LMFDB - Embedded newform 750.3.f.b.193.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [750,3,Mod(193,750)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(750, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("750.193"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	750.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,-16,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$20.4360198270$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.6879707136000000000000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 $ x^16 + 21x^12 + 86x^8 + 36*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.5
Root			$0.575212 + 0.575212i$ of defining polynomial
Character	$\chi$	$=$	750.193
Dual form			750.3.f.b.307.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} -2.44949 q^{6} +(-5.70902 + 5.70902i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -20.2091 q^{11} +(2.44949 - 2.44949i) q^{12} +(-4.72244 - 4.72244i) q^{13} -11.4180i q^{14} -4.00000 q^{16} +(17.5826 - 17.5826i) q^{17} +(-3.00000 - 3.00000i) q^{18} -15.9566i q^{19} -13.9842 q^{21} +(20.2091 - 20.2091i) q^{22} +(20.1000 + 20.1000i) q^{23} +4.89898i q^{24} +9.44488 q^{26} +(-3.67423 + 3.67423i) q^{27} +(11.4180 + 11.4180i) q^{28} -18.4148i q^{29} +40.5866 q^{31} +(4.00000 - 4.00000i) q^{32} +(-24.7510 - 24.7510i) q^{33} +35.1653i q^{34} +6.00000 q^{36} +(20.8395 - 20.8395i) q^{37} +(15.9566 + 15.9566i) q^{38} -11.5676i q^{39} -59.4029 q^{41} +(13.9842 - 13.9842i) q^{42} +(7.34188 + 7.34188i) q^{43} +40.4183i q^{44} -40.2000 q^{46} +(39.2382 - 39.2382i) q^{47} +(-4.89898 - 4.89898i) q^{48} -16.1859i q^{49} +43.0685 q^{51} +(-9.44488 + 9.44488i) q^{52} +(-66.4998 - 66.4998i) q^{53} -7.34847i q^{54} -22.8361 q^{56} +(19.5427 - 19.5427i) q^{57} +(18.4148 + 18.4148i) q^{58} -87.4131i q^{59} -66.7335 q^{61} +(-40.5866 + 40.5866i) q^{62} +(-17.1271 - 17.1271i) q^{63} +8.00000i q^{64} +49.5021 q^{66} +(71.0048 - 71.0048i) q^{67} +(-35.1653 - 35.1653i) q^{68} +49.2348i q^{69} +8.10797 q^{71} +(-6.00000 + 6.00000i) q^{72} +(40.3974 + 40.3974i) q^{73} +41.6789i q^{74} -31.9132 q^{76} +(115.374 - 115.374i) q^{77} +(11.5676 + 11.5676i) q^{78} +69.4524i q^{79} -9.00000 q^{81} +(59.4029 - 59.4029i) q^{82} +(3.71032 + 3.71032i) q^{83} +27.9684i q^{84} -14.6838 q^{86} +(22.5534 - 22.5534i) q^{87} +(-40.4183 - 40.4183i) q^{88} -32.4846i q^{89} +53.9211 q^{91} +(40.2000 - 40.2000i) q^{92} +(49.7082 + 49.7082i) q^{93} +78.4763i q^{94} +9.79796 q^{96} +(-27.6584 + 27.6584i) q^{97} +(16.1859 + 16.1859i) q^{98} -60.6274i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 24 q^{7} + 32 q^{8} - 24 q^{11} - 48 q^{13} - 64 q^{16} - 16 q^{17} - 48 q^{18} - 48 q^{21} + 24 q^{22} + 104 q^{23} + 96 q^{26} - 48 q^{28} + 200 q^{31} + 64 q^{32} - 48 q^{33} + 96 q^{36}+ \cdots - 224 q^{98}+O(q^{100}) $ 16 * q - 16 * q^2 + 24 * q^7 + 32 * q^8 - 24 * q^11 - 48 * q^13 - 64 * q^16 - 16 * q^17 - 48 * q^18 - 48 * q^21 + 24 * q^22 + 104 * q^23 + 96 * q^26 - 48 * q^28 + 200 * q^31 + 64 * q^32 - 48 * q^33 + 96 * q^36 - 128 * q^37 + 56 * q^38 - 192 * q^41 + 48 * q^42 + 88 * q^43 - 208 * q^46 + 16 * q^47 - 48 * q^51 - 96 * q^52 - 160 * q^53 + 96 * q^56 + 48 * q^57 + 24 * q^58 - 24 * q^61 - 200 * q^62 + 72 * q^63 + 96 * q^66 + 264 * q^67 + 32 * q^68 + 176 * q^71 - 96 * q^72 + 32 * q^73 - 112 * q^76 + 32 * q^77 + 96 * q^78 - 144 * q^81 + 192 * q^82 - 288 * q^83 - 176 * q^86 - 168 * q^87 - 48 * q^88 + 32 * q^91 + 208 * q^92 + 144 * q^93 - 48 * q^97 - 224 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		−	2.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	−2.44949			−0.408248
$7$	−5.70902	+	5.70902i	−0.815575	+	0.815575i	−0.985463	−	0.169888i	$-0.945659\pi$
$7$	−5.70902	+	5.70902i	−0.815575	+	0.815575i	0.169888	+	0.985463i	$0.445659\pi$
$8$	2.00000	+	2.00000i	0.250000	+	0.250000i
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−20.2091			−1.83719			−0.918597	−	0.395196i	$-0.870677\pi$
$11$	−20.2091			−1.83719			−0.918597	+	0.395196i	$0.870677\pi$
$12$	2.44949	−	2.44949i	0.204124	−	0.204124i
$13$	−4.72244	−	4.72244i	−0.363265	−	0.363265i	0.501749	−	0.865013i	$-0.332690\pi$
$13$	−4.72244	−	4.72244i	−0.363265	−	0.363265i	−0.865013	+	0.501749i	$0.832690\pi$
$14$		−	11.4180i		−	0.815575i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	17.5826	−	17.5826i	1.03427	−	1.03427i	0.0348812	−	0.999391i	$-0.488895\pi$
$17$	17.5826	−	17.5826i	1.03427	−	1.03427i	0.999391	−	0.0348812i	$-0.0111053\pi$
$18$	−3.00000	−	3.00000i	−0.166667	−	0.166667i
$19$		−	15.9566i		−	0.839820i	−0.907566	−	0.419910i	$-0.862062\pi$
$19$		−	15.9566i		−	0.839820i	0.907566	−	0.419910i	$-0.137938\pi$
$20$		0			0
$21$	−13.9842			−0.665914
$22$	20.2091	−	20.2091i	0.918597	−	0.918597i
$23$	20.1000	+	20.1000i	0.873914	+	0.873914i	0.992896	−	0.118982i	$-0.0379632\pi$
$23$	20.1000	+	20.1000i	0.873914	+	0.873914i	−0.118982	+	0.992896i	$0.537963\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$	9.44488			0.363265
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$	11.4180	+	11.4180i	0.407787	+	0.407787i
$29$		−	18.4148i		−	0.634992i	−0.948260	−	0.317496i	$-0.897158\pi$
$29$		−	18.4148i		−	0.634992i	0.948260	−	0.317496i	$-0.102842\pi$
$30$		0			0
$31$	40.5866			1.30924			0.654622	−	0.755956i	$-0.272828\pi$
$31$	40.5866			1.30924			0.654622	+	0.755956i	$0.272828\pi$
$32$	4.00000	−	4.00000i	0.125000	−	0.125000i
$33$	−24.7510	−	24.7510i	−0.750031	−	0.750031i
$34$			35.1653i			1.03427i
$35$		0			0
$36$	6.00000			0.166667
$37$	20.8395	−	20.8395i	0.563229	−	0.563229i	−0.366994	−	0.930223i	$-0.619613\pi$
$37$	20.8395	−	20.8395i	0.563229	−	0.563229i	0.930223	+	0.366994i	$0.119613\pi$
$38$	15.9566	+	15.9566i	0.419910	+	0.419910i
$39$		−	11.5676i		−	0.296604i
$40$		0			0
$41$	−59.4029			−1.44885			−0.724426	−	0.689353i	$-0.757895\pi$
$41$	−59.4029			−1.44885			−0.724426	+	0.689353i	$0.757895\pi$
$42$	13.9842	−	13.9842i	0.332957	−	0.332957i
$43$	7.34188	+	7.34188i	0.170741	+	0.170741i	0.787305	−	0.616564i	$-0.211476\pi$
$43$	7.34188	+	7.34188i	0.170741	+	0.170741i	−0.616564	+	0.787305i	$0.711476\pi$
$44$			40.4183i			0.918597i
$45$		0			0
$46$	−40.2000			−0.873914
$47$	39.2382	−	39.2382i	0.834855	−	0.834855i	−0.153322	−	0.988176i	$-0.548997\pi$
$47$	39.2382	−	39.2382i	0.834855	−	0.834855i	0.988176	+	0.153322i	$0.0489971\pi$
$48$	−4.89898	−	4.89898i	−0.102062	−	0.102062i
$49$		−	16.1859i		−	0.330325i
$50$		0			0
$51$	43.0685			0.844480
$52$	−9.44488	+	9.44488i	−0.181632	+	0.181632i
$53$	−66.4998	−	66.4998i	−1.25471	−	1.25471i	−0.953583	−	0.301129i	$-0.902636\pi$
$53$	−66.4998	−	66.4998i	−1.25471	−	1.25471i	−0.301129	−	0.953583i	$-0.597364\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	−22.8361			−0.407787
$57$	19.5427	−	19.5427i	0.342855	−	0.342855i
$58$	18.4148	+	18.4148i	0.317496	+	0.317496i
$59$		−	87.4131i		−	1.48158i	−0.671737	−	0.740789i	$-0.734452\pi$
$59$		−	87.4131i		−	1.48158i	0.671737	−	0.740789i	$-0.265548\pi$
$60$		0			0
$61$	−66.7335			−1.09399			−0.546996	−	0.837135i	$-0.684229\pi$
$61$	−66.7335			−1.09399			−0.546996	+	0.837135i	$0.684229\pi$
$62$	−40.5866	+	40.5866i	−0.654622	+	0.654622i
$63$	−17.1271	−	17.1271i	−0.271858	−	0.271858i
$64$			8.00000i			0.125000i
$65$		0			0
$66$	49.5021			0.750031
