} -899.522i q^{30} +(139.400 - 139.400i) q^{31} +(128.000 - 128.000i) q^{32} +(-200.028 - 200.028i) q^{33} +(-447.405 - 447.405i) q^{34} +1088.57 q^{35} -1552.86i q^{36} +(700.116 + 700.116i) q^{37} +1070.25i q^{38} +433.860 q^{40} +(-1872.79 + 1872.79i) q^{41} +2663.41 q^{42} -430.626i q^{43} +(96.4783 - 96.4783i) q^{44} +(2631.74 - 2631.74i) q^{45} +(1967.99 + 1967.99i) q^{46} +(59.3604 + 59.3604i) q^{47} +1061.53 q^{48} +822.164i q^{49} +(-514.708 - 514.708i) q^{50} -3710.42i q^{51} +3773.97 q^{53} +(3752.10 - 3752.10i) q^{54} +327.017 q^{55} +1284.62i q^{56} +(-4437.90 + 4437.90i) q^{57} +(1121.42 - 1121.42i) q^{58} +(-924.257 - 924.257i) q^{59} +(1799.04 + 1799.04i) q^{60} -3935.30 q^{61} +557.601i q^{62} +(7792.37 + 7792.37i) q^{63} +512.000i q^{64} +800.113 q^{66} +(2992.99 - 2992.99i) q^{67} +1789.62 q^{68} +16320.9i q^{69} +(-2177.14 + 2177.14i) q^{70} +(9.46035 - 9.46035i) q^{71} +(3105.73 + 3105.73i) q^{72} +(4922.36 + 4922.36i) q^{73} -2800.47 q^{74} -4268.57i q^{75} +(-2140.51 - 2140.51i) q^{76} +968.269i q^{77} +1624.91 q^{79} +(-867.721 + 867.721i) q^{80} +15394.2 q^{81} -7491.16i q^{82} +(4406.88 - 4406.88i) q^{83} +(-5326.82 + 5326.82i) q^{84} +(3032.99 + 3032.99i) q^{85} +(861.252 + 861.252i) q^{86} +9300.11 q^{87} +385.913i q^{88} +(3715.92 + 3715.92i) q^{89} +10527.0i q^{90} -7871.95 q^{92} +(-2312.15 + 2312.15i) q^{93} -237.441 q^{94} -7255.32i q^{95} +(-2123.06 + 2123.06i) q^{96} +(4894.57 - 4894.57i) q^{97} +(-1644.33 - 1644.33i) q^{98} +(2340.90 + 2340.90i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 30 q^{5} + 90 q^{7} + 96 q^{8} + 558 q^{9} + 66 q^{11} - 360 q^{14} + 288 q^{15} - 384 q^{16} - 1116 q^{18} + 318 q^{19} - 240 q^{20} - 756 q^{21} - 264 q^{22} - 1404 q^{27} + 720 q^{28}+ \cdots + 61074 q^{99}+O(q^{100}) $ 6 * q - 12 * q^2 + 30 * q^5 + 90 * q^7 + 96 * q^8 + 558 * q^9 + 66 * q^11 - 360 * q^14 + 288 * q^15 - 384 * q^16 - 1116 * q^18 + 318 * q^19 - 240 * q^20 - 756 * q^21 - 264 * q^22 - 1404 * q^27 + 720 * q^28 - 276 * q^29 - 3282 * q^31 + 768 * q^32 + 3240 * q^33 - 2280 * q^34 + 5424 * q^35 + 3006 * q^37 + 960 * q^40 + 894 * q^41 + 3024 * q^42 + 528 * q^44 + 17226 * q^45 + 4848 * q^46 - 1566 * q^47 + 3300 * q^50 - 1356 * q^53 + 2808 * q^54 + 22212 * q^55 - 16812 * q^57 + 552 * q^58 - 5178 * q^59 - 2304 * q^60 + 2172 * q^61 + 24210 * q^63 - 12960 * q^66 - 1134 * q^67 + 9120 * q^68 - 10848 * q^70 + 18498 * q^71 + 8928 * q^72 + 13278 * q^73 - 12024 * q^74 - 2544 * q^76 - 13596 * q^79 - 1920 * q^80 + 58158 * q^81 + 11490 * q^83 - 6048 * q^84 + 10512 * q^85 + 7128 * q^86 + 30744 * q^87 + 28038 * q^89 - 19392 * q^92 - 23364 * q^93 + 6264 * q^94 + 27378 * q^97 + 11340 * q^98 + 61074 * 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 $

$2$	−2.00000	+	2.00000i	−0.500000	+	0.500000i
$2$	−2.00000	+	2.00000i	−0.500000	+	0.500000i
$3$	−16.5864			−1.84293			−0.921465	−	0.388460i	$-0.873007\pi$
$3$	−16.5864			−1.84293			−0.921465	+	0.388460i	$0.873007\pi$
$4$		−	8.00000i		−	0.500000i
$5$	13.5581	−	13.5581i	0.542326	−	0.542326i	−0.381885	−	0.924210i	$-0.624725\pi$
$5$	13.5581	−	13.5581i	0.542326	−	0.542326i	0.924210	+	0.381885i	$0.124725\pi$
$6$	33.1728	−	33.1728i	0.921465	−	0.921465i
$7$	40.1445	+	40.1445i	0.819276	+	0.819276i	0.986003	−	0.166727i	$-0.0533199\pi$
$7$	40.1445	+	40.1445i	0.819276	+	0.819276i	−0.166727	+	0.986003i	$0.553320\pi$
$8$	16.0000	+	16.0000i	0.250000	+	0.250000i
$9$	194.108			2.39639
$10$			54.2326i			0.542326i
$11$	12.0598	+	12.0598i	0.0996677	+	0.0996677i	0.755182	−	0.655515i	$-0.227548\pi$
$11$	12.0598	+	12.0598i	0.0996677	+	0.0996677i	−0.655515	+	0.755182i	$0.727548\pi$
$12$			132.691i			0.921465i
$13$		0			0
$13$		0			0
$14$	−160.578			−0.819276
$15$	−224.880	+	224.880i	−0.999468	+	0.999468i
$16$	−64.0000			−0.250000
$17$			223.703i			0.774058i	0.922068	+	0.387029i	$0.126499\pi$
$17$			223.703i			0.774058i	−0.922068	+	0.387029i	$0.873501\pi$
$18$	−388.216	+	388.216i	−1.19820	+	1.19820i
$19$	267.563	−	267.563i	0.741172	−	0.741172i	−0.231631	−	0.972804i	$-0.574406\pi$
$19$	267.563	−	267.563i	0.741172	−	0.741172i	0.972804	+	0.231631i	$0.0744064\pi$
$20$	−108.465	−	108.465i	−0.271163	−	0.271163i
$21$	−665.852	−	665.852i	−1.50987	−	1.50987i
$22$	−48.2392			−0.0996677
$23$		−	983.993i		−	1.86010i	−0.367432	−	0.930050i	$-0.619763\pi$
$23$		−	983.993i		−	1.86010i	0.367432	−	0.930050i	$-0.380237\pi$
$24$	−265.382	−	265.382i	−0.460733	−	0.460733i
$25$			257.354i			0.411766i
$26$		0			0
$27$	−1876.05			−2.57346
$28$	321.156	−	321.156i	0.409638	−	0.409638i
$29$	−560.708			−0.666715			−0.333358	−	0.942800i	$-0.608182\pi$
$29$	−560.708			−0.666715			−0.333358	+	0.942800i	$0.608182\pi$
$30$		−	899.522i		−	0.999468i
$31$	139.400	−	139.400i	0.145058	−	0.145058i	−0.630849	−	0.775906i	$-0.717293\pi$
$31$	139.400	−	139.400i	0.145058	−	0.145058i	0.775906	+	0.630849i	$0.217293\pi$
$32$	128.000	−	128.000i	0.125000	−	0.125000i
$33$	−200.028	−	200.028i	−0.183681	−	0.183681i
$34$	−447.405	−	447.405i	−0.387029	−	0.387029i
$35$	1088.57			0.888628
$36$		−	1552.86i		−	1.19820i
$37$	700.116	+	700.116i	0.511407	+	0.511407i	0.914957	−	0.403550i	$-0.132224\pi$
$37$	700.116	+	700.116i	0.511407	+	0.511407i	−0.403550	+	0.914957i	$0.632224\pi$
$38$			1070.25i			0.741172i
$39$		0			0
$40$	433.860			0.271163
$41$	−1872.79	+	1872.79i	−1.11409	+	1.11409i	−0.121501	+	0.992591i	$0.538771\pi$
$41$	−1872.79	+	1872.79i	−1.11409	+	1.11409i	−0.992591	+	0.121501i	$0.961229\pi$
$42$	2663.41			1.50987
$43$		−	430.626i		−	0.232897i	−0.993197	−	0.116448i	$-0.962849\pi$
$43$		−	430.626i		−	0.232897i	0.993197	−	0.116448i	$-0.0371510\pi$
$44$	96.4783	−	96.4783i	0.0498339	−	0.0498339i
$45$	2631.74	−	2631.74i	1.29963	−	1.29963i
$46$	1967.99	+	1967.99i	0.930050	+	0.930050i
$47$	59.3604	+	59.3604i	0.0268721	+	0.0268721i	0.720415	−	0.693543i	$-0.243951\pi$
$47$	59.3604	+	59.3604i	0.0268721	+	0.0268721i	−0.693543	+	0.720415i	$0.743951\pi$
$48$	1061.53			0.460733
$49$			822.164i			0.342426i
$50$	−514.708	−	514.708i	−0.205883	−	0.205883i
$51$		−	3710.42i		−	1.42653i
$52$		0			0
$53$	3773.97			1.34353			0.671763	−	0.740766i	$-0.265537\pi$
$53$	3773.97			1.34353			0.671763	+	0.740766i	$0.265537\pi$
$54$	3752.10	−	3752.10i	1.28673	−	1.28673i
$55$	327.017			0.108105
$56$			1284.62i			0.409638i
$57$	−4437.90	+	4437.90i	−1.36593	+	1.36593i
$58$	1121.42	−	1121.42i	0.333358	−	0.333358i
$59$	−924.257	−	924.257i	−0.265515	−	0.265515i	0.561775	−	0.827290i	$-0.310119\pi$
$59$	−924.257	−	924.257i	−0.265515	−	0.265515i	−0.827290	+	0.561775i	$0.810119\pi$
$60$	1799.04	+	1799.04i	0.499734	+	0.499734i
$61$	−3935.30			−1.05759			−0.528796	−	0.848749i	$-0.677356\pi$
$61$	−3935.30			−1.05759			−0.528796	+	0.848749i	$0.677356\pi$
$62$			557.601i			0.145058i
$63$	7792.37	+	7792.37i	1.96331	+	1.96331i
$64$			512.000i			0.125000i
$65$		0			0
$66$	800.113			0.183681
$67$	2992.99	−	2992.99i	0.666738	−	0.666738i	−0.290222	−	0.956959i	$-0.593729\pi$
$67$	2992.99	−	2992.99i	0.666738	−	0.666738i	0.956959	+	0.290222i	$0.0937290\pi$
$68$	1789.62			0.387029
$69$			16320.9i			3.42804i
$70$	−2177.14	+	2177.14i	−0.444314	+	0.444314i
$71$	9.46035	−	9.46035i	0.00187668	−	0.00187668i	−0.706168	−	0.708044i	$-0.749578\pi$
$71$	9.46035	−	9.46035i	0.00187668	−	0.00187668i	0.708044	+	0.706168i	$0.249578\pi$
$72$	3105.73	+	3105.73i	0.599099	+	0.599099i
$73$	4922.36	+	4922.36i	0.923693	+	0.923693i	0.997288	−	0.0735956i	$-0.0234474\pi$
$73$	4922.36	+	4922.36i	0.923693	+	0.923693i	−0.0735956	+	0.997288i	$0.523447\pi$
$74$	−2800.47			−0.511407
$75$		−	4268.57i		−	0.758856i
$76$	−2140.51	−	2140.51i	−0.370586	−	0.370586i
$77$			968.269i			0.163311i
$78$		0			0
$79$	1624.91			0.260361			0.130180	−	0.991490i	$-0.458444\pi$
