LMFDB - Embedded newform 1764.4.t.a.521.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(521,1764)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.t (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 42x^{12} + 1139x^{8} + 26250x^{4} + 390625 $ x^16 + 42x^12 + 1139x^8 + 26250*x^4 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.3
Root			$-0.829861 + 2.07637i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.4.t.a.1097.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.10680 - 1.91704i) q^{5} +O(q^{10})$	$q+(-1.10680 - 1.91704i) q^{5} +(-17.6210 - 10.1735i) q^{11} -64.4624i q^{13} +(63.9455 - 110.757i) q^{17} +(-7.57978 + 4.37619i) q^{19} +(66.0596 - 38.1396i) q^{23} +(60.0500 - 104.010i) q^{25} +190.742i q^{29} +(143.078 + 82.6063i) q^{31} +(54.3250 + 94.0936i) q^{37} -312.838 q^{41} -139.875 q^{43} +(223.934 + 387.865i) q^{47} +(-460.473 - 265.854i) q^{53} +45.0402i q^{55} +(-138.959 + 240.684i) q^{59} +(-157.131 + 90.7195i) q^{61} +(-123.577 + 71.3471i) q^{65} +(88.4500 - 153.200i) q^{67} -259.314i q^{71} +(-423.316 - 244.401i) q^{73} +(-284.650 - 493.028i) q^{79} +1069.12 q^{83} -283.100 q^{85} +(-335.223 - 580.623i) q^{89} +(16.7786 + 9.68715i) q^{95} -526.369i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 40 q^{25} - 512 q^{37} + 64 q^{43} + 2336 q^{67} - 1792 q^{79} - 2688 q^{85}+O(q^{100}) $ 16 * q + 40 * q^25 - 512 * q^37 + 64 * q^43 + 2336 * q^67 - 1792 * q^79 - 2688 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.10680	−	1.91704i	−0.0989954	−	0.171465i	0.812274	−	0.583276i	$-0.198230\pi$
$5$	−1.10680	−	1.91704i	−0.0989954	−	0.171465i	−0.911269	+	0.411811i	$0.864896\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.6210	−	10.1735i	−0.482994	−	0.278857i	0.238669	−	0.971101i	$-0.423289\pi$
$11$	−17.6210	−	10.1735i	−0.482994	−	0.278857i	−0.721663	+	0.692244i	$0.756622\pi$
$12$		0			0
$13$		−	64.4624i		−	1.37528i	−0.726051	−	0.687640i	$-0.758647\pi$
$13$		−	64.4624i		−	1.37528i	0.726051	−	0.687640i	$-0.241353\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	63.9455	−	110.757i	0.912297	−	1.58015i	0.101487	−	0.994837i	$-0.467640\pi$
$17$	63.9455	−	110.757i	0.912297	−	1.58015i	0.810811	−	0.585309i	$-0.199027\pi$
$18$		0			0
$19$	−7.57978	+	4.37619i	−0.0915221	+	0.0528403i	−0.545063	−	0.838395i	$-0.683494\pi$
$19$	−7.57978	+	4.37619i	−0.0915221	+	0.0528403i	0.453540	+	0.891236i	$0.350161\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	66.0596	−	38.1396i	0.598886	−	0.345767i	−0.169717	−	0.985493i	$-0.554285\pi$
$23$	66.0596	−	38.1396i	0.598886	−	0.345767i	0.768603	+	0.639726i	$0.220952\pi$
$24$		0			0
$25$	60.0500	−	104.010i	0.480400	−	0.832077i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			190.742i			1.22138i	0.791871	+	0.610688i	$0.209107\pi$
$29$			190.742i			1.22138i	−0.791871	+	0.610688i	$0.790893\pi$
$30$		0			0
$31$	143.078	+	82.6063i	0.828955	+	0.478598i	0.853495	−	0.521101i	$-0.174479\pi$
$31$	143.078	+	82.6063i	0.828955	+	0.478598i	−0.0245394	+	0.999699i	$0.507812\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	54.3250	+	94.0936i	0.241378	+	0.418078i	0.961107	−	0.276177i	$-0.0890674\pi$
$37$	54.3250	+	94.0936i	0.241378	+	0.418078i	−0.719729	+	0.694255i	$0.755734\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−312.838			−1.19163			−0.595817	−	0.803120i	$-0.703172\pi$
$41$	−312.838			−1.19163			−0.595817	+	0.803120i	$0.703172\pi$
$42$		0			0
$43$	−139.875			−0.496063			−0.248032	−	0.968752i	$-0.579784\pi$
$43$	−139.875			−0.496063			−0.248032	+	0.968752i	$0.579784\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	223.934	+	387.865i	0.694981	+	1.20374i	0.970187	+	0.242357i	$0.0779204\pi$
$47$	223.934	+	387.865i	0.694981	+	1.20374i	−0.275207	+	0.961385i	$0.588746\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−460.473	−	265.854i	−1.19341	−	0.689017i	−0.234334	−	0.972156i	$-0.575291\pi$
$53$	−460.473	−	265.854i	−1.19341	−	0.689017i	−0.959079	+	0.283139i	$0.908624\pi$
$54$		0			0
$55$			45.0402i			0.110422i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−138.959	+	240.684i	−0.306626	+	0.531091i	−0.977622	−	0.210369i	$-0.932533\pi$
$59$	−138.959	+	240.684i	−0.306626	+	0.531091i	0.670996	+	0.741461i	$0.265867\pi$
$60$		0			0
$61$	−157.131	+	90.7195i	−0.329812	+	0.190417i	−0.655758	−	0.754971i	$-0.727651\pi$
$61$	−157.131	+	90.7195i	−0.329812	+	0.190417i	0.325946	+	0.945389i	$0.394317\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−123.577	+	71.3471i	−0.235813	+	0.136146i
$66$		0			0
$67$	88.4500	−	153.200i	0.161282	−	0.279349i	−0.774047	−	0.633128i	$-0.781770\pi$
$67$	88.4500	−	153.200i	0.161282	−	0.279349i	0.935329	+	0.353780i	$0.115104\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	259.314i		−	0.433449i	−0.976233	−	0.216724i	$-0.930463\pi$
$71$		−	259.314i		−	0.433449i	0.976233	−	0.216724i	$-0.0695373\pi$
$72$		0			0
$73$	−423.316	−	244.401i	−0.678703	−	0.391850i	0.120663	−	0.992694i	$-0.461498\pi$
$73$	−423.316	−	244.401i	−0.678703	−	0.391850i	−0.799366	+	0.600844i	$0.794831\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−284.650	−	493.028i	−0.405388	−	0.702152i	0.588979	−	0.808148i	$-0.299530\pi$
$79$	−284.650	−	493.028i	−0.405388	−	0.702152i	−0.994367	+	0.105996i	$0.966197\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1069.12			1.41386			0.706932	−	0.707282i	$-0.250079\pi$
$83$	1069.12			1.41386			0.706932	+	0.707282i	$0.250079\pi$