$67$	71.0048	−	71.0048i	1.05977	−	1.05977i	0.0616766	−	0.998096i	$-0.480355\pi$
$67$	71.0048	−	71.0048i	1.05977	−	1.05977i	0.998096	−	0.0616766i	$-0.0196448\pi$
$68$	−35.1653	−	35.1653i	−0.517136	−	0.517136i
$69$			49.2348i			0.713548i
$70$		0			0
$71$	8.10797			0.114197			0.0570984	−	0.998369i	$-0.481815\pi$
$71$	8.10797			0.114197			0.0570984	+	0.998369i	$0.481815\pi$
$72$	−6.00000	+	6.00000i	−0.0833333	+	0.0833333i
$73$	40.3974	+	40.3974i	0.553389	+	0.553389i	0.927417	−	0.374028i	$-0.122024\pi$
$73$	40.3974	+	40.3974i	0.553389	+	0.553389i	−0.374028	+	0.927417i	$0.622024\pi$
$74$			41.6789i			0.563229i
$75$		0			0
$76$	−31.9132			−0.419910
$77$	115.374	−	115.374i	1.49837	−	1.49837i
$78$	11.5676	+	11.5676i	0.148302	+	0.148302i
$79$			69.4524i			0.879144i	0.898207	+	0.439572i	$0.144870\pi$
$79$			69.4524i			0.879144i	−0.898207	+	0.439572i	$0.855130\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$	59.4029	−	59.4029i	0.724426	−	0.724426i
$83$	3.71032	+	3.71032i	0.0447027	+	0.0447027i	0.729105	−	0.684402i	$-0.239937\pi$
$83$	3.71032	+	3.71032i	0.0447027	+	0.0447027i	−0.684402	+	0.729105i	$0.739937\pi$
$84$			27.9684i			0.332957i
$85$		0			0
$86$	−14.6838			−0.170741
$87$	22.5534	−	22.5534i	0.259234	−	0.259234i
$88$	−40.4183	−	40.4183i	−0.459298	−	0.459298i
$89$		−	32.4846i		−	0.364996i	−0.983206	−	0.182498i	$-0.941582\pi$
$89$		−	32.4846i		−	0.364996i	0.983206	−	0.182498i	$-0.0584183\pi$
$90$		0			0
$91$	53.9211			0.592539
$92$	40.2000	−	40.2000i	0.436957	−	0.436957i
$93$	49.7082	+	49.7082i	0.534497	+	0.534497i
$94$			78.4763i			0.834855i
$95$		0			0
$96$	9.79796			0.102062
$97$	−27.6584	+	27.6584i	−0.285139	+	0.285139i	−0.835154	−	0.550016i	$-0.814622\pi$
$97$	−27.6584	+	27.6584i	−0.285139	+	0.285139i	0.550016	+	0.835154i	$0.314622\pi$
$98$	16.1859	+	16.1859i	0.165162	+	0.165162i
$99$		−	60.6274i		−	0.612398i
$100$		0			0
$101$	−55.2952			−0.547478			−0.273739	−	0.961804i	$-0.588260\pi$
$101$	−55.2952			−0.547478			−0.273739	+	0.961804i	$0.588260\pi$
$102$	−43.0685	+	43.0685i	−0.422240	+	0.422240i
$103$	67.1228	+	67.1228i	0.651678	+	0.651678i	0.953397	−	0.301719i	$-0.0975605\pi$
$103$	67.1228	+	67.1228i	0.651678	+	0.651678i	−0.301719	+	0.953397i	$0.597561\pi$
$104$		−	18.8898i		−	0.181632i
$105$		0			0
$106$	133.000			1.25471
$107$	−88.1875	+	88.1875i	−0.824182	+	0.824182i	−0.986705	−	0.162522i	$-0.948037\pi$
$107$	−88.1875	+	88.1875i	−0.824182	+	0.824182i	0.162522	+	0.986705i	$0.448037\pi$
$108$	7.34847	+	7.34847i	0.0680414	+	0.0680414i
$109$		−	83.1645i		−	0.762977i	−0.924374	−	0.381488i	$-0.875412\pi$
$109$		−	83.1645i		−	0.762977i	0.924374	−	0.381488i	$-0.124588\pi$
$110$		0			0
$111$	51.0461			0.459875
$112$	22.8361	−	22.8361i	0.203894	−	0.203894i
$113$	−105.407	−	105.407i	−0.932808	−	0.932808i	0.0650728	−	0.997881i	$-0.479272\pi$
$113$	−105.407	−	105.407i	−0.932808	−	0.932808i	−0.997881	+	0.0650728i	$0.979272\pi$
$114$			39.0855i			0.342855i
$115$		0			0
$116$	−36.8295			−0.317496
$117$	14.1673	−	14.1673i	0.121088	−	0.121088i
$118$	87.4131	+	87.4131i	0.740789	+	0.740789i
$119$			200.759i			1.68705i
$120$		0			0
$121$	287.409			2.37528
$122$	66.7335	−	66.7335i	0.546996	−	0.546996i
$123$	−72.7534	−	72.7534i	−0.591491	−	0.591491i
$124$		−	81.1732i		−	0.654622i
$125$		0			0
$126$	34.2541			0.271858
$127$	−16.2842	+	16.2842i	−0.128222	+	0.128222i	−0.768305	−	0.640083i	$-0.778900\pi$
$127$	−16.2842	+	16.2842i	−0.128222	+	0.128222i	0.640083	+	0.768305i	$0.278900\pi$
$128$	−8.00000	−	8.00000i	−0.0625000	−	0.0625000i
$129$			17.9839i			0.139410i
$130$		0			0
$131$	−69.7174			−0.532194			−0.266097	−	0.963946i	$-0.585734\pi$
$131$	−69.7174			−0.532194			−0.266097	+	0.963946i	$0.585734\pi$
$132$	−49.5021	+	49.5021i	−0.375016	+	0.375016i
$133$	91.0965	+	91.0965i	0.684936	+	0.684936i
$134$			142.010i			1.05977i
$135$		0			0
$136$	70.3305			0.517136
$137$	−86.2967	+	86.2967i	−0.629903	+	0.629903i	−0.948044	−	0.318141i	$-0.896942\pi$
$137$	−86.2967	+	86.2967i	−0.629903	+	0.629903i	0.318141	+	0.948044i	$0.396942\pi$
$138$	−49.2348	−	49.2348i	−0.356774	−	0.356774i
$139$		−	238.515i		−	1.71593i	−0.513707	−	0.857966i	$-0.671728\pi$
$139$		−	238.515i		−	1.71593i	0.513707	−	0.857966i	$-0.328272\pi$
$140$		0			0
$141$	96.1135			0.681656
$142$	−8.10797	+	8.10797i	−0.0570984	+	0.0570984i
$143$	95.4364	+	95.4364i	0.667388	+	0.667388i
$144$		−	12.0000i		−	0.0833333i
$145$		0			0
$146$	−80.7948			−0.553389
$147$	19.8236	−	19.8236i	0.134854	−	0.134854i
$148$	−41.6789	−	41.6789i	−0.281615	−	0.281615i
$149$		−	287.763i		−	1.93130i	−0.259853	−	0.965648i	$-0.583674\pi$
$149$		−	287.763i		−	1.93130i	0.259853	−	0.965648i	$-0.416326\pi$
$150$		0			0
$151$	168.462			1.11564			0.557820	−	0.829962i	$-0.311638\pi$
$151$	168.462			1.11564			0.557820	+	0.829962i	$0.311638\pi$
$152$	31.9132	−	31.9132i	0.209955	−	0.209955i
$153$	52.7479	+	52.7479i	0.344758	+	0.344758i
$154$			230.749i			1.49837i
$155$		0			0
$156$	−23.1351			−0.148302
$157$	56.5489	−	56.5489i	0.360184	−	0.360184i	−0.503697	−	0.863881i	$-0.668027\pi$
$157$	56.5489	−	56.5489i	0.360184	−	0.360184i	0.863881	+	0.503697i	$0.168027\pi$
$158$	−69.4524	−	69.4524i	−0.439572	−	0.439572i
$159$		−	162.890i		−	1.02447i
$160$		0			0
$161$	−229.503			−1.42548
$162$	9.00000	−	9.00000i	0.0555556	−	0.0555556i
$163$	11.4435	+	11.4435i	0.0702055	+	0.0702055i	0.741338	−	0.671132i	$-0.234192\pi$
$163$	11.4435	+	11.4435i	0.0702055	+	0.0702055i	−0.671132	+	0.741338i	$0.734192\pi$
$164$			118.806i			0.724426i
$165$		0			0
$166$	−7.42065			−0.0447027
$167$	63.2930	−	63.2930i	0.379000	−	0.379000i	−0.491741	−	0.870741i	$-0.663639\pi$
$167$	63.2930	−	63.2930i	0.379000	−	0.379000i	0.870741	+	0.491741i	$0.163639\pi$
$168$	−27.9684	−	27.9684i	−0.166479	−	0.166479i
$169$		−	124.397i		−	0.736078i
$170$		0			0
$171$	47.8697			0.279940
$172$	14.6838	−	14.6838i	0.0853707	−	0.0853707i
$173$	−43.9739	−	43.9739i	−0.254185	−	0.254185i	0.568499	−	0.822684i	$-0.307524\pi$
$173$	−43.9739	−	43.9739i	−0.254185	−	0.254185i	−0.822684	+	0.568499i	$0.807524\pi$
$174$			45.1068i			0.259234i
$175$		0			0
$176$	80.8365			0.459298
$177$	107.059	−	107.059i	0.604852	−	0.604852i
$178$	32.4846	+	32.4846i	0.182498	+	0.182498i
$179$		−	248.801i		−	1.38995i	−0.719033	−	0.694976i	$-0.755415\pi$
$179$		−	248.801i		−	1.38995i	0.719033	−	0.694976i	$-0.244585\pi$
$180$		0			0
$181$	−21.4944			−0.118754			−0.0593769	−	0.998236i	$-0.518911\pi$
$181$	−21.4944			−0.118754			−0.0593769	+	0.998236i	$0.518911\pi$
$182$	−53.9211	+	53.9211i	−0.296270	+	0.296270i
$183$	−81.7315	−	81.7315i	−0.446620	−	0.446620i
$184$			80.4001i			0.436957i
$185$		0			0
$186$	−99.4164			−0.534497
$187$	−355.330	+	355.330i	−1.90016	+	1.90016i
$188$	−78.4763	−	78.4763i	−0.417427	−	0.417427i
$189$		−	41.9526i		−	0.221971i
$190$		0			0
$191$	−342.435			−1.79285			−0.896427	−	0.443191i	$-0.853846\pi$
$191$	−342.435			−1.79285			−0.896427	+	0.443191i	$0.853846\pi$
$192$	−9.79796	+	9.79796i	−0.0510310	+	0.0510310i
$193$	60.0384	+	60.0384i	0.311080	+	0.311080i	0.845328	−	0.534248i	$-0.179405\pi$
$193$	60.0384	+	60.0384i	0.311080	+	0.311080i	−0.534248	+	0.845328i	$0.679405\pi$
$194$		−	55.3169i		−	0.285139i
$195$		0			0
$196$	−32.3718			−0.165162
$197$	−25.5854	+	25.5854i	−0.129875	+	0.129875i	−0.769056	−	0.639181i	$-0.779273\pi$
$197$	−25.5854	+	25.5854i	−0.129875	+	0.129875i	0.639181	+	0.769056i	$0.279273\pi$