$79$	1624.91			0.260361			0.130180	+	0.991490i	$0.458444\pi$
$80$	−867.721	+	867.721i	−0.135581	+	0.135581i
$81$	15394.2			2.34631
$82$		−	7491.16i		−	1.11409i
$83$	4406.88	−	4406.88i	0.639698	−	0.639698i	−0.310783	−	0.950481i	$-0.600591\pi$
$83$	4406.88	−	4406.88i	0.639698	−	0.639698i	0.950481	+	0.310783i	$0.100591\pi$
$84$	−5326.82	+	5326.82i	−0.754934	+	0.754934i
$85$	3032.99	+	3032.99i	0.419791	+	0.419791i
$86$	861.252	+	861.252i	0.116448	+	0.116448i
$87$	9300.11			1.22871
$88$			385.913i			0.0498339i
$89$	3715.92	+	3715.92i	0.469122	+	0.469122i	0.901630	−	0.432508i	$-0.142371\pi$
$89$	3715.92	+	3715.92i	0.469122	+	0.469122i	−0.432508	+	0.901630i	$0.642371\pi$
$90$			10527.0i			1.29963i
$91$		0			0
$92$	−7871.95			−0.930050
$93$	−2312.15	+	2312.15i	−0.267331	+	0.267331i
$94$	−237.441			−0.0268721
$95$		−	7255.32i		−	0.803913i
$96$	−2123.06	+	2123.06i	−0.230366	+	0.230366i
$97$	4894.57	−	4894.57i	0.520201	−	0.520201i	−0.397431	−	0.917632i	$-0.630098\pi$
$97$	4894.57	−	4894.57i	0.520201	−	0.520201i	0.917632	+	0.397431i	$0.130098\pi$
$98$	−1644.33	−	1644.33i	−0.171213	−	0.171213i
$99$	2340.90	+	2340.90i	0.238843	+	0.238843i
$100$	2058.83			0.205883
$101$		−	1445.28i		−	0.141681i	−0.997488	−	0.0708403i	$-0.977432\pi$
$101$		−	1445.28i		−	0.141681i	0.997488	−	0.0708403i	$-0.0225681\pi$
$102$	7420.83	+	7420.83i	0.713267	+	0.713267i
$103$			4448.97i			0.419358i	0.977770	+	0.209679i	$0.0672420\pi$
$103$			4448.97i			0.419358i	−0.977770	+	0.209679i	$0.932758\pi$
$104$		0			0
$105$	−18055.4			−1.63768
$106$	−7547.93	+	7547.93i	−0.671763	+	0.671763i
$107$	8104.41			0.707870			0.353935	−	0.935270i	$-0.384843\pi$
$107$	8104.41			0.707870			0.353935	+	0.935270i	$0.384843\pi$
$108$			15008.4i			1.28673i
$109$	2675.08	−	2675.08i	0.225156	−	0.225156i	−0.585509	−	0.810666i	$-0.699105\pi$
$109$	2675.08	−	2675.08i	0.225156	−	0.225156i	0.810666	+	0.585509i	$0.199105\pi$
$110$	−654.033	+	654.033i	−0.0540523	+	0.0540523i
$111$	−11612.4	−	11612.4i	−0.942488	−	0.942488i
$112$	−2569.25	−	2569.25i	−0.204819	−	0.204819i
$113$	−3008.25			−0.235590			−0.117795	−	0.993038i	$-0.537583\pi$
$113$	−3008.25			−0.235590			−0.117795	+	0.993038i	$0.537583\pi$
$114$		−	17751.6i		−	1.36593i
$115$	−13341.1	−	13341.1i	−1.00878	−	1.00878i
$116$			4485.66i			0.333358i
$117$		0			0
$118$	3697.03			0.265515
$119$	−8980.43	+	8980.43i	−0.634167	+	0.634167i
$120$	−7196.17			−0.499734
$121$		−	14350.1i		−	0.980133i
$122$	7870.60	−	7870.60i	0.528796	−	0.528796i
$123$	31062.8	−	31062.8i	2.05319	−	2.05319i
$124$	−1115.20	−	1115.20i	−0.0725288	−	0.0725288i
$125$	11963.1	+	11963.1i	0.765637	+	0.765637i
$126$	−31169.5			−1.96331
$127$		−	3624.98i		−	0.224749i	−0.993666	−	0.112374i	$-0.964154\pi$
$127$		−	3624.98i		−	0.224749i	0.993666	−	0.112374i	$-0.0358456\pi$
$128$	−1024.00	−	1024.00i	−0.0625000	−	0.0625000i
$129$			7142.53i			0.429213i
$130$		0			0
$131$	4428.23			0.258040			0.129020	−	0.991642i	$-0.458817\pi$
$131$	4428.23			0.258040			0.129020	+	0.991642i	$0.458817\pi$
$132$	−1600.23	+	1600.23i	−0.0918404	+	0.0918404i
$133$	21482.4			1.21445
$134$			11971.9i			0.666738i
$135$	−25435.8	+	25435.8i	−1.39565	+	1.39565i
$136$	−3579.24	+	3579.24i	−0.193514	+	0.193514i
$137$	−977.708	−	977.708i	−0.0520916	−	0.0520916i	0.680581	−	0.732673i	$-0.261727\pi$
$137$	−977.708	−	977.708i	−0.0520916	−	0.0520916i	−0.732673	+	0.680581i	$0.761727\pi$
$138$	−32641.8	−	32641.8i	−1.71402	−	1.71402i
$139$	36138.6			1.87043			0.935216	−	0.354079i	$-0.115206\pi$
$139$	36138.6			1.87043			0.935216	+	0.354079i	$0.115206\pi$
$140$		−	8708.56i		−	0.444314i
$141$	−984.573	−	984.573i	−0.0495233	−	0.0495233i
$142$			37.8414i			0.00187668i
$143$		0			0
$144$	−12422.9			−0.599099
$145$	−7602.15	+	7602.15i	−0.361577	+	0.361577i
$146$	−19689.4			−0.923693
$147$		−	13636.7i		−	0.631067i
$148$	5600.93	−	5600.93i	0.255704	−	0.255704i
$149$	−12263.5	+	12263.5i	−0.552383	+	0.552383i	−0.927128	−	0.374745i	$-0.877730\pi$
$149$	−12263.5	+	12263.5i	−0.552383	+	0.552383i	0.374745	+	0.927128i	$0.377730\pi$
$150$	8537.14	+	8537.14i	0.379428	+	0.379428i
$151$	−15818.3	−	15818.3i	−0.693756	−	0.693756i	0.269300	−	0.963056i	$-0.413208\pi$
$151$	−15818.3	−	15818.3i	−0.693756	−	0.693756i	−0.963056	+	0.269300i	$0.913208\pi$
$152$	8562.02			0.370586
$153$			43422.5i			1.85495i
$154$	−1936.54	−	1936.54i	−0.0816553	−	0.0816553i
$155$		−	3780.02i		−	0.157337i
$156$		0			0
$157$	17629.6			0.715225			0.357612	−	0.933870i	$-0.383591\pi$
$157$	17629.6			0.715225			0.357612	+	0.933870i	$0.383591\pi$
$158$	−3249.82	+	3249.82i	−0.130180	+	0.130180i
$159$	−62596.4			−2.47603
$160$		−	3470.88i		−	0.135581i
$161$	39501.9	−	39501.9i	1.52394	−	1.52394i
$162$	−30788.3	+	30788.3i	−1.17316	+	1.17316i
$163$	34964.7	+	34964.7i	1.31600	+	1.31600i	0.916916	+	0.399080i	$0.130670\pi$
$163$	34964.7	+	34964.7i	1.31600	+	1.31600i	0.399080	+	0.916916i	$0.369330\pi$
$164$	14982.3	+	14982.3i	0.557046	+	0.557046i
$165$	−5424.02			−0.199229
$166$			17627.5i			0.639698i
$167$	3546.29	+	3546.29i	0.127157	+	0.127157i	0.767821	−	0.640664i	$-0.221341\pi$
$167$	3546.29	+	3546.29i	0.127157	+	0.127157i	−0.640664	+	0.767821i	$0.721341\pi$
$168$		−	21307.3i		−	0.754934i
$169$		0			0
$170$	−12132.0			−0.419791
$171$	51936.1	−	51936.1i	1.77614	−	1.77614i
$172$	−3445.01			−0.116448
$173$		−	38615.8i		−	1.29025i	−0.764078	−	0.645124i	$-0.776806\pi$
$173$		−	38615.8i		−	1.29025i	0.764078	−	0.645124i	$-0.223194\pi$
$174$	−18600.2	+	18600.2i	−0.614355	+	0.614355i
$175$	−10331.3	+	10331.3i	−0.337350	+	0.337350i
$176$	−771.827	−	771.827i	−0.0249169	−	0.0249169i
$177$	15330.1	+	15330.1i	0.489326	+	0.489326i
$178$	−14863.7			−0.469122
$179$			4373.46i			0.136496i	0.997668	+	0.0682479i	$0.0217409\pi$
$179$			4373.46i			0.136496i	−0.997668	+	0.0682479i	$0.978259\pi$
$180$	−21053.9	−	21053.9i	−0.649813	−	0.649813i
$181$		−	25076.9i		−	0.765450i	−0.923862	−	0.382725i	$-0.874986\pi$
$181$		−	25076.9i		−	0.765450i	0.923862	−	0.382725i	$-0.125014\pi$
$182$		0			0
$183$	65272.4			1.94907
$184$	15743.9	−	15743.9i	0.465025	−	0.465025i
$185$	18984.5			0.554698
$186$		−	9248.58i		−	0.267331i
$187$	−2697.81	+	2697.81i	−0.0771485	+	0.0771485i
$188$	474.883	−	474.883i	0.0134360	−	0.0134360i
$189$	−75313.2	−	75313.2i	−2.10837	−	2.10837i
$190$	14510.6	+	14510.6i	0.401957	+	0.401957i
$191$	7846.01			0.215071			0.107536	−	0.994201i	$-0.465704\pi$
$191$	7846.01			0.215071			0.107536	+	0.994201i	$0.465704\pi$
$192$		−	8492.23i		−	0.230366i
$193$	26725.4	+	26725.4i	0.717481	+	0.717481i	0.968089	−	0.250608i	$-0.0806304\pi$
$193$	26725.4	+	26725.4i	0.717481	+	0.717481i	−0.250608	+	0.968089i	$0.580630\pi$
$194$			19578.3i			0.520201i
$195$		0			0
$196$	6577.32			0.171213
$197$	6777.04	−	6777.04i	0.174626	−	0.174626i	−0.614383	−	0.789008i	$-0.710595\pi$
$197$	6777.04	−	6777.04i	0.174626	−	0.174626i	0.789008	+	0.614383i	$0.210595\pi$
$198$	−9363.61			−0.238843
$199$			61087.0i			1.54256i	0.636495	+	0.771281i	$0.280384\pi$
$199$			61087.0i			1.54256i	−0.636495	+	0.771281i	$0.719616\pi$
$200$	−4117.66	+	4117.66i	−0.102942	+	0.102942i
$201$	−49642.8	+	49642.8i	−1.22875	+	1.22875i
$202$	2890.57	+	2890.57i	0.0708403	+	0.0708403i
$203$	−22509.3	−	22509.3i	−0.546224	−	0.546224i
$204$	−29683.3			−0.713267
$205$			50783.1i			1.20840i
$206$	−8897.95	−	8897.95i	−0.209679	−	0.209679i
$207$		−	191001.i		−	4.45754i
$208$		0			0
$209$	6453.51			0.147742
$210$	36110.9	−	36110.9i	0.818840	−	0.818840i
$211$	−76793.1			−1.72487			−0.862436	−	0.506165i	$-0.831063\pi$
$211$	−76793.1			−1.72487			−0.862436	+	0.506165i	$0.831063\pi$
$212$		−	30191.7i		−	0.671763i
$213$	−156.913	+	156.913i	−0.00345859	+	0.00345859i
