LMFDB - Embedded newform 576.7.h.b.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(161,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.h (of order $2$, degree $1$, 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Character	$\chi$	$=$	576.161
Dual form			576.7.h.b.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-238.767 q^{5} +590.190 q^{7} +O(q^{10})$	$q-238.767 q^{5} +590.190 q^{7} -646.721 q^{11} +1796.55i q^{13} +4420.15i q^{17} -169.523i q^{19} -15636.4i q^{23} +41384.6 q^{25} -14520.1 q^{29} +8109.94 q^{31} -140918. q^{35} +81346.0i q^{37} -30885.6i q^{41} -92332.6i q^{43} -142425. i q^{47} +230675. q^{49} +183794. q^{53} +154416. q^{55} -18321.5 q^{59} +211001. i q^{61} -428958. i q^{65} -478057. i q^{67} +582966. i q^{71} -351679. q^{73} -381688. q^{77} +444742. q^{79} +870207. q^{83} -1.05539e6i q^{85} -57220.8i q^{89} +1.06031e6i q^{91} +40476.5i q^{95} -1.53212e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) $ 32 * q + 169888 * q^25 + 829792 * q^49 - 1493888 * q^73 - 15893248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−238.767	−1.91014	−0.955068	−	0.296388i	$-0.904218\pi$
$5$	−238.767	−1.91014	−0.955068	+	0.296388i	$0.904218\pi$
$6$	0	0
$7$	590.190	1.72067	0.860335	−	0.509728i	$-0.170254\pi$
$7$	590.190	1.72067	0.860335	+	0.509728i	$0.170254\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−646.721	−0.485891	−0.242946	−	0.970040i	$-0.578114\pi$
$11$	−646.721	−0.485891	−0.242946	+	0.970040i	$0.578114\pi$
$12$	0	0
$13$	1796.55i	0.817731i	0.912595	+	0.408865i	$0.134075\pi$
$13$	1796.55i	0.817731i	−0.912595	+	0.408865i	$0.865925\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4420.15i	0.899684i	0.893108	+	0.449842i	$0.148520\pi$
$17$	4420.15i	0.899684i	−0.893108	+	0.449842i	$0.851480\pi$
$18$	0	0
$19$	− 169.523i	− 0.0247154i	−0.999924	−	0.0123577i	$-0.996066\pi$
$19$	− 169.523i	− 0.0247154i	0.999924	−	0.0123577i	$-0.00393368\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 15636.4i	− 1.28515i	−0.766223	−	0.642575i	$-0.777866\pi$
$23$	− 15636.4i	− 1.28515i	0.766223	−	0.642575i	$-0.222134\pi$
$24$	0	0
$25$	41384.6	2.64862
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−14520.1	−0.595354	−0.297677	−	0.954667i	$-0.596212\pi$
$29$	−14520.1	−0.595354	−0.297677	+	0.954667i	$0.596212\pi$
$30$	0	0
$31$	8109.94	0.272228	0.136114	−	0.990693i	$-0.456539\pi$
$31$	8109.94	0.272228	0.136114	+	0.990693i	$0.456539\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−140918.	−3.28671
$36$	0	0
$37$	81346.0i	1.60595i	0.596015	+	0.802973i	$0.296750\pi$
$37$	81346.0i	1.60595i	−0.596015	+	0.802973i	$0.703250\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 30885.6i	− 0.448131i	−0.974574	−	0.224065i	$-0.928067\pi$
$41$	− 30885.6i	− 0.448131i	0.974574	−	0.224065i	$-0.0719329\pi$
$42$	0	0
$43$	− 92332.6i	− 1.16131i	−0.814148	−	0.580657i	$-0.802796\pi$
$43$	− 92332.6i	− 1.16131i	0.814148	−	0.580657i	$-0.197204\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 142425.i	− 1.37181i	−0.727692	−	0.685904i	$-0.759407\pi$
$47$	− 142425.i	− 1.37181i	0.727692	−	0.685904i	$-0.240593\pi$
$48$	0	0
$49$	230675.	1.96071
$50$	0	0
$51$	0	0
$52$	0	0
$53$	183794.	1.23454	0.617269	−	0.786752i	$-0.288239\pi$
$53$	183794.	1.23454	0.617269	+	0.786752i	$0.288239\pi$
$54$	0	0
$55$	154416.	0.928118
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−18321.5	−0.0892082	−0.0446041	−	0.999005i	$-0.514203\pi$
$59$	−18321.5	−0.0892082	−0.0446041	+	0.999005i	$0.514203\pi$
$60$	0	0
$61$	211001.i	0.929599i	0.885416	+	0.464800i	$0.153874\pi$
$61$	211001.i	0.929599i	−0.885416	+	0.464800i	$0.846126\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 428958.i	− 1.56198i
$66$	0	0
$67$	− 478057.i	− 1.58948i	−0.606950	−	0.794740i	$-0.707607\pi$
$67$	− 478057.i	− 1.58948i	0.606950	−	0.794740i	$-0.292393\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	582966.i	1.62880i	0.580303	+	0.814401i	$0.302934\pi$
$71$	582966.i	1.62880i	−0.580303	+	0.814401i	$0.697066\pi$
$72$	0	0
$73$	−351679.	−0.904019	−0.452009	−	0.892013i	$-0.649293\pi$
$73$	−351679.	−0.904019	−0.452009	+	0.892013i	$0.649293\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−381688.	−0.836059
$78$	0	0
$79$	444742.	0.902042	0.451021	−	0.892513i	$-0.351060\pi$
$79$	444742.	0.902042	0.451021	+	0.892513i	$0.351060\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	870207.	1.52191	0.760954	−	0.648806i	$-0.224731\pi$