$198$	60.6274	+	60.6274i	0.306199	+	0.306199i
$199$			135.303i			0.679914i	0.940441	+	0.339957i	$0.110412\pi$
$199$			135.303i			0.679914i	−0.940441	+	0.339957i	$0.889588\pi$
$200$		0			0
$201$	173.925			0.865301
$202$	55.2952	−	55.2952i	0.273739	−	0.273739i
$203$	105.130	+	105.130i	0.517883	+	0.517883i
$204$		−	86.1370i		−	0.422240i
$205$		0			0
$206$	−134.246			−0.651678
$207$	−60.3001	+	60.3001i	−0.291305	+	0.291305i
$208$	18.8898	+	18.8898i	0.0908162	+	0.0908162i
$209$			322.469i			1.54291i
$210$		0			0
$211$	−280.929			−1.33142			−0.665709	−	0.746211i	$-0.731871\pi$
$211$	−280.929			−1.33142			−0.665709	+	0.746211i	$0.731871\pi$
$212$	−133.000	+	133.000i	−0.627356	+	0.627356i
$213$	9.93020	+	9.93020i	0.0466206	+	0.0466206i
$214$		−	176.375i		−	0.824182i
$215$		0			0
$216$	−14.6969			−0.0680414
$217$	−231.710	+	231.710i	−1.06779	+	1.06779i
$218$	83.1645	+	83.1645i	0.381488	+	0.381488i
$219$			98.9531i			0.451841i
$220$		0			0
$221$	−166.066			−0.751429
$222$	−51.0461	+	51.0461i	−0.229937	+	0.229937i
$223$	−213.070	−	213.070i	−0.955469	−	0.955469i	0.0435804	−	0.999050i	$-0.486124\pi$
$223$	−213.070	−	213.070i	−0.955469	−	0.955469i	−0.999050	+	0.0435804i	$0.986124\pi$
$224$			45.6722i			0.203894i
$225$		0			0
$226$	210.815			0.932808
$227$	−199.877	+	199.877i	−0.880513	+	0.880513i	−0.993587	−	0.113073i	$-0.963931\pi$
$227$	−199.877	+	199.877i	−0.880513	+	0.880513i	0.113073	+	0.993587i	$0.463931\pi$
$228$	−39.0855	−	39.0855i	−0.171428	−	0.171428i
$229$		−	107.466i		−	0.469285i	−0.972082	−	0.234643i	$-0.924608\pi$
$229$		−	107.466i		−	0.469285i	0.972082	−	0.234643i	$-0.0753920\pi$
$230$		0			0
$231$	282.608			1.22341
$232$	36.8295	−	36.8295i	0.158748	−	0.158748i
$233$	185.314	+	185.314i	0.795338	+	0.795338i	0.982356	−	0.187018i	$-0.0598824\pi$
$233$	185.314	+	185.314i	0.795338	+	0.795338i	−0.187018	+	0.982356i	$0.559882\pi$
$234$			28.3346i			0.121088i
$235$		0			0
$236$	−174.826			−0.740789
$237$	−85.0615	+	85.0615i	−0.358909	+	0.358909i
$238$	−200.759	−	200.759i	−0.843527	−	0.843527i
$239$		−	99.8633i		−	0.417838i	−0.977933	−	0.208919i	$-0.933006\pi$
$239$		−	99.8633i		−	0.417838i	0.977933	−	0.208919i	$-0.0669945\pi$
$240$		0			0
$241$	−355.694			−1.47591			−0.737955	−	0.674850i	$-0.764208\pi$
$241$	−355.694			−1.47591			−0.737955	+	0.674850i	$0.764208\pi$
$242$	−287.409	+	287.409i	−1.18764	+	1.18764i
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$			133.467i			0.546996i
$245$		0			0
$246$	145.507			0.591491
$247$	−75.3540	+	75.3540i	−0.305077	+	0.305077i
$248$	81.1732	+	81.1732i	0.327311	+	0.327311i
$249$			9.08840i			0.0364996i
$250$		0			0
$251$	319.526			1.27301			0.636505	−	0.771272i	$-0.280379\pi$
$251$	319.526			1.27301			0.636505	+	0.771272i	$0.280379\pi$
$252$	−34.2541	+	34.2541i	−0.135929	+	0.135929i
$253$	−406.204	−	406.204i	−1.60555	−	1.60555i
$254$		−	32.5684i		−	0.128222i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−32.4963	+	32.4963i	−0.126445	+	0.126445i	−0.767497	−	0.641052i	$-0.778498\pi$
$257$	−32.4963	+	32.4963i	−0.126445	+	0.126445i	0.641052	+	0.767497i	$0.278498\pi$
$258$	−17.9839	−	17.9839i	−0.0697048	−	0.0697048i
$259$			237.946i			0.918711i
$260$		0			0
$261$	55.2443			0.211664
$262$	69.7174	−	69.7174i	0.266097	−	0.266097i
$263$	−55.9709	−	55.9709i	−0.212817	−	0.212817i	0.592646	−	0.805463i	$-0.298083\pi$
$263$	−55.9709	−	55.9709i	−0.212817	−	0.212817i	−0.805463	+	0.592646i	$0.798083\pi$
$264$		−	99.0041i		−	0.375016i
$265$		0			0
$266$	−182.193			−0.684936
$267$	39.7854	−	39.7854i	0.149009	−	0.149009i
$268$	−142.010	−	142.010i	−0.529886	−	0.529886i
$269$			275.469i			1.02405i	0.858971	+	0.512025i	$0.171104\pi$
$269$			275.469i			1.02405i	−0.858971	+	0.512025i	$0.828896\pi$
$270$		0			0
$271$	265.170			0.978487			0.489243	−	0.872147i	$-0.337273\pi$
$271$	265.170			0.978487			0.489243	+	0.872147i	$0.337273\pi$
$272$	−70.3305	+	70.3305i	−0.258568	+	0.258568i
$273$	66.0395	+	66.0395i	0.241903	+	0.241903i
$274$		−	172.593i		−	0.629903i
$275$		0			0
$276$	98.4696			0.356774
$277$	−78.3029	+	78.3029i	−0.282682	+	0.282682i	−0.834178	−	0.551496i	$-0.814057\pi$
$277$	−78.3029	+	78.3029i	−0.282682	+	0.282682i	0.551496	+	0.834178i	$0.314057\pi$
$278$	238.515	+	238.515i	0.857966	+	0.857966i
$279$			121.760i			0.436415i
$280$		0			0
$281$	−280.509			−0.998253			−0.499127	−	0.866529i	$-0.666346\pi$
$281$	−280.509			−0.998253			−0.499127	+	0.866529i	$0.666346\pi$
$282$	−96.1135	+	96.1135i	−0.340828	+	0.340828i
$283$	268.891	+	268.891i	0.950144	+	0.950144i	0.998815	−	0.0486709i	$-0.0154985\pi$
$283$	268.891	+	268.891i	0.950144	+	0.950144i	−0.0486709	+	0.998815i	$0.515499\pi$
$284$		−	16.2159i		−	0.0570984i
$285$		0			0
$286$	−190.873			−0.667388
$287$	339.133	−	339.133i	1.18165	−	1.18165i
$288$	12.0000	+	12.0000i	0.0416667	+	0.0416667i
$289$		−	329.298i		−	1.13944i
$290$		0			0
$291$	−67.7491			−0.232815
$292$	80.7948	−	80.7948i	0.276695	−	0.276695i
$293$	79.4739	+	79.4739i	0.271242	+	0.271242i	0.829600	−	0.558358i	$-0.188568\pi$
$293$	79.4739	+	79.4739i	0.271242	+	0.271242i	−0.558358	+	0.829600i	$0.688568\pi$
$294$			39.6472i			0.134854i
$295$		0			0
$296$	83.3579			0.281615
$297$	74.2531	−	74.2531i	0.250010	−	0.250010i
$298$	287.763	+	287.763i	0.965648	+	0.965648i
$299$		−	189.842i		−	0.634924i
$300$		0			0
$301$	−83.8299			−0.278505
$302$	−168.462	+	168.462i	−0.557820	+	0.557820i
$303$	−67.7226	−	67.7226i	−0.223507	−	0.223507i
$304$			63.8263i			0.209955i
$305$		0			0
$306$	−105.496			−0.344758
$307$	−305.867	+	305.867i	−0.996310	+	0.996310i	−0.999993	−	0.00368298i	$-0.998828\pi$
$307$	−305.867	+	305.867i	−0.996310	+	0.996310i	0.00368298	+	0.999993i	$0.498828\pi$
$308$	−230.749	−	230.749i	−0.749184	−	0.749184i
$309$			164.417i			0.532092i
$310$		0			0
$311$	−111.946			−0.359956			−0.179978	−	0.983671i	$-0.557603\pi$
$311$	−111.946			−0.359956			−0.179978	+	0.983671i	$0.557603\pi$
$312$	23.1351	−	23.1351i	0.0741511	−	0.0741511i
$313$	155.048	+	155.048i	0.495361	+	0.495361i	0.909990	−	0.414629i	$-0.136089\pi$
$313$	155.048	+	155.048i	0.495361	+	0.495361i	−0.414629	+	0.909990i	$0.636089\pi$
$314$			113.098i			0.360184i
$315$		0			0
$316$	138.905			0.439572
$317$	224.330	−	224.330i	0.707665	−	0.707665i	−0.258378	−	0.966044i	$-0.583188\pi$
$317$	224.330	−	224.330i	0.707665	−	0.707665i	0.966044	+	0.258378i	$0.0831881\pi$
$318$	162.890	+	162.890i	0.512234	+	0.512234i
$319$			372.146i			1.16660i
$320$		0			0
$321$	−216.014			−0.672942
$322$	229.503	−	229.503i	0.712742	−	0.712742i
$323$	−280.559	−	280.559i	−0.868603	−	0.868603i
$324$			18.0000i			0.0555556i
$325$		0			0
$326$	−22.8870			−0.0702055
$327$	101.855	−	101.855i	0.311484	−	0.311484i
$328$	−118.806	−	118.806i	−0.362213	−	0.362213i
$329$			448.023i			1.36177i
$330$		0			0
$331$	314.778			0.950992			0.475496	−	0.879718i	$-0.342269\pi$
$331$	314.778			0.950992			0.475496	+	0.879718i	$0.342269\pi$
$332$	7.42065	−	7.42065i	0.0223513	−	0.0223513i
$333$	62.5184	+	62.5184i	0.187743	+	0.187743i
$334$			126.586i			0.379000i
$335$		0			0
$336$	55.9368			0.166479
$337$	27.8091	−	27.8091i	0.0825197	−	0.0825197i	−0.664642	−	0.747162i	$-0.731416\pi$
$337$	27.8091	−	27.8091i	0.0825197	−	0.0825197i	0.747162	+	0.664642i	$0.231416\pi$
$338$	124.397	+	124.397i	0.368039	+	0.368039i