$214$	−16208.8	+	16208.8i	−0.353935	+	0.353935i
$215$	−5838.49	−	5838.49i	−0.126306	−	0.126306i
$216$	−30016.8	−	30016.8i	−0.643365	−	0.643365i
$217$	11192.3			0.237684
$218$			10700.3i			0.225156i
$219$	−81644.1	−	81644.1i	−1.70230	−	1.70230i
$220$		−	2616.13i		−	0.0540523i
$221$		0			0
$222$	46449.6			0.942488
$223$	−52857.4	+	52857.4i	−1.06291	+	1.06291i	−0.0650260	+	0.997884i	$0.520713\pi$
$223$	−52857.4	+	52857.4i	−1.06291	+	1.06291i	−0.997884	+	0.0650260i	$0.979287\pi$
$224$	10277.0			0.204819
$225$			49954.4i			0.986754i
$226$	6016.51	−	6016.51i	0.117795	−	0.117795i
$227$	32344.8	−	32344.8i	0.627700	−	0.627700i	−0.319788	−	0.947489i	$-0.603612\pi$
$227$	32344.8	−	32344.8i	0.627700	−	0.627700i	0.947489	+	0.319788i	$0.103612\pi$
$228$	35503.2	+	35503.2i	0.682965	+	0.682965i
$229$	49809.4	+	49809.4i	0.949818	+	0.949818i	0.998800	−	0.0489817i	$-0.0155976\pi$
$229$	49809.4	+	49809.4i	0.949818	+	0.949818i	−0.0489817	+	0.998800i	$0.515598\pi$
$230$	53364.5			1.00878
$231$		−	16060.1i		−	0.300970i
$232$	−8971.32	−	8971.32i	−0.166679	−	0.166679i
$233$			72869.4i			1.34225i	0.741345	+	0.671124i	$0.234188\pi$
$233$			72869.4i			1.34225i	−0.741345	+	0.671124i	$0.765812\pi$
$234$		0			0
$235$	1609.63			0.0291468
$236$	−7394.06	+	7394.06i	−0.132757	+	0.132757i
$237$	−26951.4			−0.479827
$238$		−	35921.7i		−	0.634167i
$239$	14223.7	−	14223.7i	0.249010	−	0.249010i	−0.571554	−	0.820564i	$-0.693659\pi$
$239$	14223.7	−	14223.7i	0.249010	−	0.249010i	0.820564	+	0.571554i	$0.193659\pi$
$240$	14392.3	−	14392.3i	0.249867	−	0.249867i
$241$	42338.8	+	42338.8i	0.728962	+	0.728962i	0.970413	−	0.241451i	$-0.0776234\pi$
$241$	42338.8	+	42338.8i	0.728962	+	0.728962i	−0.241451	+	0.970413i	$0.577623\pi$
$242$	28700.2	+	28700.2i	0.490066	+	0.490066i
$243$	−103373.			−1.75063
$244$			31482.4i			0.528796i
$245$	11147.0	+	11147.0i	0.185706	+	0.185706i
$246$			124251.i			2.05319i
$247$		0			0
$248$	4460.81			0.0725288
$249$	−73094.1	+	73094.1i	−1.17892	+	1.17892i
$250$	−47852.3			−0.765637
$251$		−	114161.i		−	1.81206i	−0.423216	−	0.906029i	$-0.639099\pi$
$251$		−	114161.i		−	1.81206i	0.423216	−	0.906029i	$-0.360901\pi$
$252$	62339.0	−	62339.0i	0.981654	−	0.981654i
$253$	11866.8	−	11866.8i	0.185392	−	0.185392i
$254$	7249.95	+	7249.95i	0.112374	+	0.112374i
$255$	−50306.3	−	50306.3i	−0.773646	−	0.773646i
$256$	4096.00			0.0625000
$257$			28776.5i			0.435685i	0.975984	+	0.217842i	$0.0699018\pi$
$257$			28776.5i			0.435685i	−0.975984	+	0.217842i	$0.930098\pi$
$258$	−14285.1	−	14285.1i	−0.214606	−	0.214606i
$259$			56211.7i			0.837967i
$260$		0			0
$261$	−108838.			−1.59771
$262$	−8856.46	+	8856.46i	−0.129020	+	0.129020i
$263$	−41981.1			−0.606936			−0.303468	−	0.952842i	$-0.598144\pi$
$263$	−41981.1			−0.606936			−0.303468	+	0.952842i	$0.598144\pi$
$264$		−	6400.91i		−	0.0918404i
$265$	51168.0	−	51168.0i	0.728629	−	0.728629i
$266$	−42964.8	+	42964.8i	−0.607224	+	0.607224i
$267$	−61633.6	−	61633.6i	−0.864560	−	0.864560i
$268$	−23943.9	−	23943.9i	−0.333369	−	0.333369i
$269$	87860.2			1.21419			0.607096	−	0.794628i	$-0.292334\pi$
$269$	87860.2			1.21419			0.607096	+	0.794628i	$0.292334\pi$
$270$		−	101743.i		−	1.39565i
$271$	59053.1	+	59053.1i	0.804088	+	0.804088i	0.983732	−	0.179643i	$-0.0574944\pi$
$271$	59053.1	+	59053.1i	0.804088	+	0.804088i	−0.179643	+	0.983732i	$0.557494\pi$
$272$		−	14317.0i		−	0.193514i
$273$		0			0
$274$	3910.83			0.0520916
$275$	−3103.63	+	3103.63i	−0.0410398	+	0.0410398i
$276$	130567.			1.71402
$277$			59016.9i			0.769160i	0.923092	+	0.384580i	$0.125654\pi$
$277$			59016.9i			0.769160i	−0.923092	+	0.384580i	$0.874346\pi$
$278$	−72277.2	+	72277.2i	−0.935216	+	0.935216i
$279$	27058.7	−	27058.7i	0.347615	−	0.347615i
$280$	17417.1	+	17417.1i	0.222157	+	0.222157i
$281$	−73210.7	−	73210.7i	−0.927175	−	0.927175i	0.0703472	−	0.997523i	$-0.477589\pi$
$281$	−73210.7	−	73210.7i	−0.927175	−	0.927175i	−0.997523	+	0.0703472i	$0.977589\pi$
$282$	3938.29			0.0495233
$283$		−	40778.4i		−	0.509164i	−0.967051	−	0.254582i	$-0.918062\pi$
$283$		−	40778.4i		−	0.509164i	0.967051	−	0.254582i	$-0.0819379\pi$
$284$	−75.6828	−	75.6828i	−0.000938340	−	0.000938340i
$285$			120339.i			1.48156i
$286$		0			0
$287$	−150364.			−1.82550
$288$	24845.8	−	24845.8i	0.299549	−	0.299549i
$289$	33478.1			0.400835
$290$		−	30408.6i		−	0.361577i
$291$	−81183.2	+	81183.2i	−0.958694	+	0.958694i
$292$	39378.9	−	39378.9i	0.461846	−	0.461846i
$293$	−41000.3	−	41000.3i	−0.477586	−	0.477586i	0.426773	−	0.904359i	$-0.359650\pi$
$293$	−41000.3	−	41000.3i	−0.477586	−	0.477586i	−0.904359	+	0.426773i	$0.859650\pi$
$294$	27273.5	+	27273.5i	0.315534	+	0.315534i
$295$	−25062.4			−0.287991
$296$			22403.7i			0.255704i
$297$	−22624.8	−	22624.8i	−0.256491	−	0.256491i
$298$		−	49053.9i		−	0.552383i
$299$		0			0
$300$	−34148.5			−0.379428
$301$	17287.3	−	17287.3i	0.190807	−	0.190807i
$302$	63273.3			0.693756
$303$			23972.0i			0.261107i
$304$	−17124.0	+	17124.0i	−0.185293	+	0.185293i
$305$	−53355.4	+	53355.4i	−0.573559	+	0.573559i
$306$	−86844.9	−	86844.9i	−0.927474	−	0.927474i
$307$	−3442.87	−	3442.87i	−0.0365295	−	0.0365295i	0.688606	−	0.725136i	$-0.258223\pi$
$307$	−3442.87	−	3442.87i	−0.0365295	−	0.0365295i	−0.725136	+	0.688606i	$0.758223\pi$
$308$	7746.15			0.0816553
$309$		−	73792.4i		−	0.772849i
$310$	7560.03	+	7560.03i	0.0786684	+	0.0786684i
$311$			4606.97i			0.0476316i	0.999716	+	0.0238158i	$0.00758151\pi$
$311$			4606.97i			0.0476316i	−0.999716	+	0.0238158i	$0.992418\pi$
$312$		0			0
$313$	120038.			1.22527			0.612633	−	0.790368i	$-0.290111\pi$
$313$	120038.			1.22527			0.612633	+	0.790368i	$0.290111\pi$
$314$	−35259.1	+	35259.1i	−0.357612	+	0.357612i
$315$	211300.			2.12950
$316$		−	12999.3i		−	0.130180i
$317$	87032.2	−	87032.2i	0.866087	−	0.866087i	−0.125950	−	0.992037i	$-0.540198\pi$
$317$	87032.2	−	87032.2i	0.866087	−	0.866087i	0.992037	+	0.125950i	$0.0401979\pi$
$318$	125193.	−	125193.i	1.23801	−	1.23801i
$319$	−6762.02	−	6762.02i	−0.0664500	−	0.0664500i
$320$	6941.77	+	6941.77i	0.0677907	+	0.0677907i
$321$	−134423.			−1.30456
$322$			158008.i			1.52394i
$323$	59854.6	+	59854.6i	0.573710	+	0.573710i
$324$		−	123153.i		−	1.17316i
$325$		0			0
$326$	−139859.			−1.31600
$327$	−44369.9	+	44369.9i	−0.414947	+	0.414947i
$328$	−59929.2			−0.557046
$329$			4765.99i			0.0440312i
$330$	10848.0	−	10848.0i	0.0996147	−	0.0996147i
$331$	−61566.7	+	61566.7i	−0.561940	+	0.561940i	−0.929858	−	0.367918i	$-0.880071\pi$
$331$	−61566.7	+	61566.7i	−0.561940	+	0.561940i	0.367918	+	0.929858i	$0.380071\pi$
$332$	−35255.0	−	35255.0i	−0.319849	−	0.319849i
$333$	135898.	+	135898.i	1.22553	+	1.22553i
$334$	−14185.1			−0.127157
$335$		−	81158.6i		−	0.723178i
$336$	42614.5	+	42614.5i	0.377467	+	0.377467i
$337$			8069.67i			0.0710552i	0.999369	+	0.0355276i	$0.0113112\pi$
$337$			8069.67i			0.0710552i	−0.999369	+	0.0355276i	$0.988689\pi$
$338$		0			0
$339$	49896.0			0.434177
$340$	24263.9	−	24263.9i	0.209896	−	0.209896i
$341$	3362.28			0.0289151
$342$			207745.i			1.77614i
$343$	63381.6	−	63381.6i	0.538735	−	0.538735i
$344$	6890.02	−	6890.02i	0.0582242	−	0.0582242i
$345$	221281.	+	221281.i	1.85911	+	1.85911i
$346$	77231.7	+	77231.7i	0.645124	+	0.645124i
$347$	156363.			1.29860			0.649300	−	0.760533i	$-0.275062\pi$
$347$	156363.			1.29860			0.649300	+	0.760533i	$0.275062\pi$
$348$		−	74400.9i		−	0.614355i
$349$	141731.	+	141731.i	1.16363	+	1.16363i	0.983676	+	0.179951i	$0.0575938\pi$
$349$	141731.	+	141731.i	1.16363	+	1.16363i	0.179951	+	0.983676i	$0.442406\pi$
$350$		−	41325.4i		−	0.337350i
$351$		0			0
$352$	3087.31			0.0249169
$353$	125248.	−	125248.i	1.00513	−	1.00513i	0.00514429	−	0.999987i	$-0.498363\pi$