$83$	870207.	1.52191	0.760954	+	0.648806i	$0.224731\pi$
$84$	0	0
$85$	− 1.05539e6i	− 1.71852i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 57220.8i	− 0.0811678i	−0.999176	−	0.0405839i	$-0.987078\pi$
$89$	− 57220.8i	− 0.0811678i	0.999176	−	0.0405839i	$-0.0129218\pi$
$90$	0	0
$91$	1.06031e6i	1.40705i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	40476.5i	0.0472098i
$96$	0	0
$97$	−1.53212e6	−1.67872	−0.839359	−	0.543577i	$-0.817070\pi$
$97$	−1.53212e6	−1.67872	−0.839359	+	0.543577i	$0.817070\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	193735.	0.188038	0.0940189	−	0.995570i	$-0.470029\pi$
$101$	193735.	0.188038	0.0940189	+	0.995570i	$0.470029\pi$
$102$	0	0
$103$	−1.68797e6	−1.54473	−0.772365	−	0.635179i	$-0.780926\pi$
$103$	−1.68797e6	−1.54473	−0.772365	+	0.635179i	$0.780926\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−347867.	−0.283963	−0.141982	−	0.989869i	$-0.545347\pi$
$107$	−347867.	−0.283963	−0.141982	+	0.989869i	$0.545347\pi$
$108$	0	0
$109$	1.51958e6i	1.17339i	0.809808	+	0.586695i	$0.199571\pi$
$109$	1.51958e6i	1.17339i	−0.809808	+	0.586695i	$0.800429\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.89084e6i	1.31044i	0.755436	+	0.655222i	$0.227425\pi$
$113$	1.89084e6i	1.31044i	−0.755436	+	0.655222i	$0.772575\pi$
$114$	0	0
$115$	3.73346e6i	2.45481i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.60873e6i	1.54806i
$120$	0	0
$121$	−1.35331e6	−0.763910
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.15055e6	−3.14908
$126$	0	0
$127$	2.05372e6	1.00260	0.501302	−	0.865272i	$-0.332854\pi$
$127$	2.05372e6	1.00260	0.501302	+	0.865272i	$0.332854\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.91575e6	−1.29699	−0.648495	−	0.761219i	$-0.724601\pi$
$131$	−2.91575e6	−1.29699	−0.648495	+	0.761219i	$0.724601\pi$
$132$	0	0
$133$	− 100051.i	− 0.0425271i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.98810e6i	0.773173i	0.922253	+	0.386587i	$0.126346\pi$
$137$	1.98810e6i	0.773173i	−0.922253	+	0.386587i	$0.873654\pi$
$138$	0	0
$139$	− 981342.i	− 0.365406i	−0.983168	−	0.182703i	$-0.941515\pi$
$139$	− 981342.i	− 0.365406i	0.983168	−	0.182703i	$-0.0584847\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.16187e6i	− 0.397328i
$144$	0	0
$145$	3.46692e6	1.13721
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.20226e6	0.665748	0.332874	−	0.942971i	$-0.391982\pi$
$149$	2.20226e6	0.665748	0.332874	+	0.942971i	$0.391982\pi$
$150$	0	0
$151$	−2.68049e6	−0.778544	−0.389272	−	0.921123i	$-0.627273\pi$
$151$	−2.68049e6	−0.778544	−0.389272	+	0.921123i	$0.627273\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.93638e6	−0.519992
$156$	0	0
$157$	6.24891e6i	1.61475i	0.590038	+	0.807375i	$0.299113\pi$
$157$	6.24891e6i	1.61475i	−0.590038	+	0.807375i	$0.700887\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 9.22846e6i	− 2.21132i
$162$	0	0
$163$	− 2.82972e6i	− 0.653402i	−0.945128	−	0.326701i	$-0.894063\pi$
$163$	− 2.82972e6i	− 0.653402i	0.945128	−	0.326701i	$-0.105937\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.24614e6i	1.55581i	0.628380	+	0.777906i	$0.283718\pi$
$167$	7.24614e6i	1.55581i	−0.628380	+	0.777906i	$0.716282\pi$
$168$	0	0
$169$	1.59920e6	0.331317
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.59195e6	−1.08000	−0.540002	−	0.841664i	$-0.681576\pi$
$173$	−5.59195e6	−1.08000	−0.540002	+	0.841664i	$0.681576\pi$
$174$	0	0
$175$	2.44248e7	4.55740
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.51518e6	−0.787256	−0.393628	−	0.919270i	$-0.628780\pi$
$179$	−4.51518e6	−0.787256	−0.393628	+	0.919270i	$0.628780\pi$
$180$	0	0
$181$	4.61113e6i	0.777628i	0.921316	+	0.388814i	$0.127115\pi$
$181$	4.61113e6i	0.777628i	−0.921316	+	0.388814i	$0.872885\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.94227e7i	− 3.06758i
$186$	0	0
$187$	− 2.85860e6i	− 0.437149i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.24952e6i	0.322841i	0.986886	+	0.161421i	$0.0516076\pi$
$191$	2.24952e6i	0.322841i	−0.986886	+	0.161421i	$0.948392\pi$
$192$	0	0
$193$	−1.05850e7	−1.47237	−0.736185	−	0.676780i	$-0.763375\pi$
$193$	−1.05850e7	−1.47237	−0.736185	+	0.676780i	$0.763375\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.10527e6	0.144568	0.0722839	−	0.997384i	$-0.476971\pi$
$197$	1.10527e6	0.144568	0.0722839	+	0.997384i	$0.476971\pi$
$198$	0	0
$199$	−6.54586e6	−0.830629	−0.415315	−	0.909678i	$-0.636329\pi$