$339$		−	258.194i		−	0.761634i
$340$		0			0
$341$	−820.220			−2.40534
$342$	−47.8697	+	47.8697i	−0.139970	+	0.139970i
$343$	−187.336	−	187.336i	−0.546170	−	0.546170i
$344$			29.3675i			0.0853707i
$345$		0			0
$346$	87.9479			0.254185
$347$	−83.0635	+	83.0635i	−0.239376	+	0.239376i	−0.816592	−	0.577216i	$-0.804139\pi$
$347$	−83.0635	+	83.0635i	−0.239376	+	0.239376i	0.577216	+	0.816592i	$0.304139\pi$
$348$	−45.1068	−	45.1068i	−0.129617	−	0.129617i
$349$		−	170.578i		−	0.488763i	−0.969679	−	0.244382i	$-0.921415\pi$
$349$		−	170.578i		−	0.488763i	0.969679	−	0.244382i	$-0.0785850\pi$
$350$		0			0
$351$	34.7027			0.0988681
$352$	−80.8365	+	80.8365i	−0.229649	+	0.229649i
$353$	6.69721	+	6.69721i	0.0189723	+	0.0189723i	0.716529	−	0.697557i	$-0.245730\pi$
$353$	6.69721	+	6.69721i	0.0189723	+	0.0189723i	−0.697557	+	0.716529i	$0.745730\pi$
$354$			214.118i			0.604852i
$355$		0			0
$356$	−64.9693			−0.182498
$357$	−245.879	+	245.879i	−0.688737	+	0.688737i
$358$	248.801	+	248.801i	0.694976	+	0.694976i
$359$		−	297.590i		−	0.828940i	−0.910063	−	0.414470i	$-0.863967\pi$
$359$		−	297.590i		−	0.828940i	0.910063	−	0.414470i	$-0.136033\pi$
$360$		0			0
$361$	106.387			0.294702
$362$	21.4944	−	21.4944i	0.0593769	−	0.0593769i
$363$	352.003	+	352.003i	0.969704	+	0.969704i
$364$		−	107.842i		−	0.296270i
$365$		0			0
$366$	163.463			0.446620
$367$	−243.376	+	243.376i	−0.663151	+	0.663151i	−0.956121	−	0.292971i	$-0.905356\pi$
$367$	−243.376	+	243.376i	−0.663151	+	0.663151i	0.292971	+	0.956121i	$0.405356\pi$
$368$	−80.4001	−	80.4001i	−0.218478	−	0.218478i
$369$		−	178.209i		−	0.482951i
$370$		0			0
$371$	759.297			2.04662
$372$	99.4164	−	99.4164i	0.267248	−	0.267248i
$373$	31.8670	+	31.8670i	0.0854342	+	0.0854342i	0.748532	−	0.663098i	$-0.230759\pi$
$373$	31.8670	+	31.8670i	0.0854342	+	0.0854342i	−0.663098	+	0.748532i	$0.730759\pi$
$374$		−	710.660i		−	1.90016i
$375$		0			0
$376$	156.953			0.417427
$377$	−86.9626	+	86.9626i	−0.230670	+	0.230670i
$378$	41.9526	+	41.9526i	0.110986	+	0.110986i
$379$			329.670i			0.869843i	0.900468	+	0.434921i	$0.143224\pi$
$379$			329.670i			0.869843i	−0.900468	+	0.434921i	$0.856776\pi$
$380$		0			0
$381$	−39.8880			−0.104693
$382$	342.435	−	342.435i	0.896427	−	0.896427i
$383$	−377.941	−	377.941i	−0.986792	−	0.986792i	0.0131216	−	0.999914i	$-0.495823\pi$
$383$	−377.941	−	377.941i	−0.986792	−	0.986792i	−0.999914	+	0.0131216i	$0.995823\pi$
$384$		−	19.5959i		−	0.0510310i
$385$		0			0
$386$	−120.077			−0.311080
$387$	−22.0256	+	22.0256i	−0.0569138	+	0.0569138i
$388$	55.3169	+	55.3169i	0.142569	+	0.142569i
$389$		−	438.570i		−	1.12743i	−0.825969	−	0.563715i	$-0.809372\pi$
$389$		−	438.570i		−	1.12743i	0.825969	−	0.563715i	$-0.190628\pi$
$390$		0			0
$391$	706.823			1.80773
$392$	32.3718	−	32.3718i	0.0825812	−	0.0825812i
$393$	−85.3860	−	85.3860i	−0.217267	−	0.217267i
$394$		−	51.1709i		−	0.129875i
$395$		0			0
$396$	−121.255			−0.306199
$397$	−331.079	+	331.079i	−0.833952	+	0.833952i	−0.988055	−	0.154103i	$-0.950751\pi$
$397$	−331.079	+	331.079i	−0.833952	+	0.833952i	0.154103	+	0.988055i	$0.450751\pi$
$398$	−135.303	−	135.303i	−0.339957	−	0.339957i
$399$			223.140i			0.559248i
$400$		0			0
$401$	−441.641			−1.10135			−0.550674	−	0.834720i	$-0.685629\pi$
$401$	−441.641			−1.10135			−0.550674	+	0.834720i	$0.685629\pi$
$402$	−173.925	+	173.925i	−0.432650	+	0.432650i
$403$	−191.668	−	191.668i	−0.475602	−	0.475602i
$404$			110.590i			0.273739i
$405$		0			0
$406$	−210.261			−0.517883
$407$	−421.148	+	421.148i	−1.03476	+	1.03476i
$408$	86.1370	+	86.1370i	0.211120	+	0.211120i
$409$			240.811i			0.588779i	0.955686	+	0.294390i	$0.0951163\pi$
$409$			240.811i			0.588779i	−0.955686	+	0.294390i	$0.904884\pi$
$410$		0			0
$411$	−211.383			−0.514314
$412$	134.246	−	134.246i	0.325839	−	0.325839i
$413$	499.044	+	499.044i	1.20834	+	1.20834i
$414$		−	120.600i		−	0.291305i
$415$		0			0
$416$	−37.7795			−0.0908162
$417$	292.119	−	292.119i	0.700526	−	0.700526i
$418$	−322.469	−	322.469i	−0.771456	−	0.771456i
$419$			641.021i			1.52988i	0.644100	+	0.764942i	$0.277232\pi$
$419$			641.021i			1.52988i	−0.644100	+	0.764942i	$0.722768\pi$
$420$		0			0
$421$	−493.748			−1.17280			−0.586399	−	0.810023i	$-0.699455\pi$
$421$	−493.748			−1.17280			−0.586399	+	0.810023i	$0.699455\pi$
$422$	280.929	−	280.929i	0.665709	−	0.665709i
$423$	117.714	+	117.714i	0.278285	+	0.278285i
$424$		−	265.999i		−	0.627356i
$425$		0			0
$426$	−19.8604			−0.0466206
$427$	380.983	−	380.983i	0.892232	−	0.892232i
$428$	176.375	+	176.375i	0.412091	+	0.412091i
$429$			233.771i			0.544920i
$430$		0			0
$431$	117.686			0.273054			0.136527	−	0.990636i	$-0.456406\pi$
$431$	117.686			0.273054			0.136527	+	0.990636i	$0.456406\pi$
$432$	14.6969	−	14.6969i	0.0340207	−	0.0340207i
$433$	−362.842	−	362.842i	−0.837973	−	0.837973i	0.150619	−	0.988592i	$-0.451873\pi$
$433$	−362.842	−	362.842i	−0.837973	−	0.837973i	−0.988592	+	0.150619i	$0.951873\pi$
$434$		−	463.420i		−	1.06779i
$435$		0			0
$436$	−166.329			−0.381488
$437$	320.728	−	320.728i	0.733931	−	0.733931i
$438$	−98.9531	−	98.9531i	−0.225920	−	0.225920i
$439$			303.617i			0.691610i	0.938306	+	0.345805i	$0.112394\pi$
$439$			303.617i			0.691610i	−0.938306	+	0.345805i	$0.887606\pi$
$440$		0			0
$441$	48.5577			0.110108
$442$	166.066	−	166.066i	0.375715	−	0.375715i
$443$	276.171	+	276.171i	0.623411	+	0.623411i	0.946402	−	0.322991i	$-0.104688\pi$
$443$	276.171	+	276.171i	0.623411	+	0.623411i	−0.322991	+	0.946402i	$0.604688\pi$
$444$		−	102.092i		−	0.229937i
$445$		0			0
$446$	426.139			0.955469
$447$	352.436	−	352.436i	0.788448	−	0.788448i
$448$	−45.6722	−	45.6722i	−0.101947	−	0.101947i
$449$		−	314.925i		−	0.701391i	−0.936490	−	0.350696i	$-0.885945\pi$
$449$		−	314.925i		−	0.701391i	0.936490	−	0.350696i	$-0.114055\pi$
$450$		0			0
$451$	1200.48			2.66182
$452$	−210.815	+	210.815i	−0.466404	+	0.466404i
$453$	206.322	+	206.322i	0.455458	+	0.455458i
$454$		−	399.753i		−	0.880513i
$455$		0			0
$456$	78.1710			0.171428
$457$	563.709	−	563.709i	1.23350	−	1.23350i	0.270887	−	0.962611i	$-0.412683\pi$
$457$	563.709	−	563.709i	1.23350	−	1.23350i	0.962611	−	0.270887i	$-0.0873169\pi$
$458$	107.466	+	107.466i	0.234643	+	0.234643i
$459$			129.205i			0.281493i
$460$		0			0
$461$	660.573			1.43291			0.716457	−	0.697632i	$-0.245763\pi$
$461$	660.573			1.43291			0.716457	+	0.697632i	$0.245763\pi$
$462$	−282.608	+	282.608i	−0.611707	+	0.611707i
$463$	−114.255	−	114.255i	−0.246772	−	0.246772i	0.572873	−	0.819644i	$-0.305829\pi$
$463$	−114.255	−	114.255i	−0.246772	−	0.246772i	−0.819644	+	0.572873i	$0.805829\pi$
$464$			73.6590i			0.158748i
$465$		0			0
$466$	−370.628			−0.795338
$467$	497.886	−	497.886i	1.06614	−	1.06614i	0.0684851	−	0.997652i	$-0.478183\pi$
$467$	497.886	−	497.886i	1.06614	−	1.06614i	0.997652	−	0.0684851i	$-0.0218166\pi$
$468$	−28.3346	−	28.3346i	−0.0605441	−	0.0605441i
$469$			810.736i			1.72865i
$470$		0			0
$471$	138.516			0.294089
$472$	174.826	−	174.826i	0.370395	−	0.370395i
$473$	−148.373	−	148.373i	−0.313685	−	0.313685i
$474$		−	170.123i		−	0.358909i
$475$		0			0
$476$	401.519			0.843527
$477$	199.499	−	199.499i	0.418237	−	0.418237i
$478$	99.8633	+	99.8633i	0.208919	+	0.208919i
$479$		−	416.013i		−	0.868503i	−0.900792	−	0.434252i	$-0.857013\pi$