$353$	125248.	−	125248.i	1.00513	−	1.00513i	0.999987	−	0.00514429i	$-0.00163748\pi$
$354$	−61320.3			−0.489326
$355$		−	256.529i		−	0.00203554i
$356$	29727.3	−	29727.3i	0.234561	−	0.234561i
$357$	148953.	−	148953.i	1.16873	−	1.16873i
$358$	−8746.93	−	8746.93i	−0.0682479	−	0.0682479i
$359$	50595.2	+	50595.2i	0.392573	+	0.392573i	0.875604	−	0.483030i	$-0.160464\pi$
$359$	50595.2	+	50595.2i	0.392573	+	0.392573i	−0.483030	+	0.875604i	$0.660464\pi$
$360$	84215.8			0.649813
$361$		−	12859.1i		−	0.0986722i
$362$	50153.8	+	50153.8i	0.382725	+	0.382725i
$363$			238017.i			1.80632i
$364$		0			0
$365$	133476.			1.00188
$366$	−130545.	+	130545.i	−0.974535	+	0.974535i
$367$	137759.			1.02279			0.511395	−	0.859346i	$-0.329129\pi$
$367$	137759.			1.02279			0.511395	+	0.859346i	$0.329129\pi$
$368$			62975.6i			0.465025i
$369$	−363523.	+	363523.i	−2.66980	+	2.66980i
$370$	−37969.1	+	37969.1i	−0.277349	+	0.277349i
$371$	151504.	+	151504.i	1.10072	+	1.10072i
$372$	18497.2	+	18497.2i	0.133666	+	0.133666i
$373$	122430.			0.879975			0.439988	−	0.898004i	$-0.354983\pi$
$373$	122430.			0.879975			0.439988	+	0.898004i	$0.354983\pi$
$374$		−	10791.2i		−	0.0771485i
$375$	−198424.	−	198424.i	−1.41102	−	1.41102i
$376$			1899.53i			0.0134360i
$377$		0			0
$378$	301253.			2.10837
$379$	113156.	−	113156.i	0.787769	−	0.787769i	−0.193359	−	0.981128i	$-0.561938\pi$
$379$	113156.	−	113156.i	0.787769	−	0.787769i	0.981128	+	0.193359i	$0.0619381\pi$
$380$	−58042.5			−0.401957
$381$			60125.2i			0.414197i
$382$	−15692.0	+	15692.0i	−0.107536	+	0.107536i
$383$	−68319.5	+	68319.5i	−0.465744	+	0.465744i	−0.900532	−	0.434789i	$-0.856823\pi$
$383$	−68319.5	+	68319.5i	−0.465744	+	0.465744i	0.434789	+	0.900532i	$0.356823\pi$
$384$	16984.5	+	16984.5i	0.115183	+	0.115183i
$385$	13127.9	+	13127.9i	0.0885676	+	0.0885676i
$386$	−106902.			−0.717481
$387$		−	83588.0i		−	0.558113i
$388$	−39156.5	−	39156.5i	−0.260100	−	0.260100i
$389$		−	85946.1i		−	0.567972i	−0.958828	−	0.283986i	$-0.908343\pi$
$389$		−	85946.1i		−	0.567972i	0.958828	−	0.283986i	$-0.0916569\pi$
$390$		0			0
$391$	220122.			1.43983
$392$	−13154.6	+	13154.6i	−0.0856065	+	0.0856065i
$393$	−73448.3			−0.475551
$394$			27108.2i			0.174626i
$395$	22030.8	−	22030.8i	0.141200	−	0.141200i
$396$	18727.2	−	18727.2i	0.119422	−	0.119422i
$397$	49712.1	+	49712.1i	0.315414	+	0.315414i	0.847003	−	0.531589i	$-0.178405\pi$
$397$	49712.1	+	49712.1i	0.315414	+	0.315414i	−0.531589	+	0.847003i	$0.678405\pi$
$398$	−122174.	−	122174.i	−0.771281	−	0.771281i
$399$	−356315.			−2.23815
$400$		−	16470.6i		−	0.102942i
$401$	−55547.8	−	55547.8i	−0.345445	−	0.345445i	0.512965	−	0.858410i	$-0.328547\pi$
$401$	−55547.8	−	55547.8i	−0.345445	−	0.345445i	−0.858410	+	0.512965i	$0.828547\pi$
$402$		−	198571.i		−	1.22875i
$403$		0			0
$404$	−11562.3			−0.0708403
$405$	208716.	−	208716.i	1.27247	−	1.27247i
$406$	90037.3			0.546224
$407$			16886.5i			0.101942i
$408$	59366.7	−	59366.7i	0.356634	−	0.356634i
$409$	−77395.2	+	77395.2i	−0.462666	+	0.462666i	−0.899528	−	0.436862i	$-0.856090\pi$
$409$	−77395.2	+	77395.2i	−0.462666	+	0.462666i	0.436862	+	0.899528i	$0.356090\pi$
$410$	−101566.	−	101566.i	−0.604201	−	0.604201i
$411$	16216.6	+	16216.6i	0.0960013	+	0.0960013i
$412$	35591.8			0.209679
$413$		−	74207.7i		−	0.435060i
$414$	382002.	+	382002.i	2.22877	+	2.22877i
$415$		−	119498.i		−	0.693849i
$416$		0			0
$417$	−599409.			−3.44708
$418$	−12907.0	+	12907.0i	−0.0738709	+	0.0738709i
$419$	144154.			0.821106			0.410553	−	0.911837i	$-0.365336\pi$
$419$	144154.			0.821106			0.410553	+	0.911837i	$0.365336\pi$
$420$			144443.i			0.818840i
$421$	−190864.	+	190864.i	−1.07686	+	1.07686i	−0.0800732	+	0.996789i	$0.525515\pi$
$421$	−190864.	+	190864.i	−1.07686	+	1.07686i	−0.996789	+	0.0800732i	$0.974485\pi$
$422$	153586.	−	153586.i	0.862436	−	0.862436i
$423$	11522.3	+	11522.3i	0.0643960	+	0.0643960i
$424$	60383.5	+	60383.5i	0.335882	+	0.335882i
$425$	−57570.7			−0.318731
$426$		−	627.651i		−	0.00345859i
$427$	−157981.	−	157981.i	−0.866460	−	0.866460i
$428$		−	64835.2i		−	0.353935i
$429$		0			0
$430$	23354.0			0.126306
$431$	−27601.8	+	27601.8i	−0.148587	+	0.148587i	−0.777487	−	0.628899i	$-0.783506\pi$
$431$	−27601.8	+	27601.8i	−0.148587	+	0.148587i	0.628899	+	0.777487i	$0.283506\pi$
$432$	120067.			0.643365
$433$			89204.9i			0.475787i	0.971291	+	0.237894i	$0.0764570\pi$
$433$			89204.9i			0.475787i	−0.971291	+	0.237894i	$0.923543\pi$
$434$	−22384.6	+	22384.6i	−0.118842	+	0.118842i
$435$	126092.	−	126092.i	0.666361	−	0.666361i
$436$	−21400.6	−	21400.6i	−0.112578	−	0.112578i
$437$	−263280.	−	263280.i	−1.37865	−	1.37865i
$438$	326576.			1.70230
$439$			7453.00i			0.0386725i	0.999813	+	0.0193362i	$0.00615530\pi$
$439$			7453.00i			0.0386725i	−0.999813	+	0.0193362i	$0.993845\pi$
$440$	5232.27	+	5232.27i	0.0270262	+	0.0270262i
$441$			159589.i			0.820587i
$442$		0			0
$443$	−206983.			−1.05470			−0.527349	−	0.849649i	$-0.676814\pi$
$443$	−206983.			−1.05470			−0.527349	+	0.849649i	$0.676814\pi$
$444$	−92899.2	+	92899.2i	−0.471244	+	0.471244i
$445$	100762.			0.508834
$446$		−	211430.i		−	1.06291i
$447$	203406.	−	203406.i	1.01800	−	1.01800i
$448$	−20554.0	+	20554.0i	−0.102409	+	0.102409i
$449$	−201683.	−	201683.i	−1.00040	−	1.00040i	−1.00000		0.000405082i	$-0.999871\pi$
$449$	−201683.	−	201683.i	−1.00040	−	1.00040i	−0.000405082	−	1.00000i	$-0.500129\pi$
$450$	−99908.8	−	99908.8i	−0.493377	−	0.493377i
$451$	−45170.9			−0.222078
$452$			24066.0i			0.117795i
$453$	262369.	+	262369.i	1.27854	+	1.27854i
$454$			129379.i			0.627700i
$455$		0			0
$456$	−142013.			−0.682965
$457$	254320.	−	254320.i	1.21772	−	1.21772i	0.249295	−	0.968428i	$-0.419801\pi$
$457$	254320.	−	254320.i	1.21772	−	1.21772i	0.968428	−	0.249295i	$-0.0801987\pi$
$458$	−199238.			−0.949818
$459$		−	419678.i		−	1.99201i
$460$	−106729.	+	106729.i	−0.504390	+	0.504390i
$461$	3957.17	−	3957.17i	0.0186201	−	0.0186201i	−0.697735	−	0.716356i	$-0.745809\pi$
$461$	3957.17	−	3957.17i	0.0186201	−	0.0186201i	0.716356	+	0.697735i	$0.245809\pi$
$462$	32120.2	+	32120.2i	0.150485	+	0.150485i
$463$	154055.	+	154055.i	0.718642	+	0.718642i	0.968327	−	0.249685i	$-0.0803270\pi$
$463$	154055.	+	154055.i	0.718642	+	0.718642i	−0.249685	+	0.968327i	$0.580327\pi$
$464$	35885.3			0.166679
$465$			62696.8i			0.289961i
$466$	−145739.	−	145739.i	−0.671124	−	0.671124i
$467$			56541.9i			0.259261i	0.991562	+	0.129630i	$0.0413790\pi$
$467$			56541.9i			0.259261i	−0.991562	+	0.129630i	$0.958621\pi$
$468$		0			0
$469$	240304.			1.09248
$470$	−3219.26	+	3219.26i	−0.0145734	+	0.0145734i
$471$	−292411.			−1.31811
$472$		−	29576.2i		−	0.132757i
$473$	5193.26	−	5193.26i	0.0232123	−	0.0232123i
$474$	53902.8	−	53902.8i	0.239913	−	0.239913i
$475$	68858.4	+	68858.4i	0.305190	+	0.305190i
$476$	71843.5	+	71843.5i	0.317083	+	0.317083i
$477$	732557.			3.21962
$478$			56894.9i			0.249010i
$479$	47534.5	+	47534.5i	0.207175	+	0.207175i	0.803066	−	0.595891i	$-0.203201\pi$
$479$	47534.5	+	47534.5i	0.207175	+	0.207175i	−0.595891	+	0.803066i	$0.703201\pi$
$480$			57569.4i			0.249867i
$481$		0			0
$482$	−169355.			−0.728962
$483$	−655194.	+	655194.i	−2.80851	+	2.80851i
$484$	−114801.			−0.490066
$485$		−	132722.i		−	0.564236i
$486$	206746.	−	206746.i	0.875317	−	0.875317i
$487$	109924.	−	109924.i	0.463485	−	0.463485i	−0.436311	−	0.899796i	$-0.643715\pi$
$487$	109924.	−	109924.i	0.463485	−	0.463485i	0.899796	+	0.436311i	$0.143715\pi$
$488$	−62964.8	−	62964.8i	−0.264398	−	0.264398i
$489$	−579938.	−	579938.i	−2.42529	−	2.42529i
$490$	−44588.1			−0.185706
$491$		−	425487.i		−	1.76491i	−0.470396	−	0.882456i	$-0.655889\pi$
$491$		−	425487.i		−	1.76491i	0.470396	−	0.882456i	$-0.344111\pi$