$84$		0			0
$85$	−283.100			−0.361253
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−335.223	−	580.623i	−0.399253	−	0.691527i	0.594381	−	0.804184i	$-0.297397\pi$
$89$	−335.223	−	580.623i	−0.399253	−	0.691527i	−0.993634	+	0.112657i	$0.964064\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	16.7786	+	9.68715i	0.0181205	+	0.0104619i
$96$		0			0
$97$		−	526.369i		−	0.550976i	−0.961305	−	0.275488i	$-0.911161\pi$
$97$		−	526.369i		−	0.550976i	0.961305	−	0.275488i	$-0.0888394\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	179.911	−	311.614i	0.177245	−	0.306998i	−0.763691	−	0.645582i	$-0.776615\pi$
$101$	179.911	−	311.614i	0.177245	−	0.306998i	0.940936	+	0.338584i	$0.109948\pi$
$102$		0			0
$103$	730.551	−	421.784i	0.698867	−	0.403491i	−0.108058	−	0.994145i	$-0.534463\pi$
$103$	730.551	−	421.784i	0.698867	−	0.403491i	0.806925	+	0.590653i	$0.201130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1736.09	+	1002.33i	−1.56854	+	0.905599i	−0.572205	+	0.820111i	$0.693912\pi$
$107$	−1736.09	+	1002.33i	−1.56854	+	0.905599i	−0.996339	+	0.0854887i	$0.972755\pi$
$108$		0			0
$109$	167.475	−	290.075i	0.147167	−	0.254901i	−0.783012	−	0.622006i	$-0.786318\pi$
$109$	167.475	−	290.075i	0.147167	−	0.254901i	0.930179	+	0.367105i	$0.119651\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			252.225i			0.209976i	0.994473	+	0.104988i	$0.0334805\pi$
$113$			252.225i			0.209976i	−0.994473	+	0.104988i	$0.966520\pi$
$114$		0			0
$115$	−146.230	−	84.4259i	−0.118574	−	0.0684587i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−458.500	−	794.145i	−0.344478	−	0.596653i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−542.554			−0.388220
$126$		0			0
$127$	1484.70			1.03737			0.518684	−	0.854966i	$-0.326422\pi$
$127$	1484.70			1.03737			0.518684	+	0.854966i	$0.326422\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1046.12	−	1811.94i	−0.697710	−	1.20847i	−0.969258	−	0.246045i	$-0.920869\pi$
$131$	−1046.12	−	1811.94i	−0.697710	−	1.20847i	0.271548	−	0.962425i	$-0.412464\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1889.23	−	1090.75i	−1.17816	−	0.680211i	−0.222571	−	0.974916i	$-0.571445\pi$
$137$	−1889.23	−	1090.75i	−1.17816	−	0.680211i	−0.955588	+	0.294706i	$0.904778\pi$
$138$		0			0
$139$			2676.25i			1.63307i	0.577299	+	0.816533i	$0.304107\pi$
$139$			2676.25i			1.63307i	−0.577299	+	0.816533i	$0.695893\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−655.808	+	1135.89i	−0.383506	+	0.664252i
$144$		0			0
$145$	365.660	−	211.114i	0.209423	−	0.120911i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1166.85	+	673.678i	−0.641555	+	0.370402i	−0.785213	−	0.619225i	$-0.787447\pi$
$149$	−1166.85	+	673.678i	−0.641555	+	0.370402i	0.143658	+	0.989627i	$0.454113\pi$
$150$		0			0
$151$	1159.49	−	2008.29i	0.624886	−	1.08233i	−0.363677	−	0.931525i	$-0.618479\pi$
$151$	1159.49	−	2008.29i	0.624886	−	1.08233i	0.988563	−	0.150809i	$-0.0481878\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	365.715i		−	0.189516i
$156$		0			0
$157$	1860.02	+	1073.88i	0.945513	+	0.545892i	0.891684	−	0.452658i	$-0.149524\pi$
$157$	1860.02	+	1073.88i	0.945513	+	0.545892i	0.0538290	+	0.998550i	$0.482857\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1353.25	−	2343.90i	−0.650274	−	1.12631i	−0.983056	−	0.183304i	$-0.941321\pi$
$163$	−1353.25	−	2343.90i	−0.650274	−	1.12631i	0.332782	−	0.943004i	$-0.392013\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−506.417			−0.234657			−0.117329	−	0.993093i	$-0.537433\pi$
$167$	−506.417			−0.234657			−0.117329	+	0.993093i	$0.537433\pi$
$168$		0			0
$169$	−1958.40			−0.891397
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1772.88	+	3070.71i	0.779128	+	1.34949i	0.932445	+	0.361313i	$0.117671\pi$
$173$	1772.88	+	3070.71i	0.779128	+	1.34949i	−0.153316	+	0.988177i	$0.548995\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2989.56	−	1726.02i	−1.24832	−	0.720720i	−0.277549	−	0.960712i	$-0.589522\pi$
$179$	−2989.56	−	1726.02i	−1.24832	−	0.720720i	−0.970775	+	0.239992i	$0.922855\pi$
$180$		0			0
$181$		−	1318.70i		−	0.541537i	−0.962644	−	0.270769i	$-0.912722\pi$
$181$		−	1318.70i		−	0.541537i	0.962644	−	0.270769i	$-0.0872778\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	120.254	−	208.286i	0.0477905	−	0.0827756i
$186$		0			0
$187$	−2253.57	+	1301.10i	−0.881268	+	0.508801i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1089.76	+	629.175i	−0.412840	+	0.238353i	−0.692009	−	0.721889i	$-0.743274\pi$
$191$	−1089.76	+	629.175i	−0.412840	+	0.238353i	0.279169	+	0.960242i	$0.409941\pi$
$192$		0			0
$193$	2191.90	−	3796.48i	0.817494	−	1.41594i	−0.0900288	−	0.995939i	$-0.528696\pi$
$193$	2191.90	−	3796.48i	0.817494	−	1.41594i	0.907523	−	0.420002i	$-0.137971\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4678.18i			1.69191i	0.533253	+	0.845956i	$0.320969\pi$
$197$			4678.18i			1.69191i	−0.533253	+	0.845956i	$0.679031\pi$
$198$		0			0
$199$	−3763.47	−	2172.84i	−1.34063	−	0.774013i	−0.353730	−	0.935348i	$-0.615087\pi$
$199$	−3763.47	−	2172.84i	−1.34063	−	0.774013i	−0.986900	+	0.161335i	$0.948420\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	346.249	+	599.721i	0.117966	+	0.204324i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	178.085			0.0589395
$210$		0			0
$211$	−2773.22			−0.904818			−0.452409	−	0.891811i	$-0.649435\pi$