$479$		−	416.013i		−	0.868503i	0.900792	−	0.434252i	$-0.142987\pi$
$480$		0			0
$481$	−196.826			−0.409202
$482$	355.694	−	355.694i	0.737955	−	0.737955i
$483$	−281.083	−	281.083i	−0.581952	−	0.581952i
$484$		−	574.818i		−	1.18764i
$485$		0			0
$486$	22.0454			0.0453609
$487$	517.473	−	517.473i	1.06257	−	1.06257i	0.0646652	−	0.997907i	$-0.479402\pi$
$487$	517.473	−	517.473i	1.06257	−	1.06257i	0.997907	−	0.0646652i	$-0.0205980\pi$
$488$	−133.467	−	133.467i	−0.273498	−	0.273498i
$489$			28.0307i			0.0573226i
$490$		0			0
$491$	483.386			0.984492			0.492246	−	0.870456i	$-0.336176\pi$
$491$	483.386			0.984492			0.492246	+	0.870456i	$0.336176\pi$
$492$	−145.507	+	145.507i	−0.295746	+	0.295746i
$493$	−323.780	−	323.780i	−0.656755	−	0.656755i
$494$		−	150.708i		−	0.305077i
$495$		0			0
$496$	−162.346			−0.327311
$497$	−46.2886	+	46.2886i	−0.0931360	+	0.0931360i
$498$	−9.08840	−	9.08840i	−0.0182498	−	0.0182498i
$499$		−	869.520i		−	1.74252i	−0.490818	−	0.871262i	$-0.663302\pi$
$499$		−	869.520i		−	1.74252i	0.490818	−	0.871262i	$-0.336698\pi$
$500$		0			0
$501$	155.036			0.309452
$502$	−319.526	+	319.526i	−0.636505	+	0.636505i
$503$	−378.561	−	378.561i	−0.752607	−	0.752607i	0.222358	−	0.974965i	$-0.428625\pi$
$503$	−378.561	−	378.561i	−0.752607	−	0.752607i	−0.974965	+	0.222358i	$0.928625\pi$
$504$		−	68.5083i		−	0.135929i
$505$		0			0
$506$	812.408			1.60555
$507$	152.355	−	152.355i	0.300502	−	0.300502i
$508$	32.5684	+	32.5684i	0.0641111	+	0.0641111i
$509$			764.251i			1.50147i	0.660601	+	0.750737i	$0.270302\pi$
$509$			764.251i			1.50147i	−0.660601	+	0.750737i	$0.729698\pi$
$510$		0			0
$511$	−461.260			−0.902661
$512$	−16.0000	+	16.0000i	−0.0312500	+	0.0312500i
$513$	58.6282	+	58.6282i	0.114285	+	0.114285i
$514$		−	64.9927i		−	0.126445i
$515$		0			0
$516$	35.9677			0.0697048
$517$	−792.969	+	792.969i	−1.53379	+	1.53379i
$518$	−237.946	−	237.946i	−0.459355	−	0.459355i
$519$		−	107.714i		−	0.207541i
$520$		0			0
$521$	535.527			1.02788			0.513941	−	0.857825i	$-0.328185\pi$
$521$	535.527			1.02788			0.513941	+	0.857825i	$0.328185\pi$
$522$	−55.2443	+	55.2443i	−0.105832	+	0.105832i
$523$	355.245	+	355.245i	0.679244	+	0.679244i	0.959829	−	0.280585i	$-0.0905285\pi$
$523$	355.245	+	355.245i	0.679244	+	0.679244i	−0.280585	+	0.959829i	$0.590528\pi$
$524$			139.435i			0.266097i
$525$		0			0
$526$	111.942			0.212817
$527$	713.619	−	713.619i	1.35412	−	1.35412i
$528$	99.0041	+	99.0041i	0.187508	+	0.187508i
$529$			279.022i			0.527451i
$530$		0			0
$531$	262.239			0.493860
$532$	182.193	−	182.193i	0.342468	−	0.342468i
$533$	280.527	+	280.527i	0.526317	+	0.526317i
$534$			79.5708i			0.149009i
$535$		0			0
$536$	284.019			0.529886
$537$	304.718	−	304.718i	0.567446	−	0.567446i
$538$	−275.469	−	275.469i	−0.512025	−	0.512025i
$539$			327.103i			0.606870i
$540$		0			0
$541$	−1051.65			−1.94390			−0.971952	−	0.235178i	$-0.924433\pi$
$541$	−1051.65			−1.94390			−0.971952	+	0.235178i	$0.924433\pi$
$542$	−265.170	+	265.170i	−0.489243	+	0.489243i
$543$	−26.3252	−	26.3252i	−0.0484810	−	0.0484810i
$544$		−	140.661i		−	0.258568i
$545$		0			0
$546$	−132.079			−0.241903
$547$	−547.786	+	547.786i	−1.00144	+	1.00144i	−0.00143807	+	0.999999i	$0.500458\pi$
$547$	−547.786	+	547.786i	−1.00144	+	1.00144i	−0.999999	+	0.00143807i	$0.999542\pi$
$548$	172.593	+	172.593i	0.314951	+	0.314951i
$549$		−	200.201i		−	0.364664i
$550$		0			0
$551$	−293.837			−0.533279
$552$	−98.4696	+	98.4696i	−0.178387	+	0.178387i
$553$	−396.505	−	396.505i	−0.717008	−	0.717008i
$554$		−	156.606i		−	0.282682i
$555$		0			0
$556$	−477.029			−0.857966
$557$	93.3418	−	93.3418i	0.167580	−	0.167580i	−0.618335	−	0.785915i	$-0.712192\pi$
$557$	93.3418	−	93.3418i	0.167580	−	0.167580i	0.785915	+	0.618335i	$0.212192\pi$
$558$	−121.760	−	121.760i	−0.218207	−	0.218207i
$559$		−	69.3432i		−	0.124049i
$560$		0			0
$561$	−870.377			−1.55147
$562$	280.509	−	280.509i	0.499127	−	0.499127i
$563$	427.735	+	427.735i	0.759742	+	0.759742i	0.976275	−	0.216533i	$-0.0694749\pi$
$563$	427.735	+	427.735i	0.759742	+	0.759742i	−0.216533	+	0.976275i	$0.569475\pi$
$564$		−	192.227i		−	0.340828i
$565$		0			0
$566$	−537.782			−0.950144
$567$	51.3812	−	51.3812i	0.0906194	−	0.0906194i
$568$	16.2159	+	16.2159i	0.0285492	+	0.0285492i
$569$			445.486i			0.782928i	0.920193	+	0.391464i	$0.128031\pi$
$569$			445.486i			0.782928i	−0.920193	+	0.391464i	$0.871969\pi$
$570$		0			0
$571$	−820.418			−1.43681			−0.718404	−	0.695626i	$-0.755127\pi$
$571$	−820.418			−1.43681			−0.718404	+	0.695626i	$0.755127\pi$
$572$	190.873	−	190.873i	0.333694	−	0.333694i
$573$	−419.396	−	419.396i	−0.731930	−	0.731930i
$574$			678.266i			1.18165i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	601.773	−	601.773i	1.04293	−	1.04293i	0.0438982	−	0.999036i	$-0.486022\pi$
$577$	601.773	−	601.773i	1.04293	−	1.04293i	0.999036	−	0.0438982i	$-0.0139777\pi$
$578$	329.298	+	329.298i	0.569720	+	0.569720i
$579$			147.063i			0.253995i
$580$		0			0
$581$	−42.3646			−0.0729168
$582$	67.7491	−	67.7491i	0.116407	−	0.116407i
$583$	1343.90	+	1343.90i	2.30515	+	2.30515i
$584$			161.590i			0.276695i
$585$		0			0
$586$	−158.948			−0.271242
$587$	225.822	−	225.822i	0.384705	−	0.384705i	−0.488089	−	0.872794i	$-0.662306\pi$
$587$	225.822	−	225.822i	0.384705	−	0.384705i	0.872794	+	0.488089i	$0.162306\pi$
$588$	−39.6472	−	39.6472i	−0.0674272	−	0.0674272i
$589$		−	647.623i		−	1.09953i
$590$		0			0
$591$	−62.6713			−0.106043
$592$	−83.3579	+	83.3579i	−0.140807	+	0.140807i
$593$	−203.495	−	203.495i	−0.343162	−	0.343162i	0.514393	−	0.857555i	$-0.328017\pi$
$593$	−203.495	−	203.495i	−0.343162	−	0.343162i	−0.857555	+	0.514393i	$0.828017\pi$
$594$			148.506i			0.250010i
$595$		0			0
$596$	−575.526			−0.965648
$597$	−165.711	+	165.711i	−0.277574	+	0.277574i
$598$	189.842	+	189.842i	0.317462	+	0.317462i
$599$			541.417i			0.903869i	0.892051	+	0.451934i	$0.149266\pi$
$599$			541.417i			0.903869i	−0.892051	+	0.451934i	$0.850734\pi$
$600$		0			0
$601$	296.683			0.493650			0.246825	−	0.969060i	$-0.420613\pi$
$601$	296.683			0.493650			0.246825	+	0.969060i	$0.420613\pi$
$602$	83.8299	−	83.8299i	0.139252	−	0.139252i
$603$	213.014	+	213.014i	0.353258	+	0.353258i
$604$		−	336.923i		−	0.557820i
$605$		0			0
$606$	135.445			0.223507
$607$	534.614	−	534.614i	0.880748	−	0.880748i	−0.112863	−	0.993611i	$-0.536002\pi$
$607$	534.614	−	534.614i	0.880748	−	0.880748i	0.993611	+	0.112863i	$0.0360021\pi$
$608$	−63.8263	−	63.8263i	−0.104978	−	0.104978i
$609$			257.516i			0.422850i
$610$		0			0
$611$	−370.600			−0.606546
$612$	105.496	−	105.496i	0.172379	−	0.172379i
$613$	−0.733205	−	0.733205i	−0.00119609	−	0.00119609i	0.706508	−	0.707705i	$-0.250269\pi$
$613$	−0.733205	−	0.733205i	−0.00119609	−	0.00119609i	−0.707705	+	0.706508i	$0.750269\pi$
$614$		−	611.734i		−	0.996310i
$615$		0			0
$616$	461.498			0.749184
$617$	95.5753	−	95.5753i	0.154903	−	0.154903i	−0.625401	−	0.780304i	$-0.715064\pi$
$617$	95.5753	−	95.5753i	0.154903	−	0.154903i	0.780304	+	0.625401i	$0.215064\pi$
$618$	−164.417	−	164.417i	−0.266046	−	0.266046i
$619$		−	99.8888i		−	0.161371i	−0.996740	−	0.0806856i	$-0.974289\pi$
$619$		−	99.8888i		−	0.161371i	0.996740	−	0.0806856i	$-0.0257110\pi$
$620$		0			0
$621$	−147.704			−0.237849
$622$	111.946	−	111.946i	0.179978	−	0.179978i
$623$	185.456	+	185.456i	0.297681	+	0.297681i