$492$	−248502.	−	248502.i	−1.02660	−	1.02660i
$493$		−	125432.i		−	0.516076i
$494$		0			0
$495$	63476.5			0.259061
$496$	−8921.62	+	8921.62i	−0.0362644	+	0.0362644i
$497$	759.562			0.00307504
$498$		−	292377.i		−	1.17892i
$499$	136551.	−	136551.i	0.548397	−	0.548397i	−0.377580	−	0.925977i	$-0.623244\pi$
$499$	136551.	−	136551.i	0.548397	−	0.548397i	0.925977	+	0.377580i	$0.123244\pi$
$500$	95704.6	−	95704.6i	0.382818	−	0.382818i
$501$	−58820.0	−	58820.0i	−0.234342	−	0.234342i
$502$	228323.	+	228323.i	0.906029	+	0.906029i
$503$	−298633.			−1.18033			−0.590164	−	0.807284i	$-0.700937\pi$
$503$	−298633.			−1.18033			−0.590164	+	0.807284i	$0.700937\pi$
$504$			249356.i			0.981654i
$505$	−19595.3	−	19595.3i	−0.0768370	−	0.0768370i
$506$			47467.0i			0.185392i
$507$		0			0
$508$	−28999.8			−0.112374
$509$	−242694.	+	242694.i	−0.936750	+	0.936750i	−0.998115	−	0.0613650i	$-0.980455\pi$
$509$	−242694.	+	242694.i	−0.936750	+	0.936750i	0.0613650	+	0.998115i	$0.480455\pi$
$510$	201225.			0.773646
$511$			395211.i			1.51352i
$512$	−8192.00	+	8192.00i	−0.0312500	+	0.0312500i
$513$	−501962.	+	501962.i	−1.90738	+	1.90738i
$514$	−57553.1	−	57553.1i	−0.217842	−	0.217842i
$515$	60319.8	+	60319.8i	0.227429	+	0.227429i
$516$	57140.2			0.214606
$517$			1431.75i			0.00535655i
$518$	−112423.	−	112423.i	−0.418983	−	0.418983i
$519$			640497.i			2.37784i
$520$		0			0
$521$	−308799.			−1.13763			−0.568815	−	0.822465i	$-0.692598\pi$
$521$	−308799.			−1.13763			−0.568815	+	0.822465i	$0.692598\pi$
$522$	217676.	−	217676.i	0.798857	−	0.798857i
$523$	−196111.			−0.716966			−0.358483	−	0.933536i	$-0.616706\pi$
$523$	−196111.			−0.716966			−0.358483	+	0.933536i	$0.616706\pi$
$524$		−	35425.8i		−	0.129020i
$525$	171360.	−	171360.i	0.621713	−	0.621713i
$526$	83962.3	−	83962.3i	0.303468	−	0.303468i
$527$	31184.2	+	31184.2i	0.112283	+	0.112283i
$528$	12801.8	+	12801.8i	0.0459202	+	0.0459202i
$529$	−688402.			−2.45998
$530$			204672.i			0.728629i
$531$	−179406.	−	179406.i	−0.636278	−	0.636278i
$532$		−	171859.i		−	0.607224i
$533$		0			0
$534$	246535.			0.864560
$535$	109881.	−	109881.i	0.383896	−	0.383896i
$536$	95775.6			0.333369
$537$		−	72539.9i		−	0.251552i
$538$	−175720.	+	175720.i	−0.607096	+	0.607096i
$539$	−9915.13	+	9915.13i	−0.0341288	+	0.0341288i
$540$	203486.	+	203486.i	0.697826	+	0.697826i
$541$	47432.2	+	47432.2i	0.162061	+	0.162061i	0.783479	−	0.621418i	$-0.213443\pi$
$541$	47432.2	+	47432.2i	0.162061	+	0.162061i	−0.621418	+	0.783479i	$0.713443\pi$
$542$	−236212.			−0.804088
$543$			415935.i			1.41067i
$544$	28633.9	+	28633.9i	0.0967572	+	0.0967572i
$545$		−	72538.2i		−	0.244216i
$546$		0			0
$547$	478901.			1.60056			0.800278	−	0.599629i	$-0.204685\pi$
$547$	478901.			1.60056			0.800278	+	0.599629i	$0.204685\pi$
$548$	−7821.66	+	7821.66i	−0.0260458	+	0.0260458i
$549$	−763873.			−2.53441
$550$		−	12414.5i		−	0.0410398i
$551$	−150025.	+	150025.i	−0.494151	+	0.494151i
$552$	−261134.	+	261134.i	−0.857009	+	0.857009i
$553$	65231.2	+	65231.2i	0.213307	+	0.213307i
$554$	−118034.	−	118034.i	−0.384580	−	0.384580i
$555$	−314885.			−1.02227
$556$		−	289109.i		−	0.935216i
$557$	−303848.	−	303848.i	−0.979369	−	0.979369i	0.0204227	−	0.999791i	$-0.493499\pi$
$557$	−303848.	−	303848.i	−0.979369	−	0.979369i	−0.999791	+	0.0204227i	$0.993499\pi$
$558$			108235.i			0.347615i
$559$		0			0
$560$	−69668.5			−0.222157
$561$	44746.9	−	44746.9i	0.142179	−	0.142179i
$562$	292843.			0.927175
$563$			563388.i			1.77742i	0.458466	+	0.888712i	$0.348399\pi$
$563$			563388.i			1.77742i	−0.458466	+	0.888712i	$0.651601\pi$
$564$	−7876.59	+	7876.59i	−0.0247617	+	0.0247617i
$565$	−40786.3	+	40786.3i	−0.127767	+	0.127767i
$566$	81556.9	+	81556.9i	0.254582	+	0.254582i
$567$	617991.	+	617991.i	1.92228	+	1.92228i
$568$	302.731			0.000938340
$569$			73625.6i			0.227407i	0.993515	+	0.113704i	$0.0362715\pi$
$569$			73625.6i			0.227407i	−0.993515	+	0.113704i	$0.963729\pi$
$570$	−240679.	−	240679.i	−0.740778	−	0.740778i
$571$		−	551854.i		−	1.69259i	−0.532715	−	0.846295i	$-0.678828\pi$
$571$		−	551854.i		−	1.69259i	0.532715	−	0.846295i	$-0.321172\pi$
$572$		0			0
$573$	−130137.			−0.396361
$574$	300729.	−	300729.i	0.912749	−	0.912749i
$575$	253234.			0.765926
$576$			99383.3i			0.299549i
$577$	171008.	−	171008.i	0.513647	−	0.513647i	−0.401995	−	0.915642i	$-0.631683\pi$
$577$	171008.	−	171008.i	0.513647	−	0.513647i	0.915642	+	0.401995i	$0.131683\pi$
$578$	−66956.2	+	66956.2i	−0.200417	+	0.200417i
$579$	−443278.	−	443278.i	−1.32227	−	1.32227i
$580$	60817.2	+	60817.2i	0.180788	+	0.180788i
$581$	353824.			1.04818
$582$		−	324733.i		−	0.958694i
$583$	45513.3	+	45513.3i	0.133906	+	0.133906i
$584$			157515.i			0.461846i
$585$		0			0
$586$	164001.			0.477586
$587$	−184791.	+	184791.i	−0.536295	+	0.536295i	−0.922439	−	0.386143i	$-0.873807\pi$
$587$	−184791.	+	184791.i	−0.536295	+	0.536295i	0.386143	+	0.922439i	$0.373807\pi$
$588$	−109094.			−0.315534
$589$		−	74596.8i		−	0.215025i
$590$	50124.8	−	50124.8i	0.143995	−	0.143995i
$591$	−112407.	+	112407.i	−0.321823	+	0.321823i
$592$	−44807.4	−	44807.4i	−0.127852	−	0.127852i
$593$	−56990.3	−	56990.3i	−0.162066	−	0.162066i	0.621415	−	0.783481i	$-0.286558\pi$
$593$	−56990.3	−	56990.3i	−0.162066	−	0.162066i	−0.783481	+	0.621415i	$0.786558\pi$
$594$	90499.2			0.256491
$595$			243516.i			0.687850i
$596$	98107.7	+	98107.7i	0.276192	+	0.276192i
$597$		−	1.01321e6i		−	2.84284i
$598$		0			0
$599$	516301.			1.43896			0.719482	−	0.694511i	$-0.244379\pi$
$599$	516301.			1.43896			0.719482	+	0.694511i	$0.244379\pi$
$600$	68297.1	−	68297.1i	0.189714	−	0.189714i
$601$	−295648.			−0.818513			−0.409256	−	0.912419i	$-0.634212\pi$
$601$	−295648.			−0.818513			−0.409256	+	0.912419i	$0.634212\pi$
$602$			69149.1i			0.190807i
$603$	580962.	−	580962.i	1.59777	−	1.59777i
$604$	−126547.	+	126547.i	−0.346878	+	0.346878i
$605$	−194561.	−	194561.i	−0.531551	−	0.531551i
$606$	−47944.0	−	47944.0i	−0.130554	−	0.130554i
$607$	−246229.			−0.668285			−0.334142	−	0.942523i	$-0.608447\pi$
$607$	−246229.			−0.668285			−0.334142	+	0.942523i	$0.608447\pi$
$608$		−	68496.2i		−	0.185293i
$609$	373348.	+	373348.i	1.00665	+	1.00665i
$610$		−	213421.i		−	0.573559i
$611$		0			0
$612$	347380.			0.927474
$613$	291289.	−	291289.i	0.775182	−	0.775182i	−0.203825	−	0.979007i	$-0.565337\pi$
$613$	291289.	−	291289.i	0.775182	−	0.775182i	0.979007	+	0.203825i	$0.0653374\pi$
$614$	13771.5			0.0365295
$615$		−	842307.i		−	2.22700i
$616$	−15492.3	+	15492.3i	−0.0408277	+	0.0408277i
$617$	−229446.	+	229446.i	−0.602712	+	0.602712i	−0.941031	−	0.338320i	$-0.890142\pi$
$617$	−229446.	+	229446.i	−0.602712	+	0.602712i	0.338320	+	0.941031i	$0.390142\pi$
$618$	147585.	+	147585.i	0.386424	+	0.386424i
$619$	180833.	+	180833.i	0.471951	+	0.471951i	0.902546	−	0.430594i	$-0.141696\pi$
$619$	180833.	+	180833.i	0.471951	+	0.471951i	−0.430594	+	0.902546i	$0.641696\pi$
$620$	−30240.1			−0.0786684
$621$			1.84602e6i			4.78689i
$622$	−9213.94	−	9213.94i	−0.0238158	−	0.0238158i
$623$			298348.i			0.768681i
$624$		0			0
$625$	163548.			0.418683
$626$	−240076.	+	240076.i	−0.612633	+	0.612633i
$627$	−107040.			−0.272278
$628$		−	141037.i		−	0.357612i
$629$	−156618.	+	156618.i	−0.395859	+	0.395859i
$630$	−422600.	+	422600.i	−1.06475	+	1.06475i
$631$	−272824.	−	272824.i	−0.685211	−	0.685211i	0.275958	−	0.961170i	$-0.411005\pi$
$631$	−272824.	−	272824.i	−0.685211	−	0.685211i	−0.961170	+	0.275958i	$0.911005\pi$
$632$	25998.6	+	25998.6i	0.0650901	+	0.0650901i
$633$	1.27372e6			3.17882
$634$			348129.i			0.866087i
$635$	−49147.9	−	49147.9i	−0.121887	−	0.121887i
$636$			500772.i			1.23801i
$637$		0			0
$638$	27048.1			0.0664500
$639$	1836.33	−	1836.33i	0.00449727	−	0.00449727i