$211$	−2773.22			−0.904818			−0.452409	+	0.891811i	$0.649435\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	154.814	+	268.146i	0.0491080	+	0.0850575i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7139.65	−	4122.08i	−2.17314	−	1.25467i
$222$		0			0
$223$		−	4836.21i		−	1.45227i	−0.687552	−	0.726135i	$-0.741315\pi$
$223$		−	4836.21i		−	1.45227i	0.687552	−	0.726135i	$-0.258685\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1051.41	−	1821.09i	0.307420	−	0.532467i	−0.670377	−	0.742020i	$-0.733868\pi$
$227$	1051.41	−	1821.09i	0.307420	−	0.532467i	0.977797	+	0.209554i	$0.0672011\pi$
$228$		0			0
$229$	−3012.78	+	1739.43i	−0.869388	+	0.501942i	−0.867145	−	0.498056i	$-0.834047\pi$
$229$	−3012.78	+	1739.43i	−0.869388	+	0.501942i	−0.00224343	+	0.999997i	$0.500714\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3036.82	+	1753.31i	−0.853856	+	0.492974i	−0.861950	−	0.506993i	$-0.830757\pi$
$233$	−3036.82	+	1753.31i	−0.853856	+	0.492974i	0.00809395	+	0.999967i	$0.497424\pi$
$234$		0			0
$235$	495.701	−	858.579i	0.137600	−	0.238330i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			259.667i			0.0702782i	0.999382	+	0.0351391i	$0.0111874\pi$
$239$			259.667i			0.0702782i	−0.999382	+	0.0351391i	$0.988813\pi$
$240$		0			0
$241$	2496.67	+	1441.46i	0.667323	+	0.385279i	0.795062	−	0.606529i	$-0.207438\pi$
$241$	2496.67	+	1441.46i	0.667323	+	0.385279i	−0.127738	+	0.991808i	$0.540772\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	282.100	+	488.611i	0.0726703	+	0.125869i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2189.64			−0.550633			−0.275317	−	0.961354i	$-0.588783\pi$
$251$	−2189.64			−0.550633			−0.275317	+	0.961354i	$0.588783\pi$
$252$		0			0
$253$	−1552.05			−0.385678
$254$		0			0
$255$		0			0
$256$		0			0
$257$	923.543	+	1599.62i	0.224160	+	0.388256i	0.956067	−	0.293148i	$-0.0947029\pi$
$257$	923.543	+	1599.62i	0.224160	+	0.388256i	−0.731907	+	0.681404i	$0.761370\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6669.76	−	3850.79i	−1.56378	−	0.902851i	−0.996869	−	0.0790745i	$-0.974803\pi$
$263$	−6669.76	−	3850.79i	−1.56378	−	0.902851i	−0.566915	−	0.823776i	$-0.691863\pi$
$264$		0			0
$265$			1176.99i			0.272838i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2210.45	−	3828.61i	0.501017	−	0.867786i	−0.498983	−	0.866612i	$-0.666293\pi$
$269$	2210.45	−	3828.61i	0.501017	−	0.867786i	0.999999	−	0.00117434i	$-0.000373805\pi$
$270$		0			0
$271$	6235.15	−	3599.86i	1.39763	−	0.806923i	0.403487	−	0.914985i	$-0.367798\pi$
$271$	6235.15	−	3599.86i	1.39763	−	0.806923i	0.994144	+	0.108062i	$0.0344646\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2116.28	+	1221.84i	−0.464061	+	0.267925i
$276$		0			0
$277$	4174.35	−	7230.18i	0.905460	−	1.56830i	0.0851607	−	0.996367i	$-0.472860\pi$
$277$	4174.35	−	7230.18i	0.905460	−	1.56830i	0.820299	−	0.571935i	$-0.193807\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			366.212i			0.0777450i	0.999244	+	0.0388725i	$0.0123766\pi$
$281$			366.212i			0.0777450i	−0.999244	+	0.0388725i	$0.987623\pi$
$282$		0			0
$283$	6260.10	+	3614.27i	1.31493	+	0.759173i	0.982908	−	0.184099i	$-0.0589368\pi$
$283$	6260.10	+	3614.27i	1.31493	+	0.759173i	0.332019	+	0.943273i	$0.392270\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5721.55	−	9910.01i	−1.16457	−	2.01710i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−147.758			−0.0294611			−0.0147306	−	0.999891i	$-0.504689\pi$
$293$	−147.758			−0.0294611			−0.0147306	+	0.999891i	$0.504689\pi$
$294$		0			0
$295$	615.200			0.121418
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2458.57	−	4258.36i	−0.475527	−	0.823637i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	347.825	+	200.817i	0.0652998	+	0.0377008i
$306$		0			0
$307$			6995.08i			1.30042i	0.759752	+	0.650212i	$0.225320\pi$
$307$			6995.08i			1.30042i	−0.759752	+	0.650212i	$0.774680\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1820.74	+	3153.62i	−0.331977	+	0.575002i	−0.982899	−	0.184143i	$-0.941049\pi$
$311$	−1820.74	+	3153.62i	−0.331977	+	0.575002i	0.650922	+	0.759144i	$0.274382\pi$
$312$		0			0
$313$	999.673	−	577.161i	0.180527	−	0.104227i	−0.407013	−	0.913422i	$-0.633430\pi$
$313$	999.673	−	577.161i	0.180527	−	0.104227i	0.587540	+	0.809195i	$0.300096\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6319.59	+	3648.62i	−1.11970	+	0.646457i	−0.941323	−	0.337506i	$-0.890417\pi$
$317$	−6319.59	+	3648.62i	−1.11970	+	0.646457i	−0.178373	+	0.983963i	$0.557083\pi$
$318$		0			0
$319$	1940.51	−	3361.07i	0.340589	−	0.589917i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1119.35i			0.192824i
$324$		0			0
$325$	−6704.71	−	3870.96i	−1.14434	−	0.660685i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	640.037	+	1108.58i	0.106283	+	0.184087i	0.914262	−	0.405124i	$-0.132772\pi$
$331$	640.037	+	1108.58i	0.106283	+	0.184087i	−0.807979	+	0.589212i	$0.799438\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−391.587			−0.0638647
$336$		0			0
$337$	520.898			0.0841992			0.0420996	−	0.999113i	$-0.486595\pi$
$337$	520.898			0.0841992			0.0420996	+	0.999113i	$0.486595\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1680.79	−	2911.21i	−0.266920	−	0.462320i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3678.54	−	2123.80i	−0.569090	−	0.328564i	0.187696	−	0.982227i	$-0.439898\pi$
$347$	−3678.54	−	2123.80i	−0.569090	−	0.328564i	−0.756786	+	0.653663i	$0.773231\pi$
$348$		0			0
$349$		−	433.289i		−	0.0664567i	−0.999448	−	0.0332284i	$-0.989421\pi$