$199$	−6.54586e6	−0.830629	−0.415315	+	0.909678i	$0.636329\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.56962e6	−1.02441
$204$	0	0
$205$	7.37446e6i	0.855990i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	109634.i	0.0120090i
$210$	0	0
$211$	− 1.26962e7i	− 1.35153i	−0.737118	−	0.675764i	$-0.763814\pi$
$211$	− 1.26962e7i	− 1.35153i	0.737118	−	0.675764i	$-0.236186\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.20460e7i	2.21827i
$216$	0	0
$217$	4.78641e6	0.468414
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.94104e6	−0.735699
$222$	0	0
$223$	9.29853e6	0.838494	0.419247	−	0.907872i	$-0.362294\pi$
$223$	9.29853e6	0.838494	0.419247	+	0.907872i	$0.362294\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.18334e6	0.528623	0.264311	−	0.964437i	$-0.414855\pi$
$227$	6.18334e6	0.528623	0.264311	+	0.964437i	$0.414855\pi$
$228$	0	0
$229$	5.94060e6i	0.494680i	0.968929	+	0.247340i	$0.0795564\pi$
$229$	5.94060e6i	0.494680i	−0.968929	+	0.247340i	$0.920444\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.90941e6i	0.467172i	0.972336	+	0.233586i	$0.0750460\pi$
$233$	5.90941e6i	0.467172i	−0.972336	+	0.233586i	$0.924954\pi$
$234$	0	0
$235$	3.40064e7i	2.62034i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.90667e6i	0.432662i	0.976320	+	0.216331i	$0.0694091\pi$
$239$	5.90667e6i	0.432662i	−0.976320	+	0.216331i	$0.930591\pi$
$240$	0	0
$241$	−6.73503e6	−0.481159	−0.240579	−	0.970629i	$-0.577337\pi$
$241$	−6.73503e6	−0.481159	−0.240579	+	0.970629i	$0.577337\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.50776e7	−3.74522
$246$	0	0
$247$	304558.	0.0202106
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.54145e7	−0.974781	−0.487391	−	0.873184i	$-0.662051\pi$
$251$	−1.54145e7	−0.974781	−0.487391	+	0.873184i	$0.662051\pi$
$252$	0	0
$253$	1.01124e7i	0.624443i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.28205e6i	0.546820i	0.961898	+	0.273410i	$0.0881516\pi$
$257$	9.28205e6i	0.546820i	−0.961898	+	0.273410i	$0.911848\pi$
$258$	0	0
$259$	4.80096e7i	2.76331i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.21452e7i	1.76705i	0.468384	+	0.883525i	$0.344836\pi$
$263$	3.21452e7i	1.76705i	−0.468384	+	0.883525i	$0.655164\pi$
$264$	0	0
$265$	−4.38840e7	−2.35814
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.49064e6	−0.282076	−0.141038	−	0.990004i	$-0.545044\pi$
$269$	−5.49064e6	−0.282076	−0.141038	+	0.990004i	$0.545044\pi$
$270$	0	0
$271$	1.29788e6	0.0652117	0.0326058	−	0.999468i	$-0.489619\pi$
$271$	1.29788e6	0.0652117	0.0326058	+	0.999468i	$0.489619\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.67643e7	−1.28694
$276$	0	0
$277$	1.86711e7i	0.878477i	0.898370	+	0.439239i	$0.144752\pi$
$277$	1.86711e7i	0.878477i	−0.898370	+	0.439239i	$0.855248\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.05185e7i	1.37545i	0.725972	+	0.687724i	$0.241390\pi$
$281$	3.05185e7i	1.37545i	−0.725972	+	0.687724i	$0.758610\pi$
$282$	0	0
$283$	− 2.64892e6i	− 0.116872i	−0.998291	−	0.0584359i	$-0.981389\pi$
$283$	− 2.64892e6i	− 0.116872i	0.998291	−	0.0584359i	$-0.0186113\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.82284e7i	− 0.771085i
$288$	0	0
$289$	4.59985e6	0.190568
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.07293e7	−0.824103	−0.412051	−	0.911161i	$-0.635188\pi$
$293$	−2.07293e7	−0.824103	−0.412051	+	0.911161i	$0.635188\pi$
$294$	0	0
$295$	4.37457e6	0.170400
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.80917e7	1.05091
$300$	0	0
$301$	− 5.44938e7i	− 1.99824i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.03801e7i	− 1.77566i
$306$	0	0
$307$	2.07090e7i	0.715720i	0.933775	+	0.357860i	$0.116494\pi$
$307$	2.07090e7i	0.715720i	−0.933775	+	0.357860i	$0.883506\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.95055e7i	0.648449i	0.945980	+	0.324224i	$0.105103\pi$
$311$	1.95055e7i	0.648449i	−0.945980	+	0.324224i	$0.894897\pi$
$312$	0	0
$313$	9.28299e6	0.302730	0.151365	−	0.988478i	$-0.451633\pi$
$313$	9.28299e6	0.302730	0.151365	+	0.988478i	$0.451633\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.10295e6	0.0660164	0.0330082	−	0.999455i	$-0.489491\pi$
$317$	2.10295e6	0.0660164	0.0330082	+	0.999455i	$0.489491\pi$
$318$	0	0
$319$	9.39046e6	0.289277
$320$	0	0
$321$	0	0
$322$	0	0
$323$	749318.	0.0222361
$324$	0	0
$325$	7.43497e7i	2.16586i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.40579e7i	− 2.36043i
$330$	0	0
$331$	− 4.40335e7i	− 1.21422i	−0.794616	−	0.607112i	$-0.792328\pi$