$640$	−27767.1			−0.0677907
$641$		−	165860.i		−	0.403670i	−0.979420	−	0.201835i	$-0.935310\pi$
$641$		−	165860.i		−	0.403670i	0.979420	−	0.201835i	$-0.0646905\pi$
$642$	268845.	−	268845.i	0.652278	−	0.652278i
$643$	−493796.	+	493796.i	−1.19433	+	1.19433i	−0.218494	+	0.975838i	$0.570114\pi$
$643$	−493796.	+	493796.i	−1.19433	+	1.19433i	−0.975838	+	0.218494i	$0.929886\pi$
$644$	−316016.	−	316016.i	−0.761968	−	0.761968i
$645$	96839.4	+	96839.4i	0.232773	+	0.232773i
$646$	−239418.			−0.573710
$647$		−	435005.i		−	1.03917i	−0.854420	−	0.519584i	$-0.826087\pi$
$647$		−	435005.i		−	1.03917i	0.854420	−	0.519584i	$-0.173913\pi$
$648$	246306.	+	246306.i	0.586578	+	0.586578i
$649$		−	22292.7i		−	0.0529265i
$650$		0			0
$651$	−185640.			−0.438036
$652$	279718.	−	279718.i	0.657998	−	0.657998i
$653$	384248.			0.901125			0.450562	−	0.892745i	$-0.351224\pi$
$653$	384248.			0.901125			0.450562	+	0.892745i	$0.351224\pi$
$654$		−	177480.i		−	0.414947i
$655$	60038.6	−	60038.6i	0.139942	−	0.139942i
$656$	119858.	−	119858.i	0.278523	−	0.278523i
$657$	955469.	+	955469.i	2.21353	+	2.21353i
$658$	−9531.97	−	9531.97i	−0.0220156	−	0.0220156i
$659$	−549266.			−1.26477			−0.632385	−	0.774654i	$-0.717924\pi$
$659$	−549266.			−1.26477			−0.632385	+	0.774654i	$0.717924\pi$
$660$			43392.2i			0.0996147i
$661$	492501.	+	492501.i	1.12721	+	1.12721i	0.990629	+	0.136580i	$0.0436112\pi$
$661$	492501.	+	492501.i	1.12721	+	1.12721i	0.136580	+	0.990629i	$0.456389\pi$
$662$		−	246267.i		−	0.561940i
$663$		0			0
$664$	141020.			0.319849
$665$	291261.	−	291261.i	0.658627	−	0.658627i
$666$	−543593.			−1.22553
$667$			551733.i			1.24016i
$668$	28370.3	−	28370.3i	0.0635786	−	0.0635786i
$669$	876713.	−	876713.i	1.95887	−	1.95887i
$670$	162317.	+	162317.i	0.361589	+	0.361589i
$671$	−47458.9	−	47458.9i	−0.105408	−	0.105408i
$672$	−170458.			−0.377467
$673$		−	606330.i		−	1.33869i	−0.742953	−	0.669343i	$-0.766576\pi$
$673$		−	606330.i		−	1.33869i	0.742953	−	0.669343i	$-0.233424\pi$
$674$	−16139.3	−	16139.3i	−0.0355276	−	0.0355276i
$675$		−	482809.i		−	1.05966i
$676$		0			0
$677$	−364281.			−0.794802			−0.397401	−	0.917645i	$-0.630088\pi$
$677$	−364281.			−0.794802			−0.397401	+	0.917645i	$0.630088\pi$
$678$	−99792.1	+	99792.1i	−0.217088	+	0.217088i
$679$	392980.			0.852376
$680$			97055.7i			0.209896i
$681$	−536483.	+	536483.i	−1.15681	+	1.15681i
$682$	−6724.55	+	6724.55i	−0.0144576	+	0.0144576i
$683$	−332608.	−	332608.i	−0.713003	−	0.713003i	0.254160	−	0.967162i	$-0.418201\pi$
$683$	−332608.	−	332608.i	−0.713003	−	0.713003i	−0.967162	+	0.254160i	$0.918201\pi$
$684$	−415489.	−	415489.i	−0.888070	−	0.888070i
$685$	−26511.8			−0.0565013
$686$			253526.i			0.538735i
$687$	−826158.	−	826158.i	−1.75045	−	1.75045i
$688$			27560.1i			0.0582242i
$689$		0			0
$690$	−885123.			−1.85911
$691$	−28649.1	+	28649.1i	−0.0600006	+	0.0600006i	−0.736470	−	0.676470i	$-0.763509\pi$
$691$	−28649.1	+	28649.1i	−0.0600006	+	0.0600006i	0.676470	+	0.736470i	$0.263509\pi$
$692$	−308927.			−0.645124
$693$			187949.i			0.391357i
$694$	−312726.	+	312726.i	−0.649300	+	0.649300i
$695$	489972.	−	489972.i	1.01438	−	1.01438i
$696$	148802.	+	148802.i	0.307178	+	0.307178i
$697$	−418948.	−	418948.i	−0.862372	−	0.862372i
$698$	−566923.			−1.16363
$699$		−	1.20864e6i		−	2.47367i
$700$	82650.8	+	82650.8i	0.168675	+	0.168675i
$701$			164808.i			0.335383i	0.985840	+	0.167691i	$0.0536312\pi$
$701$			164808.i			0.335383i	−0.985840	+	0.167691i	$0.946369\pi$
$702$		0			0
$703$	374651.			0.758081
$704$	−6174.61	+	6174.61i	−0.0124585	+	0.0124585i
$705$	−26698.0			−0.0537155
$706$			500994.i			1.00513i
$707$	58020.2	−	58020.2i	0.116075	−	0.116075i
$708$	122641.	−	122641.i	0.244663	−	0.244663i
$709$	−157777.	−	157777.i	−0.313870	−	0.313870i	0.532537	−	0.846407i	$-0.321239\pi$
$709$	−157777.	−	157777.i	−0.313870	−	0.313870i	−0.846407	+	0.532537i	$0.821239\pi$
$710$	513.059	+	513.059i	0.00101777	+	0.00101777i
$711$	315408.			0.623927
$712$			118909.i			0.234561i
$713$	−137169.	−	137169.i	−0.269822	−	0.269822i
$714$			595812.i			1.16873i
$715$		0			0
$716$	34987.7			0.0682479
$717$	−235920.	+	235920.i	−0.458909	+	0.458909i
$718$	−202381.			−0.392573
$719$		−	701282.i		−	1.35655i	−0.734809	−	0.678274i	$-0.762728\pi$
$719$		−	701282.i		−	1.35655i	0.734809	−	0.678274i	$-0.237272\pi$
$720$	−168432.	+	168432.i	−0.324906	+	0.324906i
$721$	−178602.	+	178602.i	−0.343570	+	0.343570i
$722$	25718.1	+	25718.1i	0.0493361	+	0.0493361i
$723$	−702248.	−	702248.i	−1.34343	−	1.34343i
$724$	−200615.			−0.382725
$725$		−	144300.i		−	0.274531i
$726$	−476033.	−	476033.i	−0.903158	−	0.903158i
$727$		−	665420.i		−	1.25900i	−0.776999	−	0.629502i	$-0.783259\pi$
$727$		−	665420.i		−	1.25900i	0.776999	−	0.629502i	$-0.216741\pi$
$728$		0			0
$729$	467659.			0.879983
$730$	−266952.	+	266952.i	−0.500942	+	0.500942i
$731$	96332.2			0.180276
$732$		−	522179.i		−	0.974535i
$733$	−322452.	+	322452.i	−0.600146	+	0.600146i	−0.940351	−	0.340205i	$-0.889503\pi$
$733$	−322452.	+	322452.i	−0.600146	+	0.600146i	0.340205	+	0.940351i	$0.389503\pi$
$734$	−275517.	+	275517.i	−0.511395	+	0.511395i
$735$	−184889.	−	184889.i	−0.342244	−	0.342244i
$736$	−125951.	−	125951.i	−0.232513	−	0.232513i
$737$	72189.6			0.132904
$738$		−	1.45409e6i		−	2.66980i
$739$	−50393.5	−	50393.5i	−0.0922753	−	0.0922753i	0.659462	−	0.751738i	$-0.270784\pi$
$739$	−50393.5	−	50393.5i	−0.0922753	−	0.0922753i	−0.751738	+	0.659462i	$0.770784\pi$
$740$		−	151876.i		−	0.277349i
$741$		0			0
$742$	−606016.			−1.10072
$743$	585418.	−	585418.i	1.06045	−	1.06045i	0.0623941	−	0.998052i	$-0.480126\pi$
$743$	585418.	−	585418.i	1.06045	−	1.06045i	0.998052	−	0.0623941i	$-0.0198736\pi$
$744$	−73988.7			−0.133666
$745$			332540.i			0.599143i
$746$	−244860.	+	244860.i	−0.439988	+	0.439988i
$747$	855410.	−	855410.i	1.53297	−	1.53297i
$748$	21582.5	+	21582.5i	0.0385743	+	0.0385743i
$749$	325347.	+	325347.i	0.579941	+	0.579941i
$750$	793696.			1.41102
$751$			854251.i			1.51463i	0.653051	+	0.757314i	$0.273489\pi$
$751$			854251.i			1.51463i	−0.653051	+	0.757314i	$0.726511\pi$
$752$	−3799.06	−	3799.06i	−0.00671801	−	0.00671801i
$753$			1.89353e6i			3.33950i
$754$		0			0
$755$	−428934.			−0.752483
$756$	−602505.	+	602505.i	−1.05419	+	1.05419i
$757$	−1110.21			−0.00193737			−0.000968684	−	1.00000i	$-0.500308\pi$
$757$	−1110.21			−0.00193737			−0.000968684		1.00000i	$0.500308\pi$
$758$			452624.i			0.787769i
$759$	−196827.	+	196827.i	−0.341665	+	0.341665i
$760$	116085.	−	116085.i	0.200978	−	0.200978i
$761$	676371.	+	676371.i	1.16793	+	1.16793i	0.982695	+	0.185232i	$0.0593037\pi$
$761$	676371.	+	676371.i	1.16793	+	1.16793i	0.185232	+	0.982695i	$0.440696\pi$
$762$	−120250.	−	120250.i	−0.207098	−	0.207098i
$763$	214780.			0.368930
$764$		−	62768.1i		−	0.107536i
$765$	588728.	+	588728.i	1.00599	+	1.00599i
$766$		−	273278.i		−	0.465744i
$767$		0			0
$768$	−67937.8			−0.115183
$769$	−255979.	+	255979.i	−0.432864	+	0.432864i	−0.889602	−	0.456737i	$-0.849018\pi$
$769$	−255979.	+	255979.i	−0.432864	+	0.432864i	0.456737	+	0.889602i	$0.349018\pi$
$770$	−52511.7			−0.0885676
$771$		−	477298.i		−	0.802937i
$772$	213804.	−	213804.i	0.358740	−	0.358740i
$773$	674317.	−	674317.i	1.12851	−	1.12851i	0.138090	−	0.990420i	$-0.455904\pi$
$773$	674317.	−	674317.i	1.12851	−	1.12851i	0.990420	−	0.138090i	$-0.0440962\pi$
$774$	167176.	+	167176.i	0.279056	+	0.279056i
$775$	35875.2	+	35875.2i	0.0597298	+	0.0597298i
$776$	156626.			0.260100
$777$		−	932348.i		−	1.54432i
$778$	171892.	+	171892.i	0.283986	+	0.283986i
$779$			1.00218e6i			1.65147i
$780$		0			0
$781$	228.180			0.000374089
$782$	−440244.	+	440244.i	−0.719913	+	0.719913i
$783$	1.05192e6			1.71576