$624$			46.2703i			0.0741511i
$625$		0			0
$626$	−310.096			−0.495361
$627$	−394.942	+	394.942i	−0.629891	+	0.629891i
$628$	−113.098	−	113.098i	−0.180092	−	0.180092i
$629$		−	732.826i		−	1.16506i
$630$		0			0
$631$	235.641			0.373441			0.186720	−	0.982413i	$-0.440214\pi$
$631$	235.641			0.373441			0.186720	+	0.982413i	$0.440214\pi$
$632$	−138.905	+	138.905i	−0.219786	+	0.219786i
$633$	−344.067	−	344.067i	−0.543549	−	0.543549i
$634$			448.660i			0.707665i
$635$		0			0
$636$	−325.781			−0.512234
$637$	−76.4370	+	76.4370i	−0.119995	+	0.119995i
$638$	−372.146	−	372.146i	−0.583301	−	0.583301i
$639$			24.3239i			0.0380656i
$640$		0			0
$641$	354.178			0.552540			0.276270	−	0.961080i	$-0.410902\pi$
$641$	354.178			0.552540			0.276270	+	0.961080i	$0.410902\pi$
$642$	216.014	−	216.014i	0.336471	−	0.336471i
$643$	703.400	+	703.400i	1.09393	+	1.09393i	0.995104	+	0.0988302i	$0.0315101\pi$
$643$	703.400	+	703.400i	1.09393	+	1.09393i	0.0988302	+	0.995104i	$0.468490\pi$
$644$			459.006i			0.712742i
$645$		0			0
$646$	561.118			0.868603
$647$	−187.002	+	187.002i	−0.289030	+	0.289030i	−0.836696	−	0.547667i	$-0.815516\pi$
$647$	−187.002	+	187.002i	−0.289030	+	0.289030i	0.547667	+	0.836696i	$0.315516\pi$
$648$	−18.0000	−	18.0000i	−0.0277778	−	0.0277778i
$649$			1766.54i			2.72195i
$650$		0			0
$651$	−567.571			−0.871844
$652$	22.8870	−	22.8870i	0.0351028	−	0.0351028i
$653$	329.263	+	329.263i	0.504231	+	0.504231i	0.912750	−	0.408519i	$-0.133955\pi$
$653$	329.263	+	329.263i	0.504231	+	0.504231i	−0.408519	+	0.912750i	$0.633955\pi$
$654$			203.711i			0.311484i
$655$		0			0
$656$	237.612			0.362213
$657$	−121.192	+	121.192i	−0.184463	+	0.184463i
$658$	−448.023	−	448.023i	−0.680886	−	0.680886i
$659$			708.388i			1.07494i	0.843282	+	0.537472i	$0.180620\pi$
$659$			708.388i			1.07494i	−0.843282	+	0.537472i	$0.819380\pi$
$660$		0			0
$661$	−462.496			−0.699691			−0.349845	−	0.936807i	$-0.613766\pi$
$661$	−462.496			−0.699691			−0.349845	+	0.936807i	$0.613766\pi$
$662$	−314.778	+	314.778i	−0.475496	+	0.475496i
$663$	−203.388	−	203.388i	−0.306770	−	0.306770i
$664$			14.8413i			0.0223513i
$665$		0			0
$666$	−125.037			−0.187743
$667$	370.137	−	370.137i	0.554928	−	0.554928i
$668$	−126.586	−	126.586i	−0.189500	−	0.189500i
$669$		−	521.912i		−	0.780138i
$670$		0			0
$671$	1348.63			2.00988
$672$	−55.9368	+	55.9368i	−0.0832393	+	0.0832393i
$673$	−409.524	−	409.524i	−0.608505	−	0.608505i	0.334050	−	0.942555i	$-0.391584\pi$
$673$	−409.524	−	409.524i	−0.608505	−	0.608505i	−0.942555	+	0.334050i	$0.891584\pi$
$674$			55.6183i			0.0825197i
$675$		0			0
$676$	−248.794			−0.368039
$677$	360.165	−	360.165i	0.532001	−	0.532001i	−0.389166	−	0.921167i	$-0.627237\pi$
$677$	360.165	−	360.165i	0.532001	−	0.532001i	0.921167	+	0.389166i	$0.127237\pi$
$678$	258.194	+	258.194i	0.380817	+	0.380817i
$679$		−	315.805i		−	0.465104i
$680$		0			0
$681$	−489.595			−0.718936
$682$	820.220	−	820.220i	1.20267	−	1.20267i
$683$	−533.712	−	533.712i	−0.781423	−	0.781423i	0.198648	−	0.980071i	$-0.436345\pi$
$683$	−533.712	−	533.712i	−0.781423	−	0.781423i	−0.980071	+	0.198648i	$0.936345\pi$
$684$		−	95.7395i		−	0.139970i
$685$		0			0
$686$	374.673			0.546170
$687$	131.619	−	131.619i	0.191585	−	0.191585i
$688$	−29.3675	−	29.3675i	−0.0426853	−	0.0426853i
$689$			628.082i			0.911585i
$690$		0			0
$691$	−258.392			−0.373940			−0.186970	−	0.982366i	$-0.559867\pi$
$691$	−258.392			−0.373940			−0.186970	+	0.982366i	$0.559867\pi$
$692$	−87.9479	+	87.9479i	−0.127092	+	0.127092i
$693$	346.123	+	346.123i	0.499456	+	0.499456i
$694$		−	166.127i		−	0.239376i
$695$		0			0
$696$	90.2135			0.129617
$697$	−1044.46	+	1044.46i	−1.49851	+	1.49851i
$698$	170.578	+	170.578i	0.244382	+	0.244382i
$699$			453.924i			0.649391i
$700$		0			0
$701$	1067.20			1.52239			0.761195	−	0.648523i	$-0.224613\pi$
$701$	1067.20			1.52239			0.761195	+	0.648523i	$0.224613\pi$
$702$	−34.7027	+	34.7027i	−0.0494341	+	0.0494341i
$703$	−332.527	−	332.527i	−0.473011	−	0.473011i
$704$		−	161.673i		−	0.229649i
$705$		0			0
$706$	−13.3944			−0.0189723
$707$	315.682	−	315.682i	0.446509	−	0.446509i
$708$	−214.118	−	214.118i	−0.302426	−	0.302426i
$709$		−	1234.17i		−	1.74072i	−0.492418	−	0.870359i	$-0.663887\pi$
$709$		−	1234.17i		−	1.74072i	0.492418	−	0.870359i	$-0.336113\pi$
$710$		0			0
$711$	−208.357			−0.293048
$712$	64.9693	−	64.9693i	0.0912490	−	0.0912490i
$713$	815.791	+	815.791i	1.14417	+	1.14417i
$714$		−	491.758i		−	0.688737i
$715$		0			0
$716$	−497.603			−0.694976
$717$	122.307	−	122.307i	0.170582	−	0.170582i
$718$	297.590	+	297.590i	0.414470	+	0.414470i
$719$		−	196.739i		−	0.273629i	−0.990597	−	0.136814i	$-0.956314\pi$
$719$		−	196.739i		−	0.273629i	0.990597	−	0.136814i	$-0.0436864\pi$
$720$		0			0
$721$	−766.411			−1.06298
$722$	−106.387	+	106.387i	−0.147351	+	0.147351i
$723$	−435.635	−	435.635i	−0.602538	−	0.602538i
$724$			42.9888i			0.0593769i
$725$		0			0
$726$	−704.005			−0.969704
$727$	−383.339	+	383.339i	−0.527288	+	0.527288i	−0.919763	−	0.392474i	$-0.871619\pi$
$727$	−383.339	+	383.339i	−0.527288	+	0.527288i	0.392474	+	0.919763i	$0.371619\pi$
$728$	107.842	+	107.842i	0.148135	+	0.148135i
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	258.179			0.353186
$732$	−163.463	+	163.463i	−0.223310	+	0.223310i
$733$	−647.864	−	647.864i	−0.883852	−	0.883852i	0.110071	−	0.993924i	$-0.464892\pi$
$733$	−647.864	−	647.864i	−0.883852	−	0.883852i	−0.993924	+	0.110071i	$0.964892\pi$
$734$		−	486.753i		−	0.663151i
$735$		0			0
$736$	160.800			0.218478
$737$	−1434.94	+	1434.94i	−1.94701	+	1.94701i
$738$	178.209	+	178.209i	0.241475	+	0.241475i
$739$			1205.22i			1.63088i	0.578839	+	0.815442i	$0.303506\pi$
$739$			1205.22i			1.63088i	−0.578839	+	0.815442i	$0.696494\pi$
$740$		0			0
$741$	−184.579			−0.249094
$742$	−759.297	+	759.297i	−1.02331	+	1.02331i
$743$	−513.926	−	513.926i	−0.691691	−	0.691691i	0.270913	−	0.962604i	$-0.412674\pi$
$743$	−513.926	−	513.926i	−0.691691	−	0.691691i	−0.962604	+	0.270913i	$0.912674\pi$
$744$			198.833i			0.267248i
$745$		0			0
$746$	−63.7339			−0.0854342
$747$	−11.1310	+	11.1310i	−0.0149009	+	0.0149009i
$748$	710.660	+	710.660i	0.950080	+	0.950080i
$749$		−	1006.93i		−	1.34436i
$750$		0			0
$751$	−391.376			−0.521140			−0.260570	−	0.965455i	$-0.583910\pi$
$751$	−391.376			−0.521140			−0.260570	+	0.965455i	$0.583910\pi$
$752$	−156.953	+	156.953i	−0.208714	+	0.208714i
$753$	391.337	+	391.337i	0.519704	+	0.519704i
$754$		−	173.925i		−	0.230670i
$755$		0			0
$756$	−83.9052			−0.110986
$757$	−543.625	+	543.625i	−0.718131	+	0.718131i	−0.968222	−	0.250091i	$-0.919539\pi$
$757$	−543.625	+	543.625i	−0.718131	+	0.718131i	0.250091	+	0.968222i	$0.419539\pi$
$758$	−329.670	−	329.670i	−0.434921	−	0.434921i
$759$		−	994.992i		−	1.31093i
$760$		0			0
$761$	−413.173			−0.542934			−0.271467	−	0.962448i	$-0.587509\pi$
$761$	−413.173			−0.542934			−0.271467	+	0.962448i	$0.587509\pi$
$762$	39.8880	−	39.8880i	0.0523465	−	0.0523465i
$763$	474.788	+	474.788i	0.622265	+	0.622265i
$764$			684.870i			0.896427i
$765$		0			0
$766$	755.883			0.986792
$767$	−412.803	+	412.803i	−0.538205	+	0.538205i
$768$	19.5959	+	19.5959i	0.0255155	+	0.0255155i
$769$		−	54.1412i		−	0.0704047i	−0.999380	−	0.0352024i	$-0.988792\pi$
$769$		−	54.1412i		−	0.0704047i	0.999380	−	0.0352024i	$-0.0112076\pi$