$349$		−	433.289i		−	0.0664567i	0.999448	−	0.0332284i	$-0.0105789\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4072.56	+	7053.88i	−0.614052	+	1.06357i	0.376497	+	0.926418i	$0.377128\pi$
$353$	−4072.56	+	7053.88i	−0.614052	+	1.06357i	−0.990550	+	0.137152i	$0.956205\pi$
$354$		0			0
$355$	−497.114	+	287.009i	−0.0743213	+	0.0429094i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3836.80	+	2215.18i	−0.564063	+	0.325662i	−0.754775	−	0.655984i	$-0.772254\pi$
$359$	−3836.80	+	2215.18i	−0.564063	+	0.325662i	0.190712	+	0.981646i	$0.438921\pi$
$360$		0			0
$361$	−3391.20	+	5873.73i	−0.494416	+	0.856353i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1082.02i			0.155165i
$366$		0			0
$367$	−1182.10	−	682.484i	−0.168133	−	0.0970718i	0.413572	−	0.910471i	$-0.364281\pi$
$367$	−1182.10	−	682.484i	−0.168133	−	0.0970718i	−0.581705	+	0.813400i	$0.697614\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2007.70	−	3477.44i	−0.278699	−	0.482721i	0.692363	−	0.721550i	$-0.256570\pi$
$373$	−2007.70	−	3477.44i	−0.278699	−	0.482721i	−0.971062	+	0.238829i	$0.923237\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12295.7			1.67973
$378$		0			0
$379$	7623.47			1.03322			0.516611	−	0.856220i	$-0.327193\pi$
$379$	7623.47			1.03322			0.516611	+	0.856220i	$0.327193\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5028.23	+	8709.15i	0.670837	+	1.16192i	0.977667	+	0.210159i	$0.0673982\pi$
$383$	5028.23	+	8709.15i	0.670837	+	1.16192i	−0.306830	+	0.951764i	$0.599268\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1422.45	+	821.251i	0.185401	+	0.107041i	0.589828	−	0.807529i	$-0.299196\pi$
$389$	1422.45	+	821.251i	0.185401	+	0.107041i	−0.404427	+	0.914570i	$0.632529\pi$
$390$		0			0
$391$		−	9755.41i		−	1.26177i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−630.102	+	1091.37i	−0.0802630	+	0.139020i
$396$		0			0
$397$	2337.71	−	1349.68i	0.295533	−	0.170626i	−0.344902	−	0.938639i	$-0.612088\pi$
$397$	2337.71	−	1349.68i	0.295533	−	0.170626i	0.640434	+	0.768013i	$0.278754\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10900.3	+	6293.28i	−1.35744	+	0.783720i	−0.989279	−	0.146041i	$-0.953347\pi$
$401$	−10900.3	+	6293.28i	−1.35744	+	0.783720i	−0.368164	+	0.929761i	$0.620013\pi$
$402$		0			0
$403$	5325.00	−	9223.17i	0.658206	−	1.14005i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2210.70i		−	0.269239i
$408$		0			0
$409$	−12897.4	−	7446.29i	−1.55925	−	0.900233i	−0.997329	−	0.0730427i	$-0.976729\pi$
$409$	−12897.4	−	7446.29i	−1.55925	−	0.900233i	−0.561921	−	0.827191i	$-0.689938\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1183.30	−	2049.53i	−0.139966	−	0.242428i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8282.97			0.965751			0.482876	−	0.875689i	$-0.339592\pi$
$419$	8282.97			0.965751			0.482876	+	0.875689i	$0.339592\pi$
$420$		0			0
$421$	−3991.70			−0.462098			−0.231049	−	0.972942i	$-0.574216\pi$
$421$	−3991.70			−0.462098			−0.231049	+	0.972942i	$0.574216\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7679.85	−	13301.9i	−0.876535	−	1.51820i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8913.95	−	5146.47i	−0.996218	−	0.575167i	−0.0890908	−	0.996024i	$-0.528396\pi$
$431$	−8913.95	−	5146.47i	−0.996218	−	0.575167i	−0.907127	+	0.420857i	$0.861729\pi$
$432$		0			0
$433$			9849.52i			1.09316i	0.837407	+	0.546579i	$0.184070\pi$
$433$			9849.52i			1.09316i	−0.837407	+	0.546579i	$0.815930\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−333.812	+	578.179i	−0.0365409	+	0.0632907i
$438$		0			0
$439$	−9795.58	+	5655.48i	−1.06496	+	0.614855i	−0.926800	−	0.375555i	$-0.877452\pi$
$439$	−9795.58	+	5655.48i	−1.06496	+	0.614855i	−0.138160	+	0.990410i	$0.544119\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10874.8	−	6278.55i	1.16631	−	0.673370i	0.213503	−	0.976942i	$-0.431513\pi$
$443$	10874.8	−	6278.55i	1.16631	−	0.673370i	0.952808	+	0.303573i	$0.0981795\pi$
$444$		0			0
$445$	−742.050	+	1285.27i	−0.0790484	+	0.136916i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6763.33i		−	0.710871i	−0.934701	−	0.355436i	$-0.884332\pi$
$449$		−	6763.33i		−	0.710871i	0.934701	−	0.355436i	$-0.115668\pi$
$450$		0			0
$451$	5512.51	+	3182.65i	0.575552	+	0.332295i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2456.00	−	4253.92i	−0.251394	−	0.435426i	0.712516	−	0.701656i	$-0.247555\pi$
$457$	−2456.00	−	4253.92i	−0.251394	−	0.435426i	−0.963910	+	0.266229i	$0.914222\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16772.7			−1.69454			−0.847269	−	0.531163i	$-0.821755\pi$
$461$	−16772.7			−1.69454			−0.847269	+	0.531163i	$0.821755\pi$
$462$		0			0
$463$	−11917.3			−1.19621			−0.598103	−	0.801419i	$-0.704079\pi$
$463$	−11917.3			−1.19621			−0.598103	+	0.801419i	$0.704079\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6034.81	+	10452.6i	0.597982	+	1.03574i	0.993119	+	0.117113i	$0.0373640\pi$
$467$	6034.81	+	10452.6i	0.597982	+	1.03574i	−0.395136	+	0.918622i	$0.629303\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2464.74	+	1423.02i	0.239596	+	0.138331i
$474$		0			0
$475$			1051.16i			0.101538i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5251.44	−	9095.77i	0.500928	−	0.867633i	−0.499071	−	0.866561i	$-0.666325\pi$
$479$	5251.44	−	9095.77i	0.500928	−	0.867633i	0.999999	−	0.00107200i	$-0.000341229\pi$
$480$		0			0
$481$	6065.50	−	3501.92i	0.574975	−	0.331962i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1009.07	+	582.586i	−0.0944731	+	0.0545441i