$784$		−	52618.5i		−	0.0856065i
$785$	239024.	−	239024.i	0.387885	−	0.387885i
$786$	146897.	−	146897.i	0.237775	−	0.237775i
$787$	−90287.8	−	90287.8i	−0.145774	−	0.145774i	0.630453	−	0.776227i	$-0.282869\pi$
$787$	−90287.8	−	90287.8i	−0.145774	−	0.145774i	−0.776227	+	0.630453i	$0.782869\pi$
$788$	−54216.3	−	54216.3i	−0.0873128	−	0.0873128i
$789$	696315.			1.11854
$790$			88123.0i			0.141200i
$791$	−120765.	−	120765.i	−0.193014	−	0.193014i
$792$			74908.9i			0.119422i
$793$		0			0
$794$	−198848.			−0.315414
$795$	−848691.	+	848691.i	−1.34281	+	1.34281i
$796$	488696.			0.771281
$797$			54653.7i			0.0860406i	0.999074	+	0.0430203i	$0.0136980\pi$
$797$			54653.7i			0.0860406i	−0.999074	+	0.0430203i	$0.986302\pi$
$798$	712630.	−	712630.i	1.11907	−	1.11907i
$799$	−13279.1	+	13279.1i	−0.0208005	+	0.0208005i
$800$	32941.3	+	32941.3i	0.0514708	+	0.0514708i
$801$	721289.	+	721289.i	1.12420	+	1.12420i
$802$	222191.			0.345445
$803$			118725.i			0.184125i
$804$	397142.	+	397142.i	0.614376	+	0.614376i
$805$		−	1.07115e6i		−	1.65294i
$806$		0			0
$807$	−1.45728e6			−2.23767
$808$	23124.5	−	23124.5i	0.0354201	−	0.0354201i
$809$	−596802.			−0.911870			−0.455935	−	0.890013i	$-0.650695\pi$
$809$	−596802.			−0.911870			−0.455935	+	0.890013i	$0.650695\pi$
$810$			834864.i			1.27247i
$811$	−198281.	+	198281.i	−0.301466	+	0.301466i	−0.841587	−	0.540121i	$-0.818378\pi$
$811$	−198281.	+	198281.i	−0.301466	+	0.301466i	0.540121	+	0.841587i	$0.318378\pi$
$812$	−180075.	+	180075.i	−0.273112	+	0.273112i
$813$	−979476.	−	979476.i	−1.48188	−	1.48188i
$814$	−33773.0	−	33773.0i	−0.0509708	−	0.0509708i
$815$	948112.			1.42740
$816$			237467.i			0.356634i
$817$	−115220.	−	115220.i	−0.172617	−	0.172617i
$818$		−	309581.i		−	0.462666i
$819$		0			0
$820$	406264.			0.604201
$821$	−170383.	+	170383.i	−0.252779	+	0.252779i	−0.822109	−	0.569330i	$-0.807203\pi$
$821$	−170383.	+	170383.i	−0.252779	+	0.252779i	0.569330	+	0.822109i	$0.307203\pi$
$822$	−64866.5			−0.0960013
$823$		−	494592.i		−	0.730209i	−0.930967	−	0.365105i	$-0.881033\pi$
$823$		−	494592.i		−	0.730209i	0.930967	−	0.365105i	$-0.118967\pi$
$824$	−71183.6	+	71183.6i	−0.104840	+	0.104840i
$825$	51478.0	−	51478.0i	0.0756335	−	0.0756335i
$826$	148415.	+	148415.i	0.217530	+	0.217530i
$827$	290301.	+	290301.i	0.424461	+	0.424461i	0.886737	−	0.462275i	$-0.152967\pi$
$827$	290301.	+	290301.i	0.424461	+	0.424461i	−0.462275	+	0.886737i	$0.652967\pi$
$828$	−1.52801e6			−2.22877
$829$			731661.i			1.06464i	0.846545	+	0.532318i	$0.178679\pi$
$829$			731661.i			1.06464i	−0.846545	+	0.532318i	$0.821321\pi$
$830$	238996.	+	238996.i	0.346924	+	0.346924i
$831$		−	978876.i		−	1.41751i
$832$		0			0
$833$	−183920.			−0.265057
$834$	1.19882e6	−	1.19882e6i	1.72354	−	1.72354i
$835$	96162.1			0.137921
$836$		−	51628.1i		−	0.0738709i
$837$	−261522.	+	261522.i	−0.373300	+	0.373300i
$838$	−288308.	+	288308.i	−0.410553	+	0.410553i
$839$	580044.	+	580044.i	0.824019	+	0.824019i	0.986682	−	0.162663i	$-0.0520084\pi$
$839$	580044.	+	580044.i	0.824019	+	0.824019i	−0.162663	+	0.986682i	$0.552008\pi$
$840$	−288887.	−	288887.i	−0.409420	−	0.409420i
$841$	−392888.			−0.555491
$842$		−	763456.i		−	1.07686i
$843$	1.21430e6	+	1.21430e6i	1.70872	+	1.70872i
$844$			614345.i			0.862436i
$845$		0			0
$846$	−46089.3			−0.0643960
$847$	576079.	−	576079.i	0.802999	−	0.802999i
$848$	−241534.			−0.335882
$849$			676367.i			0.938354i
$850$	115141.	−	115141.i	0.159365	−	0.159365i
$851$	688910.	−	688910.i	0.951269	−	0.951269i
$852$	1255.30	+	1255.30i	0.00172930	+	0.00172930i
$853$	278531.	+	278531.i	0.382803	+	0.382803i	0.872111	−	0.489308i	$-0.162751\pi$
$853$	278531.	+	278531.i	0.382803	+	0.382803i	−0.489308	+	0.872111i	$0.662751\pi$
$854$	631923.			0.866460
$855$		−	1.40831e6i		−	1.92649i
$856$	129670.	+	129670.i	0.176968	+	0.176968i
$857$			866031.i			1.17916i	0.807711	+	0.589579i	$0.200706\pi$
$857$			866031.i			1.17916i	−0.807711	+	0.589579i	$0.799294\pi$
$858$		0			0
$859$	422018.			0.571932			0.285966	−	0.958240i	$-0.407686\pi$
$859$	422018.			0.571932			0.285966	+	0.958240i	$0.407686\pi$
$860$	−46707.9	+	46707.9i	−0.0631529	+	0.0631529i
$861$	2.49400e6			3.36427
$862$		−	110407.i		−	0.148587i
$863$	−996621.	+	996621.i	−1.33816	+	1.33816i	−0.440320	+	0.897841i	$0.645135\pi$
$863$	−996621.	+	996621.i	−1.33816	+	1.33816i	−0.897841	+	0.440320i	$0.854865\pi$
$864$	−240135.	+	240135.i	−0.321682	+	0.321682i
$865$	−523559.	−	523559.i	−0.699734	−	0.699734i
$866$	−178410.	−	178410.i	−0.237894	−	0.237894i
$867$	−555281.			−0.738711
$868$		−	89538.5i		−	0.118842i
$869$	19596.1	+	19596.1i	0.0259495	+	0.0259495i
$870$			504369.i			0.666361i
$871$		0			0
$872$	85602.6			0.112578
$873$	950075.	−	950075.i	1.24661	−	1.24661i
$874$	1.05312e6			1.37865
$875$			960504.i			1.25454i
$876$	−653153.	+	653153.i	−0.851151	+	0.851151i
$877$	−913030.	+	913030.i	−1.18710	+	1.18710i	−0.209229	+	0.977867i	$0.567096\pi$
$877$	−913030.	+	913030.i	−1.18710	+	1.18710i	−0.977867	+	0.209229i	$0.932904\pi$
$878$	−14906.0	−	14906.0i	−0.0193362	−	0.0193362i
$879$	680046.	+	680046.i	0.880158	+	0.880158i
$880$	−20929.1			−0.0270262
$881$			1.11724e6i			1.43945i	0.694260	+	0.719724i	$0.255732\pi$
$881$			1.11724e6i			1.43945i	−0.694260	+	0.719724i	$0.744268\pi$
$882$	−319177.	−	319177.i	−0.410294	−	0.410294i
$883$		−	839397.i		−	1.07658i	−0.842760	−	0.538289i	$-0.819071\pi$
$883$		−	839397.i		−	1.07658i	0.842760	−	0.538289i	$-0.180929\pi$
$884$		0			0
$885$	415695.			0.530747
$886$	413966.	−	413966.i	0.527349	−	0.527349i
$887$	163002.			0.207179			0.103590	−	0.994620i	$-0.466967\pi$
$887$	163002.			0.207179			0.103590	+	0.994620i	$0.466967\pi$
$888$		−	371597.i		−	0.471244i
$889$	145523.	−	145523.i	0.184131	−	0.184131i
$890$	−201524.	+	201524.i	−0.254417	+	0.254417i
$891$	185650.	+	185650.i	0.233852	+	0.233852i
$892$	422859.	+	422859.i	0.531455	+	0.531455i
$893$	31765.3			0.0398336
$894$			813626.i			1.01800i
$895$	59296.0	+	59296.0i	0.0740252	+	0.0740252i
$896$		−	82216.0i		−	0.102409i
$897$		0			0
$898$	806731.			1.00040
$899$	−78162.8	+	78162.8i	−0.0967121	+	0.0967121i
$900$	399635.			0.493377
$901$			844246.i			1.03997i
$902$	90341.8	−	90341.8i	0.111039	−	0.111039i
$903$	−286733.	+	286733.i	−0.351644	+	0.351644i
$904$	−48132.1	−	48132.1i	−0.0588976	−	0.0588976i
$905$	−339996.	−	339996.i	−0.415123	−	0.415123i
$906$	−1.04948e6			−1.27854
$907$			637788.i			0.775285i	0.921810	+	0.387643i	$0.126711\pi$
$907$			637788.i			0.775285i	−0.921810	+	0.387643i	$0.873289\pi$
$908$	−258758.	−	258758.i	−0.313850	−	0.313850i
$909$		−	280541.i		−	0.339522i
$910$		0			0
$911$	1.34739e6			1.62352			0.811759	−	0.583993i	$-0.198510\pi$
$911$	1.34739e6			1.62352			0.811759	+	0.583993i	$0.198510\pi$
$912$	284026.	−	284026.i	0.341482	−	0.341482i
$913$	106292.			0.127514
$914$			1.01728e6i			1.21772i
$915$	884972.	−	884972.i	1.05703	−	1.05703i
$916$	398475.	−	398475.i	0.474909	−	0.474909i
$917$	177769.	+	177769.i	0.211406	+	0.211406i
$918$	839355.	+	839355.i	0.996003	+	0.996003i
$919$	−548840.			−0.649853			−0.324926	−	0.945739i	$-0.605340\pi$
$919$	−548840.			−0.649853			−0.324926	+	0.945739i	$0.605340\pi$
$920$		−	426916.i		−	0.504390i
$921$	57104.7	+	57104.7i	0.0673214	+	0.0673214i
$922$			15828.7i			0.0186201i
$923$		0			0
$924$	−128481.			−0.150485
$925$	−180178.	+	180178.i	−0.210580	+	0.210580i
$926$	−616219.			−0.718642
$927$			863581.i			1.00495i
$928$	−71770.6	+	71770.6i	−0.0833394	+	0.0833394i
$929$	−879177.	+	879177.i	−1.01870	+	1.01870i	−0.0188751	+	0.999822i	$0.506008\pi$
$929$	−879177.	+	879177.i	−1.01870	+	1.01870i	−0.999822	+	0.0188751i	$0.993992\pi$
$930$	−125394.	−	125394.i	−0.144980	−	0.144980i