$331$	− 4.40335e7i	− 1.21422i	0.794616	−	0.607112i	$-0.207672\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.14144e8i	3.03612i
$336$	0	0
$337$	−3.26403e7	−0.852834	−0.426417	−	0.904527i	$-0.640224\pi$
$337$	−3.26403e7	−0.852834	−0.426417	+	0.904527i	$0.640224\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.24487e6	−0.132273
$342$	0	0
$343$	6.67070e7	1.65306
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.69657e7	−1.60274	−0.801372	−	0.598166i	$-0.795896\pi$
$347$	−6.69657e7	−1.60274	−0.801372	+	0.598166i	$0.795896\pi$
$348$	0	0
$349$	4.33353e7i	1.01945i	0.860338	+	0.509725i	$0.170253\pi$
$349$	4.33353e7i	1.01945i	−0.860338	+	0.509725i	$0.829747\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.57632e7i	0.585700i	0.956158	+	0.292850i	$0.0946037\pi$
$353$	2.57632e7i	0.585700i	−0.956158	+	0.292850i	$0.905396\pi$
$354$	0	0
$355$	− 1.39193e8i	− 3.11123i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 8.26701e7i	− 1.78676i	−0.449307	−	0.893378i	$-0.648329\pi$
$359$	− 8.26701e7i	− 1.78676i	0.449307	−	0.893378i	$-0.351671\pi$
$360$	0	0
$361$	4.70171e7	0.999389
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.39692e7	1.72680
$366$	0	0
$367$	1.13644e7	0.229905	0.114953	−	0.993371i	$-0.463328\pi$
$367$	1.13644e7	0.229905	0.114953	+	0.993371i	$0.463328\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.08474e8	2.12423
$372$	0	0
$373$	− 3.83544e7i	− 0.739076i	−0.929216	−	0.369538i	$-0.879516\pi$
$373$	− 3.83544e7i	− 0.739076i	0.929216	−	0.369538i	$-0.120484\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.60861e7i	− 0.486840i
$378$	0	0
$379$	2.78439e7i	0.511461i	0.966748	+	0.255730i	$0.0823159\pi$
$379$	2.78439e7i	0.511461i	−0.966748	+	0.255730i	$0.917684\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.38593e7i	− 1.49264i	−0.665588	−	0.746320i	$-0.731819\pi$
$383$	− 8.38593e7i	− 1.49264i	0.665588	−	0.746320i	$-0.268181\pi$
$384$	0	0
$385$	9.11346e7	1.59699
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−507274.	−0.00861775	−0.00430887	−	0.999991i	$-0.501372\pi$
$389$	−507274.	−0.00861775	−0.00430887	+	0.999991i	$0.501372\pi$
$390$	0	0
$391$	6.91153e7	1.15623
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.06190e8	−1.72302
$396$	0	0
$397$	− 3.10540e6i	− 0.0496302i	−0.999692	−	0.0248151i	$-0.992100\pi$
$397$	− 3.10540e6i	− 0.0496302i	0.999692	−	0.0248151i	$-0.00789970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6.40669e7i	− 0.993574i	−0.867872	−	0.496787i	$-0.834513\pi$
$401$	− 6.40669e7i	− 0.993574i	0.867872	−	0.496787i	$-0.165487\pi$
$402$	0	0
$403$	1.45699e7i	0.222609i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.26082e7i	− 0.780315i
$408$	0	0
$409$	−1.79661e7	−0.262594	−0.131297	−	0.991343i	$-0.541914\pi$
$409$	−1.79661e7	−0.262594	−0.131297	+	0.991343i	$0.541914\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.08132e7	−0.153498
$414$	0	0
$415$	−2.07777e8	−2.90705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.91661e7	−1.34810	−0.674049	−	0.738687i	$-0.735446\pi$
$419$	−9.91661e7	−1.34810	−0.674049	+	0.738687i	$0.735446\pi$
$420$	0	0
$421$	− 7.40801e6i	− 0.0992785i	−0.998767	−	0.0496393i	$-0.984193\pi$
$421$	− 7.40801e6i	− 0.0992785i	0.998767	−	0.0496393i	$-0.0158072\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.82926e8i	2.38292i
$426$	0	0
$427$	1.24531e8i	1.59953i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.30488e8i	1.62981i	0.579593	+	0.814906i	$0.303211\pi$
$431$	1.30488e8i	1.62981i	−0.579593	+	0.814906i	$0.696789\pi$
$432$	0	0
$433$	2.50725e7	0.308841	0.154420	−	0.988005i	$-0.450649\pi$
$433$	2.50725e7	0.308841	0.154420	+	0.988005i	$0.450649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.65074e6	−0.0317630
$438$	0	0
$439$	3.81651e6	0.0451100	0.0225550	−	0.999746i	$-0.492820\pi$
$439$	3.81651e6	0.0451100	0.0225550	+	0.999746i	$0.492820\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.30818e8	1.50472	0.752359	−	0.658753i	$-0.228916\pi$
$443$	1.30818e8	1.50472	0.752359	+	0.658753i	$0.228916\pi$
$444$	0	0
$445$	1.36624e7i	0.155042i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.11063e8i	1.22696i	0.789711	+	0.613479i	$0.210231\pi$
$449$	1.11063e8i	1.22696i	−0.789711	+	0.613479i	$0.789769\pi$
$450$	0	0
$451$	1.99744e7i	0.217743i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 2.53167e8i	− 2.68765i
$456$	0	0
$457$	8.34023e6	0.0873835	0.0436918	−	0.999045i	$-0.486088\pi$