$770$		0			0
$771$	−79.5994			−0.103242
$772$	120.077	−	120.077i	0.155540	−	0.155540i
$773$	−579.780	−	579.780i	−0.750038	−	0.750038i	0.224448	−	0.974486i	$-0.427942\pi$
$773$	−579.780	−	579.780i	−0.750038	−	0.750038i	−0.974486	+	0.224448i	$0.927942\pi$
$774$		−	44.0513i		−	0.0569138i
$775$		0			0
$776$	−110.634			−0.142569
$777$	−291.423	+	291.423i	−0.375062	+	0.375062i
$778$	438.570	+	438.570i	0.563715	+	0.563715i
$779$			947.868i			1.21678i
$780$		0			0
$781$	−163.855			−0.209802
$782$	−706.823	+	706.823i	−0.903865	+	0.903865i
$783$	67.6602	+	67.6602i	0.0864114	+	0.0864114i
$784$			64.7436i			0.0825812i
$785$		0			0
$786$	170.772			0.217267
$787$	8.08450	−	8.08450i	0.0102725	−	0.0102725i	−0.701952	−	0.712224i	$-0.747688\pi$
$787$	8.08450	−	8.08450i	0.0102725	−	0.0102725i	0.712224	+	0.701952i	$0.247688\pi$
$788$	51.1709	+	51.1709i	0.0649377	+	0.0649377i
$789$		−	137.100i		−	0.173765i
$790$		0			0
$791$	1203.55			1.52155
$792$	121.255	−	121.255i	0.153099	−	0.153099i
$793$	315.145	+	315.145i	0.397409	+	0.397409i
$794$		−	662.158i		−	0.833952i
$795$		0			0
$796$	270.606			0.339957
$797$	892.181	−	892.181i	1.11942	−	1.11942i	0.127598	−	0.991826i	$-0.459273\pi$
$797$	892.181	−	892.181i	1.11942	−	1.11942i	0.991826	−	0.127598i	$-0.0407266\pi$
$798$	−223.140	−	223.140i	−0.279624	−	0.279624i
$799$		−	1379.82i		−	1.72693i
$800$		0			0
$801$	97.4539			0.121665
$802$	441.641	−	441.641i	0.550674	−	0.550674i
$803$	−816.397	−	816.397i	−1.01668	−	1.01668i
$804$		−	347.851i		−	0.432650i
$805$		0			0
$806$	383.335			0.475602
$807$	−337.380	+	337.380i	−0.418066	+	0.418066i
$808$	−110.590	−	110.590i	−0.136869	−	0.136869i
$809$		−	239.445i		−	0.295977i	−0.988989	−	0.147988i	$-0.952720\pi$
$809$		−	239.445i		−	0.295977i	0.988989	−	0.147988i	$-0.0472798\pi$
$810$		0			0
$811$	309.741			0.381925			0.190963	−	0.981597i	$-0.438839\pi$
$811$	309.741			0.381925			0.190963	+	0.981597i	$0.438839\pi$
$812$	210.261	−	210.261i	0.258942	−	0.258942i
$813$	324.765	+	324.765i	0.399465	+	0.399465i
$814$		−	842.295i		−	1.03476i
$815$		0			0
$816$	−172.274			−0.211120
$817$	117.151	−	117.151i	0.143392	−	0.143392i
$818$	−240.811	−	240.811i	−0.294390	−	0.294390i
$819$			161.763i			0.197513i
$820$		0			0
$821$	−745.573			−0.908127			−0.454064	−	0.890969i	$-0.650026\pi$
$821$	−745.573			−0.908127			−0.454064	+	0.890969i	$0.650026\pi$
$822$	211.383	−	211.383i	0.257157	−	0.257157i
$823$	898.365	+	898.365i	1.09157	+	1.09157i	0.995361	+	0.0962126i	$0.0306729\pi$
$823$	898.365	+	898.365i	1.09157	+	1.09157i	0.0962126	+	0.995361i	$0.469327\pi$
$824$			268.491i			0.325839i
$825$		0			0
$826$	−998.087			−1.20834
$827$	−475.096	+	475.096i	−0.574481	+	0.574481i	−0.933378	−	0.358896i	$-0.883153\pi$
$827$	−475.096	+	475.096i	−0.574481	+	0.574481i	0.358896	+	0.933378i	$0.383153\pi$
$828$	120.600	+	120.600i	0.145652	+	0.145652i
$829$			818.385i			0.987196i	0.869690	+	0.493598i	$0.164319\pi$
$829$			818.385i			0.987196i	−0.869690	+	0.493598i	$0.835681\pi$
$830$		0			0
$831$	−191.802			−0.230809
$832$	37.7795	−	37.7795i	0.0454081	−	0.0454081i
$833$	−284.591	−	284.591i	−0.341646	−	0.341646i
$834$			584.239i			0.700526i
$835$		0			0
$836$	644.937			0.771456
$837$	−149.125	+	149.125i	−0.178166	+	0.178166i
$838$	−641.021	−	641.021i	−0.764942	−	0.764942i
$839$			160.825i			0.191687i	0.995396	+	0.0958433i	$0.0305548\pi$
$839$			160.825i			0.191687i	−0.995396	+	0.0958433i	$0.969445\pi$
$840$		0			0
$841$	501.897			0.596785
$842$	493.748	−	493.748i	0.586399	−	0.586399i
$843$	−343.552	−	343.552i	−0.407535	−	0.407535i
$844$			561.859i			0.665709i
$845$		0			0
$846$	−235.429			−0.278285
$847$	−1640.82	+	1640.82i	−1.93722	+	1.93722i
$848$	265.999	+	265.999i	0.313678	+	0.313678i
$849$			658.645i			0.775789i
$850$		0			0
$851$	837.748			0.984427
$852$	19.8604	−	19.8604i	0.0233103	−	0.0233103i
$853$	−163.525	−	163.525i	−0.191706	−	0.191706i	0.604727	−	0.796433i	$-0.293282\pi$
$853$	−163.525	−	163.525i	−0.191706	−	0.191706i	−0.796433	+	0.604727i	$0.793282\pi$
$854$			761.967i			0.892232i
$855$		0			0
$856$	−352.750			−0.412091
$857$	−575.315	+	575.315i	−0.671313	+	0.671313i	−0.958019	−	0.286706i	$-0.907440\pi$
$857$	−575.315	+	575.315i	−0.671313	+	0.671313i	0.286706	+	0.958019i	$0.407440\pi$
$858$	−233.771	−	233.771i	−0.272460	−	0.272460i
$859$		−	831.484i		−	0.967967i	−0.875077	−	0.483984i	$-0.839189\pi$
$859$		−	831.484i		−	0.967967i	0.875077	−	0.483984i	$-0.160811\pi$
$860$		0			0
$861$	830.702			0.964811
$862$	−117.686	+	117.686i	−0.136527	+	0.136527i
$863$	−816.894	−	816.894i	−0.946574	−	0.946574i	0.0520691	−	0.998643i	$-0.483418\pi$
$863$	−816.894	−	816.894i	−0.946574	−	0.946574i	−0.998643	+	0.0520691i	$0.983418\pi$
$864$			29.3939i			0.0340207i
$865$		0			0
$866$	725.685			0.837973
$867$	403.306	−	403.306i	0.465174	−	0.465174i
$868$	463.420	+	463.420i	0.533893	+	0.533893i
$869$		−	1403.57i		−	1.61516i
$870$		0			0
$871$	−670.632			−0.769956
$872$	166.329	−	166.329i	0.190744	−	0.190744i
$873$	−82.9753	−	82.9753i	−0.0950462	−	0.0950462i
$874$			641.455i			0.733931i
$875$		0			0
$876$	197.906			0.225920
$877$	73.5839	−	73.5839i	0.0839041	−	0.0839041i	−0.663909	−	0.747813i	$-0.731104\pi$
$877$	73.5839	−	73.5839i	0.0839041	−	0.0839041i	0.747813	+	0.663909i	$0.231104\pi$
$878$	−303.617	−	303.617i	−0.345805	−	0.345805i
$879$			194.670i			0.221468i
$880$		0			0
$881$	−1040.98			−1.18159			−0.590794	−	0.806822i	$-0.701186\pi$
$881$	−1040.98			−1.18159			−0.590794	+	0.806822i	$0.701186\pi$
$882$	−48.5577	+	48.5577i	−0.0550541	+	0.0550541i
$883$	840.426	+	840.426i	0.951785	+	0.951785i	0.998890	−	0.0471052i	$-0.0149996\pi$
$883$	840.426	+	840.426i	0.951785	+	0.951785i	−0.0471052	+	0.998890i	$0.515000\pi$
$884$			332.132i			0.375715i
$885$		0			0
$886$	−552.343			−0.623411
$887$	301.035	−	301.035i	0.339386	−	0.339386i	−0.516750	−	0.856136i	$-0.672858\pi$
$887$	301.035	−	301.035i	0.339386	−	0.339386i	0.856136	+	0.516750i	$0.172858\pi$
$888$	102.092	+	102.092i	0.114969	+	0.114969i
$889$		−	185.934i		−	0.209150i
$890$		0			0
$891$	181.882			0.204133
$892$	−426.139	+	426.139i	−0.477735	+	0.477735i
$893$	−626.107	−	626.107i	−0.701128	−	0.701128i
$894$			704.873i			0.788448i
$895$		0			0
$896$	91.3444			0.101947
$897$	232.508	−	232.508i	0.259207	−	0.259207i
$898$	314.925	+	314.925i	0.350696	+	0.350696i
$899$		−	747.392i		−	0.831360i
$900$		0			0
$901$	−2338.48			−2.59543
$902$	−1200.48	+	1200.48i	−1.33091	+	1.33091i
$903$	−102.670	−	102.670i	−0.113699	−	0.113699i
$904$		−	421.629i		−	0.466404i
$905$		0			0
$906$	−412.645			−0.455458
$907$	1120.69	−	1120.69i	1.23560	−	1.23560i	0.273825	−	0.961779i	$-0.411711\pi$
$907$	1120.69	−	1120.69i	1.23560	−	1.23560i	0.961779	−	0.273825i	$-0.0882890\pi$
$908$	399.753	+	399.753i	0.440257	+	0.440257i
$909$		−	165.886i		−	0.182493i
$910$		0			0
$911$	1739.91			1.90989			0.954946	−	0.296780i	$-0.0959127\pi$
$911$	1739.91			1.90989			0.954946	+	0.296780i	$0.0959127\pi$
$912$	−78.1710	+	78.1710i	−0.0857138	+	0.0857138i
$913$	−74.9824	−	74.9824i	−0.0821275	−	0.0821275i
$914$			1127.42i			1.23350i
$915$		0			0
$916$	−214.933			−0.234643
$917$	398.018	−	398.018i	0.434044	−	0.434044i
$918$	−129.205	−	129.205i	−0.140747	−	0.140747i
$919$		−	1602.98i		−	1.74427i	−0.489269	−	0.872133i	$-0.662736\pi$