$486$		0			0
$487$	−585.288	+	1013.75i	−0.0544598	+	0.0943271i	−0.891970	−	0.452094i	$-0.850677\pi$
$487$	−585.288	+	1013.75i	−0.0544598	+	0.0943271i	0.837510	+	0.546422i	$0.184010\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			4816.93i			0.442739i	0.975190	+	0.221370i	$0.0710528\pi$
$491$			4816.93i			0.442739i	−0.975190	+	0.221370i	$0.928947\pi$
$492$		0			0
$493$	21126.0	+	12197.1i	1.92995	+	1.11426i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6856.61	+	11876.0i	0.615118	+	1.06542i	0.990364	+	0.138491i	$0.0442251\pi$
$499$	6856.61	+	11876.0i	0.615118	+	1.06542i	−0.375245	+	0.926926i	$0.622442\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18753.9			−1.66242			−0.831209	−	0.555960i	$-0.812351\pi$
$503$	−18753.9			−1.66242			−0.831209	+	0.555960i	$0.812351\pi$
$504$		0			0
$505$	−796.502			−0.0701859
$506$		0			0
$507$		0			0
$508$		0			0
$509$	732.537	+	1268.79i	0.0637900	+	0.110488i	0.896157	−	0.443738i	$-0.146348\pi$
$509$	732.537	+	1268.79i	0.0637900	+	0.110488i	−0.832367	+	0.554225i	$0.813015\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1617.15	−	933.662i	−0.138369	−	0.0798875i
$516$		0			0
$517$		−	9112.75i		−	0.775200i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	493.883	−	855.430i	0.0415305	−	0.0719330i	−0.844513	−	0.535535i	$-0.820110\pi$
$521$	493.883	−	855.430i	0.0415305	−	0.0719330i	0.886043	+	0.463602i	$0.153443\pi$
$522$		0			0
$523$	−1491.42	+	861.072i	−0.124695	+	0.0719925i	−0.561050	−	0.827782i	$-0.689602\pi$
$523$	−1491.42	+	861.072i	−0.124695	+	0.0719925i	0.436355	+	0.899774i	$0.356269\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18298.4	−	10564.6i	1.51251	−	0.873247i
$528$		0			0
$529$	−3174.25	+	5497.96i	−0.260890	+	0.451875i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			20166.3i			1.63883i
$534$		0			0
$535$	3843.02	+	2218.77i	0.310557	+	0.179300i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4169.15	+	7221.18i	0.331323	+	0.573868i	0.982772	−	0.184825i	$-0.0591717\pi$
$541$	4169.15	+	7221.18i	0.331323	+	0.573868i	−0.651449	+	0.758693i	$0.725838\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−741.447			−0.0582754
$546$		0			0
$547$	4888.70			0.382131			0.191066	−	0.981577i	$-0.438806\pi$
$547$	4888.70			0.382131			0.191066	+	0.981577i	$0.438806\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−834.723	−	1445.78i	−0.0645379	−	0.111783i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−174.002	−	100.460i	−0.0132364	−	0.00764205i	0.493367	−	0.869821i	$-0.335766\pi$
$557$	−174.002	−	100.460i	−0.0132364	−	0.00764205i	−0.506604	+	0.862179i	$0.669099\pi$
$558$		0			0
$559$			9016.67i			0.682227i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8320.58	−	14411.7i	0.622861	−	1.07883i	−0.366090	−	0.930579i	$-0.619304\pi$
$563$	8320.58	−	14411.7i	0.622861	−	1.07883i	0.988950	−	0.148247i	$-0.0473629\pi$
$564$		0			0
$565$	483.525	−	279.163i	0.0360036	−	0.0207867i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13306.7	+	7682.64i	−0.980399	+	0.566034i	−0.902391	−	0.430919i	$-0.858189\pi$
$569$	−13306.7	+	7682.64i	−0.980399	+	0.566034i	−0.0780083	+	0.996953i	$0.524856\pi$
$570$		0			0
$571$	146.148	−	253.136i	0.0107112	−	0.0185524i	−0.860620	−	0.509247i	$-0.829924\pi$
$571$	146.148	−	253.136i	0.0107112	−	0.0185524i	0.871331	+	0.490695i	$0.163257\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9161.12i		−	0.664426i
$576$		0			0
$577$	1699.14	+	980.997i	0.122593	+	0.0707789i	0.560042	−	0.828464i	$-0.310785\pi$
$577$	1699.14	+	980.997i	0.122593	+	0.0707789i	−0.437450	+	0.899243i	$0.644118\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	5409.34	+	9369.25i	0.384274	+	0.665582i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7387.52			0.519447			0.259724	−	0.965683i	$-0.416369\pi$
$587$	7387.52			0.519447			0.259724	+	0.965683i	$0.416369\pi$
$588$		0			0
$589$	−1446.00			−0.101157
$590$		0			0
$591$		0			0
$592$		0			0
$593$	359.295	+	622.318i	0.0248811	+	0.0430953i	0.878198	−	0.478298i	$-0.158746\pi$
$593$	359.295	+	622.318i	0.0248811	+	0.0430953i	−0.853317	+	0.521393i	$0.825413\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15234.0	+	8795.37i	1.03914	+	0.599949i	0.919590	−	0.392879i	$-0.128521\pi$
$599$	15234.0	+	8795.37i	1.03914	+	0.599949i	0.119551	+	0.992828i	$0.461854\pi$
$600$		0			0
$601$		−	4614.84i		−	0.313217i	−0.987661	−	0.156608i	$-0.949944\pi$
$601$		−	4614.84i		−	0.313217i	0.987661	−	0.156608i	$-0.0500560\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1014.94	+	1757.92i	−0.0682034	+	0.118132i
$606$		0			0
$607$	−17146.9	+	9899.77i	−1.14658	+	0.661976i	−0.948050	−	0.318120i	$-0.896948\pi$
$607$	−17146.9	+	9899.77i	−1.14658	+	0.661976i	−0.198525	+	0.980096i	$0.563615\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25002.7	−	14435.3i	1.65548	−	0.955793i
$612$		0			0
$613$	360.697	−	624.746i	0.0237658	−	0.0411635i	−0.853898	−	0.520441i	$-0.825768\pi$
$613$	360.697	−	624.746i	0.0237658	−	0.0411635i	0.877664	+	0.479277i	$0.159101\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	2582.07i		−	0.168477i	−0.996446	−	0.0842384i	$-0.973154\pi$
$617$		−	2582.07i		−	0.168477i	0.996446	−	0.0842384i	$-0.0268457\pi$
$618$		0			0
$619$	−21590.0	−	12465.0i	−1.40190	−	0.809387i	−0.407311	−	0.913289i	$-0.633534\pi$
$619$	−21590.0	−	12465.0i	−1.40190	−	0.809387i	−0.994587	+	0.103903i	$0.966867\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−6905.75	−	11961.1i	−0.441968	−	0.765511i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13895.3			0.880833