$457$	8.34023e6	0.0873835	0.0436918	+	0.999045i	$0.486088\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.45708e8	−1.48724	−0.743622	−	0.668601i	$-0.766894\pi$
$461$	−1.45708e8	−1.48724	−0.743622	+	0.668601i	$0.766894\pi$
$462$	0	0
$463$	−1.00372e8	−1.01128	−0.505639	−	0.862745i	$-0.668743\pi$
$463$	−1.00372e8	−1.01128	−0.505639	+	0.862745i	$0.668743\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.22734e7	0.709623	0.354812	−	0.934938i	$-0.384545\pi$
$467$	7.22734e7	0.709623	0.354812	+	0.934938i	$0.384545\pi$
$468$	0	0
$469$	− 2.82144e8i	− 2.73497i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.97134e7i	0.564272i
$474$	0	0
$475$	− 7.01566e6i	− 0.0654617i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 3.88182e7i	− 0.353206i	−0.984282	−	0.176603i	$-0.943489\pi$
$479$	− 3.88182e7i	− 0.353206i	0.984282	−	0.176603i	$-0.0565109\pi$
$480$	0	0
$481$	−1.46143e8	−1.31323
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.65820e8	3.20658
$486$	0	0
$487$	−8.98605e7	−0.778004	−0.389002	−	0.921237i	$-0.627180\pi$
$487$	−8.98605e7	−0.778004	−0.389002	+	0.921237i	$0.627180\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.45638e8	1.23036	0.615179	−	0.788388i	$-0.289084\pi$
$491$	1.45638e8	1.23036	0.615179	+	0.788388i	$0.289084\pi$
$492$	0	0
$493$	− 6.41810e7i	− 0.535631i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.44061e8i	2.80263i
$498$	0	0
$499$	1.51012e7i	0.121537i	0.998152	+	0.0607687i	$0.0193552\pi$
$499$	1.51012e7i	0.121537i	−0.998152	+	0.0607687i	$0.980645\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.85483e7i	0.381479i	0.981641	+	0.190739i	$0.0610886\pi$
$503$	4.85483e7i	0.381479i	−0.981641	+	0.190739i	$0.938911\pi$
$504$	0	0
$505$	−4.62576e7	−0.359178
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.05154e8	0.797394	0.398697	−	0.917083i	$-0.369463\pi$
$509$	1.05154e8	0.797394	0.398697	+	0.917083i	$0.369463\pi$
$510$	0	0
$511$	−2.07557e8	−1.55552
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.03031e8	2.95064
$516$	0	0
$517$	9.21094e7i	0.666549i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 2.07206e8i	− 1.46517i	−0.680674	−	0.732587i	$-0.738313\pi$
$521$	− 2.07206e8i	− 1.46517i	0.680674	−	0.732587i	$-0.261687\pi$
$522$	0	0
$523$	− 8.33420e7i	− 0.582585i	−0.956634	−	0.291292i	$-0.905915\pi$
$523$	− 8.33420e7i	− 0.582585i	0.956634	−	0.291292i	$-0.0940852\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.58471e7i	0.244919i
$528$	0	0
$529$	−9.64616e7	−0.651610
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.54877e7	0.366450
$534$	0	0
$535$	8.30591e7	0.542408
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.49183e8	−0.952691
$540$	0	0
$541$	− 9.97551e7i	− 0.630004i	−0.949091	−	0.315002i	$-0.897995\pi$
$541$	− 9.97551e7i	− 0.630004i	0.949091	−	0.315002i	$-0.102005\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.62824e8i	− 2.24134i
$546$	0	0
$547$	− 1.52135e8i	− 0.929539i	−0.885432	−	0.464770i	$-0.846137\pi$
$547$	− 1.52135e8i	− 0.929539i	0.885432	−	0.464770i	$-0.153863\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.46149e6i	0.0147144i
$552$	0	0
$553$	2.62482e8	1.55212
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.69842e7	−0.214018	−0.107009	−	0.994258i	$-0.534127\pi$
$557$	−3.69842e7	−0.214018	−0.107009	+	0.994258i	$0.534127\pi$
$558$	0	0
$559$	1.65881e8	0.949642
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.43613e7	0.472735	0.236368	−	0.971664i	$-0.424043\pi$
$563$	8.43613e7	0.472735	0.236368	+	0.971664i	$0.424043\pi$
$564$	0	0
$565$	− 4.51469e8i	− 2.50313i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.70924e7i	0.472763i	0.971660	+	0.236382i	$0.0759615\pi$
$569$	8.70924e7i	0.472763i	−0.971660	+	0.236382i	$0.924038\pi$
$570$	0	0
$571$	− 7.09499e7i	− 0.381104i	−0.981677	−	0.190552i	$-0.938972\pi$
$571$	− 7.09499e7i	− 0.381104i	0.981677	−	0.190552i	$-0.0610278\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 6.47107e8i	− 3.40387i
$576$	0	0
$577$	1.73063e8	0.900900	0.450450	−	0.892802i	$-0.351263\pi$
$577$	1.73063e8	0.900900	0.450450	+	0.892802i	$0.351263\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.13588e8	2.61870
$582$	0	0
$583$	−1.18864e8	−0.599851
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.04359e8	−1.50478	−0.752388	−	0.658721i	$-0.771098\pi$
$587$	−3.04359e8	−1.50478	−0.752388	+	0.658721i	$0.771098\pi$
$588$	0	0
$589$	− 1.37482e6i	− 0.00672823i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.45408e7i	0.213596i	0.994281	+	0.106798i	$0.0340599\pi$