$931$	219981.	+	219981.i	0.253796	+	0.253796i
$932$	582955.			0.671124
$933$		−	76413.0i		−	0.0877817i
$934$	−113084.	−	113084.i	−0.129630	−	0.129630i
$935$			73154.5i			0.0836793i
$936$		0			0
$937$	−406144.			−0.462595			−0.231298	−	0.972883i	$-0.574297\pi$
$937$	−406144.			−0.462595			−0.231298	+	0.972883i	$0.574297\pi$
$938$	−480608.	+	480608.i	−0.546242	+	0.546242i
$939$	−1.99100e6			−2.25808
$940$		−	12877.1i		−	0.0145734i
$941$	903964.	−	903964.i	1.02087	−	1.02087i	0.0210962	−	0.999777i	$-0.493284\pi$
$941$	903964.	−	903964.i	1.02087	−	1.02087i	0.999777	−	0.0210962i	$-0.00671562\pi$
$942$	584822.	−	584822.i	0.659055	−	0.659055i
$943$	1.84281e6	+	1.84281e6i	2.07232	+	2.07232i
$944$	59152.5	+	59152.5i	0.0663787	+	0.0663787i
$945$	−2.04221e6			−2.28685
$946$			20773.0i			0.0232123i
$947$	105869.	+	105869.i	0.118050	+	0.118050i	0.763664	−	0.645614i	$-0.223398\pi$
$947$	105869.	+	105869.i	0.118050	+	0.118050i	−0.645614	+	0.763664i	$0.723398\pi$
$948$			215611.i			0.239913i
$949$		0			0
$950$	−275434.			−0.305190
$951$	−1.44355e6	+	1.44355e6i	−1.59614	+	1.59614i
$952$	−287374.			−0.317083
$953$		−	857742.i		−	0.944432i	−0.881483	−	0.472216i	$-0.843454\pi$
$953$		−	857742.i		−	0.944432i	0.881483	−	0.472216i	$-0.156546\pi$
$954$	−1.46511e6	+	1.46511e6i	−1.60981	+	1.60981i
$955$	106377.	−	106377.i	0.116639	−	0.116639i
$956$	−113790.	−	113790.i	−0.124505	−	0.124505i
$957$	112157.	+	112157.i	0.122463	+	0.122463i
$958$	−190138.			−0.207175
$959$		−	78499.2i		−	0.0853548i
$960$	−115139.	−	115139.i	−0.124934	−	0.124934i
$961$			884656.i			0.957917i
$962$		0			0
$963$	1.57313e6			1.69634
$964$	338711.	−	338711.i	0.364481	−	0.364481i
$965$	724695.			0.778216
$966$		−	2.62078e6i		−	2.80851i
$967$	620036.	−	620036.i	0.663077	−	0.663077i	−0.293027	−	0.956104i	$-0.594663\pi$
$967$	620036.	−	620036.i	0.663077	−	0.663077i	0.956104	+	0.293027i	$0.0946625\pi$
$968$	229602.	−	229602.i	0.245033	−	0.245033i
$969$	−992771.	−	992771.i	−1.05731	−	1.05731i
$970$	265445.	+	265445.i	0.282118	+	0.282118i
$971$	623835.			0.661654			0.330827	−	0.943691i	$-0.392672\pi$
$971$	623835.			0.661654			0.330827	+	0.943691i	$0.392672\pi$
$972$			826985.i			0.875317i
$973$	1.45077e6	+	1.45077e6i	1.53240	+	1.53240i
$974$			439697.i			0.463485i
$975$		0			0
$976$	251859.			0.264398
$977$	214254.	−	214254.i	0.224461	−	0.224461i	−0.585913	−	0.810374i	$-0.699264\pi$
$977$	214254.	−	214254.i	0.224461	−	0.224461i	0.810374	+	0.585913i	$0.199264\pi$
$978$	2.31975e6			2.42529
$979$			89626.4i			0.0935127i
$980$	89176.1	−	89176.1i	0.0928531	−	0.0928531i
$981$	519254.	−	519254.i	0.539563	−	0.539563i
$982$	850973.	+	850973.i	0.882456	+	0.882456i
$983$	−1.30014e6	−	1.30014e6i	−1.34550	−	1.34550i	−0.890479	−	0.455024i	$-0.849631\pi$
$983$	−1.30014e6	−	1.30014e6i	−1.34550	−	1.34550i	−0.455024	−	0.890479i	$-0.650369\pi$
$984$	994009.			1.02660
$985$		−	183768.i		−	0.189408i
$986$	250864.	+	250864.i	0.258038	+	0.258038i
$987$		−	79050.5i		−	0.0811466i
$988$		0			0
$989$	−423733.			−0.433211
$990$	−126953.	+	126953.i	−0.129531	+	0.129531i
$991$	−1.48766e6			−1.51481			−0.757403	−	0.652948i	$-0.773532\pi$
$991$	−1.48766e6			−1.51481			−0.757403	+	0.652948i	$0.773532\pi$
$992$		−	35686.5i		−	0.0362644i
$993$	1.02117e6	−	1.02117e6i	1.03562	−	1.03562i
$994$	−1519.12	+	1519.12i	−0.00153752	+	0.00153752i
$995$	828226.	+	828226.i	0.836571	+	0.836571i
$996$	584753.	+	584753.i	0.589459	+	0.589459i
$997$	1.29351e6			1.30130			0.650651	−	0.759377i	$-0.274496\pi$
$997$	1.29351e6			1.30130			0.650651	+	0.759377i	$0.274496\pi$
$998$			546205.i			0.548397i
$999$	−1.31345e6	−	1.31345e6i	−1.31609	−	1.31609i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.5.d.c.239.1		6
13.5	odd	4		26.5.d.b.21.1	yes	6
13.8	odd	4	inner	338.5.d.c.99.1		6
13.12	even	2		26.5.d.b.5.1	✓	6
39.5	even	4		234.5.i.a.73.2		6
39.38	odd	2		234.5.i.a.109.2		6
52.31	even	4		208.5.t.a.177.3		6
52.51	odd	2		208.5.t.a.161.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.5.d.b.5.1	✓	6	13.12	even	2
26.5.d.b.21.1	yes	6	13.5	odd	4
208.5.t.a.161.3		6	52.51	odd	2
208.5.t.a.177.3		6	52.31	even	4
234.5.i.a.73.2		6	39.5	even	4
234.5.i.a.109.2		6	39.38	odd	2
338.5.d.c.99.1		6	13.8	odd	4	inner
338.5.d.c.239.1		6	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	338.d (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 2.00000i) q^{2} -16.5864 q^{3} -8.00000i q^{4} +(13.5581 - 13.5581i) q^{5} +(33.1728 - 33.1728i) q^{6} +(40.1445 + 40.1445i) q^{7} +(16.0000 + 16.0000i) q^{8} +194.108 q^{9} +54.2326i q^{10} +(12.0598 + 12.0598i) q^{11} +132.691i q^{12} -160.578 q^{14} +(-224.880 + 224.880i) q^{15} -64.0000 q^{16} +223.703i q^{17} +(-388.216 + 388.216i) q^{18} +(267.563 - 267.563i) q^{19} +(-108.465 - 108.465i) q^{20} +(-665.852 - 665.852i) q^{21} -48.2392 q^{22} -983.993i q^{23} +(-265.382 - 265.382i) q^{24} +257.354i q^{25} -1876.05 q^{27} +(321.156 - 321.156i) q^{28} -560.708 q^{29} -899.522i q^{30} +(139.400 - 139.400i) q^{31} +(128.000 - 128.000i) q^{32} +(-200.028 - 200.028i) q^{33} +(-447.405 - 447.405i) q^{34} +1088.57 q^{35} -1552.86i q^{36} +(700.116 + 700.116i) q^{37} +1070.25i q^{38} +433.860 q^{40} +(-1872.79 + 1872.79i) q^{41} +2663.41 q^{42} -430.626i q^{43} +(96.4783 - 96.4783i) q^{44} +(2631.74 - 2631.74i) q^{45} +(1967.99 + 1967.99i) q^{46} +(59.3604 + 59.3604i) q^{47} +1061.53 q^{48} +822.164i q^{49} +(-514.708 - 514.708i) q^{50} -3710.42i q^{51} +3773.97 q^{53} +(3752.10 - 3752.10i) q^{54} +327.017 q^{55} +1284.62i q^{56} +(-4437.90 + 4437.90i) q^{57} +(1121.42 - 1121.42i) q^{58} +(-924.257 - 924.257i) q^{59} +(1799.04 + 1799.04i) q^{60} -3935.30 q^{61} +557.601i q^{62} +(7792.37 + 7792.37i) q^{63} +512.000i q^{64} +800.113 q^{66} +(2992.99 - 2992.99i) q^{67} +1789.62 q^{68} +16320.9i q^{69} +(-2177.14 + 2177.14i) q^{70} +(9.46035 - 9.46035i) q^{71} +(3105.73 + 3105.73i) q^{72} +(4922.36 + 4922.36i) q^{73} -2800.47 q^{74} -4268.57i q^{75} +(-2140.51 - 2140.51i) q^{76} +968.269i q^{77} +1624.91 q^{79} +(-867.721 + 867.721i) q^{80} +15394.2 q^{81} -7491.16i q^{82} +(4406.88 - 4406.88i) q^{83} +(-5326.82 + 5326.82i) q^{84} +(3032.99 + 3032.99i) q^{85} +(861.252 + 861.252i) q^{86} +9300.11 q^{87} +385.913i q^{88} +(3715.92 + 3715.92i) q^{89} +10527.0i q^{90} -7871.95 q^{92} +(-2312.15 + 2312.15i) q^{93} -237.441 q^{94} -7255.32i q^{95} +(-2123.06 + 2123.06i) q^{96} +(4894.57 - 4894.57i) q^{97} +(-1644.33 - 1644.33i) q^{98} +(2340.90 + 2340.90i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 30 q^{5} + 90 q^{7} + 96 q^{8} + 558 q^{9} + 66 q^{11} - 360 q^{14} + 288 q^{15} - 384 q^{16} - 1116 q^{18} + 318 q^{19} - 240 q^{20} - 756 q^{21} - 264 q^{22} - 1404 q^{27} + 720 q^{28}+ \cdots + 61074 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 30 * q^5 + 90 * q^7 + 96 * q^8 + 558 * q^9 + 66 * q^11 - 360 * q^14 + 288 * q^15 - 384 * q^16 - 1116 * q^18 + 318 * q^19 - 240 * q^20 - 756 * q^21 - 264 * q^22 - 1404 * q^27 + 720 * q^28 - 276 * q^29 - 3282 * q^31 + 768 * q^32 + 3240 * q^33 - 2280 * q^34 + 5424 * q^35 + 3006 * q^37 + 960 * q^40 + 894 * q^41 + 3024 * q^42 + 528 * q^44 + 17226 * q^45 + 4848 * q^46 - 1566 * q^47 + 3300 * q^50 - 1356 * q^53 + 2808 * q^54 + 22212 * q^55 - 16812 * q^57 + 552 * q^58 - 5178 * q^59 - 2304 * q^60 + 2172 * q^61 + 24210 * q^63 - 12960 * q^66 - 1134 * q^67 + 9120 * q^68 - 10848 * q^70 + 18498 * q^71 + 8928 * q^72 + 13278 * q^73 - 12024 * q^74 - 2544 * q^76 - 13596 * q^79 - 1920 * q^80 + 58158 * q^81 + 11490 * q^83 - 6048 * q^84 + 10512 * q^85 + 7128 * q^86 + 30744 * q^87 + 28038 * q^89 - 19392 * q^92 - 23364 * q^93 + 6264 * q^94 + 27378 * q^97 + 11340 * q^98 + 61074 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more