$630$		0			0
$631$	13510.1			0.852343			0.426172	−	0.904642i	$-0.359862\pi$
$631$	13510.1			0.852343			0.426172	+	0.904642i	$0.359862\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1643.27	−	2846.22i	−0.102695	−	0.177872i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20481.3	+	11824.9i	1.26204	+	0.728636i	0.973468	−	0.228823i	$-0.0734878\pi$
$641$	20481.3	+	11824.9i	1.26204	+	0.728636i	0.288567	+	0.957460i	$0.406821\pi$
$642$		0			0
$643$		−	23421.3i		−	1.43647i	−0.695803	−	0.718233i	$-0.744951\pi$
$643$		−	23421.3i		−	1.43647i	0.695803	−	0.718233i	$-0.255049\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3673.81	+	6363.22i	−0.223234	+	0.386652i	−0.955788	−	0.294056i	$-0.904995\pi$
$647$	−3673.81	+	6363.22i	−0.223234	+	0.386652i	0.732554	+	0.680709i	$0.238328\pi$
$648$		0			0
$649$	4897.20	−	2827.40i	0.296197	−	0.171009i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5306.54	−	3063.73i	0.318011	−	0.183604i	−0.332495	−	0.943105i	$-0.607890\pi$
$653$	5306.54	−	3063.73i	0.318011	−	0.183604i	0.650506	+	0.759502i	$0.274557\pi$
$654$		0			0
$655$	−2315.70	+	4010.91i	−0.138140	+	0.239266i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	23048.8i		−	1.36245i	−0.732076	−	0.681223i	$-0.761448\pi$
$659$		−	23048.8i		−	1.36245i	0.732076	−	0.681223i	$-0.238552\pi$
$660$		0			0
$661$	−11133.0	−	6427.65i	−0.655105	−	0.378225i	0.135304	−	0.990804i	$-0.456799\pi$
$661$	−11133.0	−	6427.65i	−0.655105	−	0.378225i	−0.790409	+	0.612579i	$0.790132\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7274.81	+	12600.3i	0.422312	+	0.731465i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3691.74			0.212396
$672$		0			0
$673$	11239.1			0.643738			0.321869	−	0.946784i	$-0.395689\pi$
$673$	11239.1			0.643738			0.321869	+	0.946784i	$0.395689\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14273.1	+	24721.7i	0.810280	+	1.40345i	0.912668	+	0.408702i	$0.134018\pi$
$677$	14273.1	+	24721.7i	0.810280	+	1.40345i	−0.102388	+	0.994745i	$0.532648\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20083.5	+	11595.2i	1.12514	+	0.649603i	0.942709	−	0.333615i	$-0.108269\pi$
$683$	20083.5	+	11595.2i	1.12514	+	0.649603i	0.182435	+	0.983218i	$0.441602\pi$
$684$		0			0
$685$			4828.97i			0.269351i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17137.6	+	29683.2i	−0.947592	+	1.64128i
$690$		0			0
$691$	19369.4	−	11182.9i	1.06635	−	0.615656i	0.139166	−	0.990269i	$-0.455558\pi$
$691$	19369.4	−	11182.9i	1.06635	−	0.615656i	0.927181	+	0.374613i	$0.122224\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5130.46	−	2962.07i	0.280014	−	0.161666i
$696$		0			0
$697$	−20004.5	+	34648.9i	−1.08713	+	1.88296i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			9206.85i			0.496060i	0.968752	+	0.248030i	$0.0797831\pi$
$701$			9206.85i			0.496060i	−0.968752	+	0.248030i	$0.920217\pi$
$702$		0			0
$703$	−823.543	−	475.473i	−0.0441828	−	0.0255089i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3680.07	−	6374.08i	−0.194934	−	0.337635i	0.751945	−	0.659226i	$-0.229116\pi$
$709$	−3680.07	−	6374.08i	−0.194934	−	0.337635i	−0.946879	+	0.321591i	$0.895783\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12602.3			0.661934
$714$		0			0
$715$	2903.40			0.151861
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17880.3	−	30969.6i	−0.927432	−	1.60636i	−0.787602	−	0.616184i	$-0.788678\pi$
$719$	−17880.3	−	30969.6i	−0.927432	−	1.60636i	−0.139830	−	0.990176i	$-0.544656\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	19839.0	+	11454.1i	1.01628	+	0.586749i
$726$		0			0
$727$			30728.8i			1.56763i	0.620995	+	0.783815i	$0.286729\pi$
$727$			30728.8i			1.56763i	−0.620995	+	0.783815i	$0.713271\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8944.37	+	15492.1i	−0.452557	+	0.783852i
$732$		0			0
$733$	−6875.55	+	3969.60i	−0.346459	+	0.200028i	−0.663124	−	0.748509i	$-0.730770\pi$
$733$	−6875.55	+	3969.60i	−0.346459	+	0.200028i	0.316666	+	0.948537i	$0.397437\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3117.16	+	1799.69i	−0.155796	+	0.0899491i
$738$		0			0
$739$	−17807.0	+	30842.7i	−0.886391	+	1.53527i	−0.0422795	+	0.999106i	$0.513462\pi$
$739$	−17807.0	+	30842.7i	−0.886391	+	1.53527i	−0.844111	+	0.536168i	$0.819871\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	22961.5i		−	1.13375i	−0.823804	−	0.566875i	$-0.808152\pi$
$743$		−	22961.5i		−	1.13375i	0.823804	−	0.566875i	$-0.191848\pi$
$744$		0			0
$745$	2582.93	+	1491.26i	0.127022	+	0.0733362i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12475.4	+	21608.0i	0.606168	+	1.04991i	0.991866	+	0.127289i	$0.0406276\pi$
$751$	12475.4	+	21608.0i	0.606168	+	1.04991i	−0.385697	+	0.922625i	$0.626039\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5133.29			−0.247443
$756$		0			0
$757$	−32155.5			−1.54387			−0.771937	−	0.635699i	$-0.780712\pi$
$757$	−32155.5			−1.54387			−0.771937	+	0.635699i	$0.780712\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6911.51	−	11971.1i	−0.329227	−	0.570238i	0.653131	−	0.757245i	$-0.273455\pi$
$761$	−6911.51	−	11971.1i	−0.329227	−	0.570238i	−0.982359	+	0.187006i	$0.940122\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	15515.1	+	8957.63i	0.730400	+	0.421696i
$768$		0			0
$769$			36150.4i			1.69521i	0.530628	+	0.847605i	$0.321956\pi$