$593$	4.45408e7i	0.213596i	−0.994281	+	0.106798i	$0.965940\pi$
$594$	0	0
$595$	− 6.22878e8i	− 2.95701i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.27363e8i	− 0.592604i	−0.955094	−	0.296302i	$-0.904247\pi$
$599$	− 1.27363e8i	− 0.592604i	0.955094	−	0.296302i	$-0.0957535\pi$
$600$	0	0
$601$	−3.64008e7	−0.167683	−0.0838413	−	0.996479i	$-0.526719\pi$
$601$	−3.64008e7	−0.167683	−0.0838413	+	0.996479i	$0.526719\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.23126e8	1.45917
$606$	0	0
$607$	−2.47832e8	−1.10813	−0.554066	−	0.832473i	$-0.686924\pi$
$607$	−2.47832e8	−1.10813	−0.554066	+	0.832473i	$0.686924\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.55875e8	1.12177
$612$	0	0
$613$	4.19790e8i	1.82243i	0.411930	+	0.911216i	$0.364855\pi$
$613$	4.19790e8i	1.82243i	−0.411930	+	0.911216i	$0.635145\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.11993e8i	1.32828i	0.747608	+	0.664140i	$0.231202\pi$
$617$	3.11993e8i	1.32828i	−0.747608	+	0.664140i	$0.768798\pi$
$618$	0	0
$619$	1.79424e8i	0.756499i	0.925704	+	0.378249i	$0.123474\pi$
$619$	1.79424e8i	0.756499i	−0.925704	+	0.378249i	$0.876526\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.37711e7i	− 0.139663i
$624$	0	0
$625$	8.21913e8	3.36656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.59562e8	−1.44485
$630$	0	0
$631$	4.03924e7	0.160772	0.0803861	−	0.996764i	$-0.474385\pi$
$631$	4.03924e7	0.160772	0.0803861	+	0.996764i	$0.474385\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.90360e8	−1.91511
$636$	0	0
$637$	4.14421e8i	1.60333i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.28385e7i	− 0.0487461i	−0.999703	−	0.0243731i	$-0.992241\pi$
$641$	− 1.28385e7i	− 0.0487461i	0.999703	−	0.0243731i	$-0.00775896\pi$
$642$	0	0
$643$	1.66008e8i	0.624448i	0.950009	+	0.312224i	$0.101074\pi$
$643$	1.66008e8i	0.624448i	−0.950009	+	0.312224i	$0.898926\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.06615e7i	− 0.260898i	−0.991455	−	0.130449i	$-0.958358\pi$
$647$	− 7.06615e7i	− 0.260898i	0.991455	−	0.130449i	$-0.0416418\pi$
$648$	0	0
$649$	1.18489e7	0.0433455
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.41532e8	−0.508295	−0.254148	−	0.967165i	$-0.581795\pi$
$653$	−1.41532e8	−0.508295	−0.254148	+	0.967165i	$0.581795\pi$
$654$	0	0
$655$	6.96185e8	2.47743
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.53876e6	0.0298358	0.0149179	−	0.999889i	$-0.495251\pi$
$659$	8.53876e6	0.0298358	0.0149179	+	0.999889i	$0.495251\pi$
$660$	0	0
$661$	− 1.15189e8i	− 0.398846i	−0.979913	−	0.199423i	$-0.936093\pi$
$661$	− 1.15189e8i	− 0.398846i	0.979913	−	0.199423i	$-0.0639068\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.38889e7i	0.0812326i
$666$	0	0
$667$	2.27042e8i	0.765120i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.36459e8i	− 0.451684i
$672$	0	0
$673$	−2.13024e8	−0.698849	−0.349424	−	0.936965i	$-0.613623\pi$
$673$	−2.13024e8	−0.698849	−0.349424	+	0.936965i	$0.613623\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.72624e8	−0.556333	−0.278166	−	0.960533i	$-0.589727\pi$
$677$	−1.72624e8	−0.556333	−0.278166	+	0.960533i	$0.589727\pi$
$678$	0	0
$679$	−9.04243e8	−2.88852
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9.02634e7	−0.283302	−0.141651	−	0.989917i	$-0.545241\pi$
$683$	−9.02634e7	−0.283302	−0.141651	+	0.989917i	$0.545241\pi$
$684$	0	0
$685$	− 4.74693e8i	− 1.47687i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.30196e8i	1.00952i
$690$	0	0
$691$	− 2.54280e8i	− 0.770688i	−0.922773	−	0.385344i	$-0.874083\pi$
$691$	− 2.54280e8i	− 0.770688i	0.922773	−	0.385344i	$-0.125917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.34312e8i	0.697976i
$696$	0	0
$697$	1.36519e8	0.403176
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.04081e8	1.75364	0.876821	−	0.480817i	$-0.159660\pi$
$701$	6.04081e8	1.75364	0.876821	+	0.480817i	$0.159660\pi$
$702$	0	0
$703$	1.37900e7	0.0396917
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.14341e8	0.323551
$708$	0	0
$709$	2.59938e8i	0.729341i	0.931137	+	0.364670i	$0.118818\pi$
$709$	2.59938e8i	0.729341i	−0.931137	+	0.364670i	$0.881182\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.26810e8i	− 0.349853i
$714$	0	0
$715$	2.77416e8i	0.758950i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.66558e7i	0.233137i	0.993183	+	0.116568i	$0.0371895\pi$
$719$	8.66558e7i	0.233137i	−0.993183	+	0.116568i	$0.962811\pi$