$919$		−	1602.98i		−	1.74427i	0.489269	−	0.872133i	$-0.337264\pi$
$920$		0			0
$921$	−749.219			−0.813484
$922$	−660.573	+	660.573i	−0.716457	+	0.716457i
$923$	−38.2894	−	38.2894i	−0.0414837	−	0.0414837i
$924$		−	565.217i		−	0.611707i
$925$		0			0
$926$	228.510			0.246772
$927$	−201.368	+	201.368i	−0.217226	+	0.217226i
$928$	−73.6590	−	73.6590i	−0.0793740	−	0.0793740i
$929$		−	1838.90i		−	1.97944i	−0.143003	−	0.989722i	$-0.545676\pi$
$929$		−	1838.90i		−	1.97944i	0.143003	−	0.989722i	$-0.454324\pi$
$930$		0			0
$931$	−258.272			−0.277413
$932$	370.628	−	370.628i	0.397669	−	0.397669i
$933$	−137.106	−	137.106i	−0.146951	−	0.146951i
$934$			995.772i			1.06614i
$935$		0			0
$936$	56.6693			0.0605441
$937$	603.150	−	603.150i	0.643703	−	0.643703i	−0.307761	−	0.951464i	$-0.599580\pi$
$937$	603.150	−	603.150i	0.643703	−	0.643703i	0.951464	+	0.307761i	$0.0995796\pi$
$938$	−810.736	−	810.736i	−0.864324	−	0.864324i
$939$			379.789i			0.404461i
$940$		0			0
$941$	1311.13			1.39334			0.696669	−	0.717393i	$-0.254665\pi$
$941$	1311.13			1.39334			0.696669	+	0.717393i	$0.254665\pi$
$942$	−138.516	+	138.516i	−0.147045	+	0.147045i
$943$	−1194.00	−	1194.00i	−1.26617	−	1.26617i
$944$			349.653i			0.370395i
$945$		0			0
$946$	296.746			0.313685
$947$	−736.685	+	736.685i	−0.777915	+	0.777915i	−0.979476	−	0.201561i	$-0.935399\pi$
$947$	−736.685	+	736.685i	−0.777915	+	0.777915i	0.201561	+	0.979476i	$0.435399\pi$
$948$	170.123	+	170.123i	0.179455	+	0.179455i
$949$		−	381.549i		−	0.402054i
$950$		0			0
$951$	549.494			0.577806
$952$	−401.519	+	401.519i	−0.421763	+	0.421763i
$953$	1131.73	+	1131.73i	1.18755	+	1.18755i	0.977743	+	0.209805i	$0.0672831\pi$
$953$	1131.73	+	1131.73i	1.18755	+	1.18755i	0.209805	+	0.977743i	$0.432717\pi$
$954$			398.999i			0.418237i
$955$		0			0
$956$	−199.727			−0.208919
$957$	−455.784	+	455.784i	−0.476264	+	0.476264i
$958$	416.013	+	416.013i	0.434252	+	0.434252i
$959$		−	985.340i		−	1.02747i
$960$		0			0
$961$	686.271			0.714121
$962$	196.826	−	196.826i	0.204601	−	0.204601i
$963$	−264.563	−	264.563i	−0.274727	−	0.274727i
$964$			711.389i			0.737955i
$965$		0			0
$966$	562.165			0.581952
$967$	−712.879	+	712.879i	−0.737207	+	0.737207i	−0.972037	−	0.234830i	$-0.924547\pi$
$967$	−712.879	+	712.879i	−0.737207	+	0.737207i	0.234830	+	0.972037i	$0.424547\pi$
$968$	574.818	+	574.818i	0.593820	+	0.593820i
$969$		−	687.226i		−	0.709211i
$970$		0			0
$971$	1321.75			1.36123			0.680613	−	0.732643i	$-0.261713\pi$
$971$	1321.75			1.36123			0.680613	+	0.732643i	$0.261713\pi$
$972$	−22.0454	+	22.0454i	−0.0226805	+	0.0226805i
$973$	1361.69	+	1361.69i	1.39947	+	1.39947i
$974$			1034.95i			1.06257i
$975$		0			0
$976$	266.934			0.273498
$977$	−472.998	+	472.998i	−0.484134	+	0.484134i	−0.906449	−	0.422315i	$-0.861217\pi$
$977$	−472.998	+	472.998i	−0.484134	+	0.484134i	0.422315	+	0.906449i	$0.361217\pi$
$978$	−28.0307	−	28.0307i	−0.0286613	−	0.0286613i
$979$			656.486i			0.670568i
$980$		0			0
$981$	249.493			0.254326
$982$	−483.386	+	483.386i	−0.492246	+	0.492246i
$983$	8.96317	+	8.96317i	0.00911818	+	0.00911818i	0.711651	−	0.702533i	$-0.247948\pi$
$983$	8.96317	+	8.96317i	0.00911818	+	0.00911818i	−0.702533	+	0.711651i	$0.747948\pi$
$984$		−	291.014i		−	0.295746i
$985$		0			0
$986$	647.560			0.656755
$987$	−548.714	+	548.714i	−0.555941	+	0.555941i
$988$	150.708	+	150.708i	0.152538	+	0.152538i
$989$			295.144i			0.298426i
$990$		0			0
$991$	824.728			0.832218			0.416109	−	0.909315i	$-0.363393\pi$
$991$	824.728			0.832218			0.416109	+	0.909315i	$0.363393\pi$
$992$	162.346	−	162.346i	0.163656	−	0.163656i
$993$	385.523	+	385.523i	0.388241	+	0.388241i
$994$		−	92.5772i		−	0.0931360i
$995$		0			0
$996$	18.1768			0.0182498
$997$	517.929	−	517.929i	0.519487	−	0.519487i	−0.397929	−	0.917416i	$-0.630271\pi$
$997$	517.929	−	517.929i	0.519487	−	0.519487i	0.917416	+	0.397929i	$0.130271\pi$
$998$	869.520	+	869.520i	0.871262	+	0.871262i
$999$			153.138i			0.153292i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.3.f.b.193.5	✓	16
5.2	odd	4	inner	750.3.f.b.307.5	yes	16
5.3	odd	4		750.3.f.c.307.4	yes	16
5.4	even	2		750.3.f.c.193.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.3.f.b.193.5	✓	16	1.1	even	1	trivial
750.3.f.b.307.5	yes	16	5.2	odd	4	inner
750.3.f.c.193.4	yes	16	5.4	even	2
750.3.f.c.307.4	yes	16	5.3	odd	4

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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	750.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} -2.44949 q^{6} +(-5.70902 + 5.70902i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -20.2091 q^{11} +(2.44949 - 2.44949i) q^{12} +(-4.72244 - 4.72244i) q^{13} -11.4180i q^{14} -4.00000 q^{16} +(17.5826 - 17.5826i) q^{17} +(-3.00000 - 3.00000i) q^{18} -15.9566i q^{19} -13.9842 q^{21} +(20.2091 - 20.2091i) q^{22} +(20.1000 + 20.1000i) q^{23} +4.89898i q^{24} +9.44488 q^{26} +(-3.67423 + 3.67423i) q^{27} +(11.4180 + 11.4180i) q^{28} -18.4148i q^{29} +40.5866 q^{31} +(4.00000 - 4.00000i) q^{32} +(-24.7510 - 24.7510i) q^{33} +35.1653i q^{34} +6.00000 q^{36} +(20.8395 - 20.8395i) q^{37} +(15.9566 + 15.9566i) q^{38} -11.5676i q^{39} -59.4029 q^{41} +(13.9842 - 13.9842i) q^{42} +(7.34188 + 7.34188i) q^{43} +40.4183i q^{44} -40.2000 q^{46} +(39.2382 - 39.2382i) q^{47} +(-4.89898 - 4.89898i) q^{48} -16.1859i q^{49} +43.0685 q^{51} +(-9.44488 + 9.44488i) q^{52} +(-66.4998 - 66.4998i) q^{53} -7.34847i q^{54} -22.8361 q^{56} +(19.5427 - 19.5427i) q^{57} +(18.4148 + 18.4148i) q^{58} -87.4131i q^{59} -66.7335 q^{61} +(-40.5866 + 40.5866i) q^{62} +(-17.1271 - 17.1271i) q^{63} +8.00000i q^{64} +49.5021 q^{66} +(71.0048 - 71.0048i) q^{67} +(-35.1653 - 35.1653i) q^{68} +49.2348i q^{69} +8.10797 q^{71} +(-6.00000 + 6.00000i) q^{72} +(40.3974 + 40.3974i) q^{73} +41.6789i q^{74} -31.9132 q^{76} +(115.374 - 115.374i) q^{77} +(11.5676 + 11.5676i) q^{78} +69.4524i q^{79} -9.00000 q^{81} +(59.4029 - 59.4029i) q^{82} +(3.71032 + 3.71032i) q^{83} +27.9684i q^{84} -14.6838 q^{86} +(22.5534 - 22.5534i) q^{87} +(-40.4183 - 40.4183i) q^{88} -32.4846i q^{89} +53.9211 q^{91} +(40.2000 - 40.2000i) q^{92} +(49.7082 + 49.7082i) q^{93} +78.4763i q^{94} +9.79796 q^{96} +(-27.6584 + 27.6584i) q^{97} +(16.1859 + 16.1859i) q^{98} -60.6274i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 24 q^{7} + 32 q^{8} - 24 q^{11} - 48 q^{13} - 64 q^{16} - 16 q^{17} - 48 q^{18} - 48 q^{21} + 24 q^{22} + 104 q^{23} + 96 q^{26} - 48 q^{28} + 200 q^{31} + 64 q^{32} - 48 q^{33} + 96 q^{36}+ \cdots - 224 q^{98}+O(q^{100}) \) 16 * q - 16 * q^2 + 24 * q^7 + 32 * q^8 - 24 * q^11 - 48 * q^13 - 64 * q^16 - 16 * q^17 - 48 * q^18 - 48 * q^21 + 24 * q^22 + 104 * q^23 + 96 * q^26 - 48 * q^28 + 200 * q^31 + 64 * q^32 - 48 * q^33 + 96 * q^36 - 128 * q^37 + 56 * q^38 - 192 * q^41 + 48 * q^42 + 88 * q^43 - 208 * q^46 + 16 * q^47 - 48 * q^51 - 96 * q^52 - 160 * q^53 + 96 * q^56 + 48 * q^57 + 24 * q^58 - 24 * q^61 - 200 * q^62 + 72 * q^63 + 96 * q^66 + 264 * q^67 + 32 * q^68 + 176 * q^71 - 96 * q^72 + 32 * q^73 - 112 * q^76 + 32 * q^77 + 96 * q^78 - 144 * q^81 + 192 * q^82 - 288 * q^83 - 176 * q^86 - 168 * q^87 - 48 * q^88 + 32 * q^91 + 208 * q^92 + 144 * q^93 - 48 * q^97 - 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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