$769$			36150.4i			1.69521i	−0.530628	+	0.847605i	$0.678044\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16905.6	−	29281.3i	0.786612	−	1.36245i	−0.141419	−	0.989950i	$-0.545167\pi$
$773$	16905.6	−	29281.3i	0.786612	−	1.36245i	0.928031	−	0.372502i	$-0.121500\pi$
$774$		0			0
$775$	17183.7	−	9921.01i	0.796460	−	0.459836i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2371.24	−	1369.04i	0.109061	−	0.0629664i
$780$		0			0
$781$	−2638.13	+	4569.37i	−0.120870	+	0.209353i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	4754.30i		−	0.216163i
$786$		0			0
$787$	12730.6	+	7349.99i	0.576614	+	0.332908i	0.759787	−	0.650172i	$-0.225303\pi$
$787$	12730.6	+	7349.99i	0.576614	+	0.332908i	−0.183173	+	0.983081i	$0.558637\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5848.00	+	10129.0i	0.261877	+	0.453584i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−23535.5			−1.04601			−0.523005	−	0.852330i	$-0.675189\pi$
$797$	−23535.5			−1.04601			−0.523005	+	0.852330i	$0.675189\pi$
$798$		0			0
$799$	57278.2			2.53612
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4972.83	+	8613.20i	0.218540	+	0.378522i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34763.8	+	20070.9i	1.51079	+	0.872255i	0.999921	+	0.0125963i	$0.00400964\pi$
$809$	34763.8	+	20070.9i	1.51079	+	0.872255i	0.510869	+	0.859659i	$0.329324\pi$
$810$		0			0
$811$		−	29238.5i		−	1.26597i	−0.774164	−	0.632985i	$-0.781829\pi$
$811$		−	29238.5i		−	1.26597i	0.774164	−	0.632985i	$-0.218171\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2995.56	+	5188.46i	−0.128748	+	0.222999i
$816$		0			0
$817$	1060.22	−	612.119i	0.0454008	−	0.0262122i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16065.1	−	9275.20i	0.682919	−	0.394284i	−0.118035	−	0.993009i	$-0.537659\pi$
$821$	16065.1	−	9275.20i	0.682919	−	0.394284i	0.800954	+	0.598726i	$0.204326\pi$
$822$		0			0
$823$	−5105.05	+	8842.21i	−0.216222	+	0.374508i	−0.953650	−	0.300918i	$-0.902707\pi$
$823$	−5105.05	+	8842.21i	−0.216222	+	0.374508i	0.737428	+	0.675426i	$0.236040\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	24070.9i		−	1.01212i	−0.862497	−	0.506062i	$-0.831101\pi$
$827$		−	24070.9i		−	1.01212i	0.862497	−	0.506062i	$-0.168899\pi$
$828$		0			0
$829$	18775.8	+	10840.2i	0.786624	+	0.454158i	0.838773	−	0.544481i	$-0.183274\pi$
$829$	18775.8	+	10840.2i	0.786624	+	0.454158i	−0.0521484	+	0.998639i	$0.516607\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	560.504	+	970.821i	0.0232300	+	0.0402355i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33063.8			1.36053			0.680267	−	0.732964i	$-0.261864\pi$
$839$	33063.8			1.36053			0.680267	+	0.732964i	$0.261864\pi$
$840$		0			0
$841$	−11993.5			−0.491759
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2167.56	+	3754.32i	0.0882442	+	0.152843i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7177.38	+	4143.86i	0.289115	+	0.166921i
$852$		0			0
$853$		−	26065.5i		−	1.04627i	−0.852250	−	0.523134i	$-0.824763\pi$
$853$		−	26065.5i		−	1.04627i	0.852250	−	0.523134i	$-0.175237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19825.6	−	34339.0i	0.790234	−	1.36872i	−0.135589	−	0.990765i	$-0.543293\pi$
$857$	19825.6	−	34339.0i	0.790234	−	1.36872i	0.925822	−	0.377959i	$-0.123374\pi$
$858$		0			0
$859$	33125.4	−	19125.0i	1.31575	−	0.759646i	0.332704	−	0.943031i	$-0.392039\pi$
$859$	33125.4	−	19125.0i	1.31575	−	0.759646i	0.983041	+	0.183386i	$0.0587057\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37423.3	−	21606.3i	1.47613	−	0.852246i	0.476496	−	0.879176i	$-0.341907\pi$
$863$	37423.3	−	21606.3i	1.47613	−	0.852246i	0.999637	+	0.0269303i	$0.00857321\pi$
$864$		0			0
$865$	3924.44	−	6797.34i	0.154260	−	0.267187i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			11583.5i			0.452180i
$870$		0			0
$871$	−9875.63	−	5701.70i	−0.384183	−	0.221808i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2677.40	+	4637.39i	0.103089	+	0.178556i	0.912956	−	0.408058i	$-0.133794\pi$
$877$	2677.40	+	4637.39i	0.103089	+	0.178556i	−0.809867	+	0.586614i	$0.800461\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10812.1			−0.413471			−0.206735	−	0.978397i	$-0.566284\pi$
$881$	−10812.1			−0.413471			−0.206735	+	0.978397i	$0.566284\pi$
$882$		0			0
$883$	−25249.8			−0.962313			−0.481156	−	0.876635i	$-0.659783\pi$
$883$	−25249.8			−0.962313			−0.481156	+	0.876635i	$0.659783\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9226.55	−	15980.9i	−0.349264	−	0.604943i	0.636855	−	0.770984i	$-0.280235\pi$
$887$	−9226.55	−	15980.9i	−0.349264	−	0.604943i	−0.986119	+	0.166041i	$0.946902\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3394.74	−	1959.95i	−0.127212	−	0.0734460i
$894$		0			0
$895$			7641.46i			0.285392i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15756.5	+	27291.0i	−0.584548	+	1.01247i
$900$		0			0
$901$	−58890.4	+	34000.4i	−2.17749	+	1.25718i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2528.00	+	1459.54i	−0.0928547	+	0.0536097i
$906$		0			0
$907$	−5542.96	+	9600.69i	−0.202923	+	0.351473i	−0.949469	−	0.313861i	$-0.898377\pi$
$907$	−5542.96	+	9600.69i	−0.202923	+	0.351473i	0.746546	+	0.665334i	$0.231711\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13044.0i		−	0.474388i	−0.971462	−	0.237194i	$-0.923772\pi$
$911$		−	13044.0i		−	0.474388i	0.971462	−	0.237194i	$-0.0762278\pi$
$912$		0			0
$913$	−18838.9	−	10876.6i	−0.682888	−	0.394265i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	12604.9	+	21832.3i	0.452446	+	0.783659i	0.998537	−	0.0540664i	$-0.0172183\pi$