$720$	0	0
$721$	−9.96222e8	−2.65797
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.00909e8	−1.57687
$726$	0	0
$727$	−5.70508e8	−1.48477	−0.742384	−	0.669974i	$-0.766305\pi$
$727$	−5.70508e8	−1.48477	−0.742384	+	0.669974i	$0.766305\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.08124e8	1.04482
$732$	0	0
$733$	− 3.40430e8i	− 0.864402i	−0.901777	−	0.432201i	$-0.857737\pi$
$733$	− 3.40430e8i	− 0.864402i	0.901777	−	0.432201i	$-0.142263\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.09169e8i	0.772314i
$738$	0	0
$739$	− 9.72463e6i	− 0.0240957i	−0.999927	−	0.0120479i	$-0.996165\pi$
$739$	− 9.72463e6i	− 0.0240957i	0.999927	−	0.0120479i	$-0.00383504\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.76673e8i	1.40593i	0.711225	+	0.702964i	$0.248141\pi$
$743$	5.76673e8i	1.40593i	−0.711225	+	0.702964i	$0.751859\pi$
$744$	0	0
$745$	−5.25827e8	−1.27167
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.05308e8	−0.488607
$750$	0	0
$751$	7.70108e8	1.81816	0.909080	−	0.416622i	$-0.136786\pi$
$751$	7.70108e8	1.81816	0.909080	+	0.416622i	$0.136786\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.40012e8	1.48712
$756$	0	0
$757$	− 6.67412e8i	− 1.53853i	−0.638929	−	0.769266i	$-0.720622\pi$
$757$	− 6.67412e8i	− 1.53853i	0.638929	−	0.769266i	$-0.279378\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 4.38027e8i	− 0.993909i	−0.867777	−	0.496954i	$-0.834452\pi$
$761$	− 4.38027e8i	− 0.993909i	0.867777	−	0.496954i	$-0.165548\pi$
$762$	0	0
$763$	8.96838e8i	2.01902i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.29156e7i	− 0.0729483i
$768$	0	0
$769$	−2.45896e8	−0.540719	−0.270360	−	0.962759i	$-0.587143\pi$
$769$	−2.45896e8	−0.540719	−0.270360	+	0.962759i	$0.587143\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.54246e8	−0.550447	−0.275224	−	0.961380i	$-0.588752\pi$
$773$	−2.54246e8	−0.550447	−0.275224	+	0.961380i	$0.588752\pi$
$774$	0	0
$775$	3.35627e8	0.721027
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.23583e6	−0.0110757
$780$	0	0
$781$	− 3.77016e8i	− 0.791420i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.49203e9i	− 3.08439i
$786$	0	0
$787$	6.65529e8i	1.36535i	0.730724	+	0.682673i	$0.239183\pi$
$787$	6.65529e8i	1.36535i	−0.730724	+	0.682673i	$0.760817\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.11595e9i	2.25484i
$792$	0	0
$793$	−3.79075e8	−0.760162
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.36922e8	−0.270456	−0.135228	−	0.990815i	$-0.543177\pi$
$797$	−1.36922e8	−0.270456	−0.135228	+	0.990815i	$0.543177\pi$
$798$	0	0
$799$	6.29541e8	1.23419
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.27438e8	0.439255
$804$	0	0
$805$	2.20345e9i	4.22392i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.17033e8i	0.787634i	0.919189	+	0.393817i	$0.128846\pi$
$809$	4.17033e8i	0.787634i	−0.919189	+	0.393817i	$0.871154\pi$
$810$	0	0
$811$	− 3.01202e8i	− 0.564670i	−0.959316	−	0.282335i	$-0.908891\pi$
$811$	− 3.01202e8i	− 0.564670i	0.959316	−	0.282335i	$-0.0911090\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.75643e8i	1.24809i
$816$	0	0
$817$	−1.56525e7	−0.0287024
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.68430e8	−0.665771	−0.332886	−	0.942967i	$-0.608022\pi$
$821$	−3.68430e8	−0.665771	−0.332886	+	0.942967i	$0.608022\pi$
$822$	0	0
$823$	8.03918e7	0.144216	0.0721078	−	0.997397i	$-0.477027\pi$
$823$	8.03918e7	0.144216	0.0721078	+	0.997397i	$0.477027\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.03268e9	−1.82578	−0.912891	−	0.408203i	$-0.866155\pi$
$827$	−1.03268e9	−1.82578	−0.912891	+	0.408203i	$0.866155\pi$
$828$	0	0
$829$	1.25526e7i	0.0220329i	0.999939	+	0.0110164i	$0.00350671\pi$
$829$	1.25526e7i	0.0220329i	−0.999939	+	0.0110164i	$0.996493\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.01962e9i	1.76402i
$834$	0	0
$835$	− 1.73014e9i	− 2.97181i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 4.43019e8i	− 0.750129i	−0.926999	−	0.375065i	$-0.877620\pi$
$839$	− 4.43019e8i	− 0.750129i	0.926999	−	0.375065i	$-0.122380\pi$
$840$	0	0
$841$	−3.83990e8	−0.645553
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.81837e8	−0.632860
$846$	0	0
$847$	−7.98712e8	−1.31444
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.27196e9	2.06388
$852$	0	0
$853$	− 6.34252e8i	− 1.02192i	−0.859606	−	0.510958i	$-0.829291\pi$
$853$	− 6.34252e8i	− 1.02192i	0.859606	−	0.510958i	$-0.170709\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.13764e8i	− 0.339620i	−0.985477	−	0.169810i	$-0.945685\pi$