$919$	12604.9	+	21832.3i	0.452446	+	0.783659i	−0.546092	+	0.837726i	$0.683885\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16716.0			−0.596114
$924$		0			0
$925$	13048.9			0.463831
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8444.32	+	14626.0i	0.298223	+	0.516537i	0.975729	−	0.218980i	$-0.0702728\pi$
$929$	8444.32	+	14626.0i	0.298223	+	0.516537i	−0.677507	+	0.735517i	$0.736939\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4988.51	+	2880.12i	0.174483	+	0.100738i
$936$		0			0
$937$		−	56372.9i		−	1.96545i	−0.185085	−	0.982723i	$-0.559256\pi$
$937$		−	56372.9i		−	1.96545i	0.185085	−	0.982723i	$-0.440744\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16701.6	−	28928.1i	0.578595	−	1.00216i	−0.417046	−	0.908885i	$-0.636935\pi$
$941$	16701.6	−	28928.1i	0.578595	−	1.00216i	0.995641	−	0.0932705i	$-0.0297321\pi$
$942$		0			0
$943$	−20665.9	+	11931.5i	−0.713654	+	0.412028i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	37231.3	−	21495.5i	1.27757	−	0.737603i	0.301166	−	0.953572i	$-0.402624\pi$
$947$	37231.3	−	21495.5i	1.27757	−	0.737603i	0.976400	+	0.215969i	$0.0692910\pi$
$948$		0			0
$949$	−15754.7	+	27287.9i	−0.538903	+	0.933408i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			52997.6i			1.80143i	0.434414	+	0.900713i	$0.356956\pi$
$953$			52997.6i			1.80143i	−0.434414	+	0.900713i	$0.643044\pi$
$954$		0			0
$955$	2412.30	+	1392.74i	0.0817386	+	0.0471918i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1247.90	−	2161.43i	−0.0418885	−	0.0725531i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9704.00			−0.323713
$966$		0			0
$967$	8034.70			0.267196			0.133598	−	0.991036i	$-0.457347\pi$
$967$	8034.70			0.267196			0.133598	+	0.991036i	$0.457347\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	17988.7	+	31157.3i	0.594526	+	1.02975i	0.993614	+	0.112836i	$0.0359935\pi$
$971$	17988.7	+	31157.3i	0.594526	+	1.02975i	−0.399088	+	0.916913i	$0.630673\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4624.89	+	2670.18i	0.151446	+	0.0874377i	0.573808	−	0.818990i	$-0.305465\pi$
$977$	4624.89	+	2670.18i	0.151446	+	0.0874377i	−0.422362	+	0.906427i	$0.638799\pi$
$978$		0			0
$979$			13641.5i			0.445338i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	6543.94	−	11334.4i	0.212329	−	0.367764i	−0.740114	−	0.672481i	$-0.765229\pi$
$983$	6543.94	−	11334.4i	0.212329	−	0.367764i	0.952443	+	0.304717i	$0.0985619\pi$
$984$		0			0
$985$	8968.25	−	5177.82i	0.290104	−	0.167491i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−9240.09	+	5334.77i	−0.297086	+	0.171522i
$990$		0			0
$991$	5412.43	−	9374.61i	0.173493	−	0.300499i	−0.766146	−	0.642667i	$-0.777828\pi$
$991$	5412.43	−	9374.61i	0.173493	−	0.300499i	0.939639	+	0.342168i	$0.111161\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			9619.62i			0.306495i
$996$		0			0
$997$	−9649.69	−	5571.25i	−0.306528	−	0.176974i	0.338844	−	0.940843i	$-0.389964\pi$
$997$	−9649.69	−	5571.25i	−0.306528	−	0.176974i	−0.645372	+	0.763869i	$0.723298\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.a.521.3		16
3.2	odd	2	inner	1764.4.t.a.521.6		16
7.2	even	3	inner	1764.4.t.a.1097.4		16
7.3	odd	6		252.4.f.a.125.3	✓	8
7.4	even	3		252.4.f.a.125.6	yes	8
7.5	odd	6	inner	1764.4.t.a.1097.6		16
7.6	odd	2	inner	1764.4.t.a.521.5		16
21.2	odd	6	inner	1764.4.t.a.1097.5		16
21.5	even	6	inner	1764.4.t.a.1097.3		16
21.11	odd	6		252.4.f.a.125.4	yes	8
21.17	even	6		252.4.f.a.125.5	yes	8
21.20	even	2	inner	1764.4.t.a.521.4		16
28.3	even	6		1008.4.k.d.881.4		8
28.11	odd	6		1008.4.k.d.881.5		8
84.11	even	6		1008.4.k.d.881.3		8
84.59	odd	6		1008.4.k.d.881.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.f.a.125.3	✓	8	7.3	odd	6
252.4.f.a.125.4	yes	8	21.11	odd	6
252.4.f.a.125.5	yes	8	21.17	even	6
252.4.f.a.125.6	yes	8	7.4	even	3
1008.4.k.d.881.3		8	84.11	even	6
1008.4.k.d.881.4		8	28.3	even	6
1008.4.k.d.881.5		8	28.11	odd	6
1008.4.k.d.881.6		8	84.59	odd	6
1764.4.t.a.521.3		16	1.1	even	1	trivial
1764.4.t.a.521.4		16	21.20	even	2	inner
1764.4.t.a.521.5		16	7.6	odd	2	inner
1764.4.t.a.521.6		16	3.2	odd	2	inner
1764.4.t.a.1097.3		16	21.5	even	6	inner
1764.4.t.a.1097.4		16	7.2	even	3	inner
1764.4.t.a.1097.5		16	21.2	odd	6	inner
1764.4.t.a.1097.6		16	7.5	odd	6	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.10680 - 1.91704i) q^{5} +O(q^{10})\)	\(q+(-1.10680 - 1.91704i) q^{5} +(-17.6210 - 10.1735i) q^{11} -64.4624i q^{13} +(63.9455 - 110.757i) q^{17} +(-7.57978 + 4.37619i) q^{19} +(66.0596 - 38.1396i) q^{23} +(60.0500 - 104.010i) q^{25} +190.742i q^{29} +(143.078 + 82.6063i) q^{31} +(54.3250 + 94.0936i) q^{37} -312.838 q^{41} -139.875 q^{43} +(223.934 + 387.865i) q^{47} +(-460.473 - 265.854i) q^{53} +45.0402i q^{55} +(-138.959 + 240.684i) q^{59} +(-157.131 + 90.7195i) q^{61} +(-123.577 + 71.3471i) q^{65} +(88.4500 - 153.200i) q^{67} -259.314i q^{71} +(-423.316 - 244.401i) q^{73} +(-284.650 - 493.028i) q^{79} +1069.12 q^{83} -283.100 q^{85} +(-335.223 - 580.623i) q^{89} +(16.7786 + 9.68715i) q^{95} -526.369i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 40 q^{25} - 512 q^{37} + 64 q^{43} + 2336 q^{67} - 1792 q^{79} - 2688 q^{85}+O(q^{100}) \) 16 * q + 40 * q^25 - 512 * q^37 + 64 * q^43 + 2336 * q^67 - 1792 * q^79 - 2688 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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