$857$	− 2.13764e8i	− 0.339620i	0.985477	−	0.169810i	$-0.0543154\pi$
$858$	0	0
$859$	8.67632e8i	1.36885i	0.729083	+	0.684425i	$0.239947\pi$
$859$	8.67632e8i	1.36885i	−0.729083	+	0.684425i	$0.760053\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 6.24048e7i	− 0.0970924i	−0.998821	−	0.0485462i	$-0.984541\pi$
$863$	− 6.24048e7i	− 0.0970924i	0.998821	−	0.0485462i	$-0.0154588\pi$
$864$	0	0
$865$	1.33517e9	2.06295
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.87624e8	−0.438294
$870$	0	0
$871$	8.58855e8	1.29977
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.62999e9	−5.41853
$876$	0	0
$877$	4.21172e8i	0.624397i	0.950017	+	0.312198i	$0.101065\pi$
$877$	4.21172e8i	0.624397i	−0.950017	+	0.312198i	$0.898935\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.66452e8i	− 1.12087i	−0.828197	−	0.560437i	$-0.810633\pi$
$881$	− 7.66452e8i	− 1.12087i	0.828197	−	0.560437i	$-0.189367\pi$
$882$	0	0
$883$	− 2.73414e8i	− 0.397136i	−0.980087	−	0.198568i	$-0.936371\pi$
$883$	− 2.73414e8i	− 0.397136i	0.980087	−	0.198568i	$-0.0636290\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.87773e7i	0.0985541i	0.998785	+	0.0492770i	$0.0156917\pi$
$887$	6.87773e7i	0.0985541i	−0.998785	+	0.0492770i	$0.984308\pi$
$888$	0	0
$889$	1.21208e9	1.72515
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.41444e7	−0.0339048
$894$	0	0
$895$	1.07808e9	1.50377
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.17757e8	−0.162072
$900$	0	0
$901$	8.12398e8i	1.11069i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.10099e9i	− 1.48537i
$906$	0	0
$907$	1.04981e9i	1.40699i	0.710701	+	0.703494i	$0.248378\pi$
$907$	1.04981e9i	1.40699i	−0.710701	+	0.703494i	$0.751622\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 5.13949e6i	− 0.00679774i	−0.999994	−	0.00339887i	$-0.998918\pi$
$911$	− 5.13949e6i	− 0.00679774i	0.999994	−	0.00339887i	$-0.00108190\pi$
$912$	0	0
$913$	−5.62781e8	−0.739482
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.72085e9	−2.23169
$918$	0	0
$919$	3.64964e8	0.470223	0.235111	−	0.971968i	$-0.424454\pi$
$919$	3.64964e8	0.470223	0.235111	+	0.971968i	$0.424454\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.04733e9	−1.33192
$924$	0	0
$925$	3.36648e9i	4.25354i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 6.21554e8i	− 0.775232i	−0.921821	−	0.387616i	$-0.873299\pi$
$929$	− 6.21554e8i	− 0.775232i	0.921821	−	0.387616i	$-0.126701\pi$
$930$	0	0
$931$	− 3.91048e7i	− 0.0484598i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.82540e8i	0.835013i
$936$	0	0
$937$	8.53346e8	1.03730	0.518652	−	0.854985i	$-0.326434\pi$
$937$	8.53346e8	1.03730	0.518652	+	0.854985i	$0.326434\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−7.53422e8	−0.904211	−0.452105	−	0.891965i	$-0.649327\pi$
$941$	−7.53422e8	−0.904211	−0.452105	+	0.891965i	$0.649327\pi$
$942$	0	0
$943$	−4.82940e8	−0.575915
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.47389e9	−1.73547	−0.867734	−	0.497030i	$-0.834424\pi$
$947$	−1.47389e9	−1.73547	−0.867734	+	0.497030i	$0.834424\pi$
$948$	0	0
$949$	− 6.31810e8i	− 0.739244i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.95131e8i	− 0.340986i	−0.985359	−	0.170493i	$-0.945464\pi$
$953$	− 2.95131e8i	− 0.340986i	0.985359	−	0.170493i	$-0.0545360\pi$
$954$	0	0
$955$	− 5.37110e8i	− 0.616670i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.17336e9i	1.33038i
$960$	0	0
$961$	−8.21733e8	−0.925892
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.52734e9	2.81243
$966$	0	0
$967$	1.53482e9	1.69738	0.848688	−	0.528893i	$-0.177393\pi$
$967$	1.53482e9	1.69738	0.848688	+	0.528893i	$0.177393\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.12558e9	1.22947	0.614733	−	0.788735i	$-0.289264\pi$
$971$	1.12558e9	1.22947	0.614733	+	0.788735i	$0.289264\pi$
$972$	0	0
$973$	− 5.79179e8i	− 0.628744i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.41886e9i	1.52144i	0.649081	+	0.760720i	$0.275154\pi$
$977$	1.41886e9i	1.52144i	−0.649081	+	0.760720i	$0.724846\pi$
$978$	0	0
$979$	3.70059e7i	0.0394387i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.60096e9i	− 1.68547i	−0.538329	−	0.842735i	$-0.680944\pi$
$983$	− 1.60096e9i	− 1.68547i	0.538329	−	0.842735i	$-0.319056\pi$
$984$	0	0
$985$	−2.63903e8	−0.276144
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.44375e9	−1.49246
$990$	0	0
$991$	−5.85578e8	−0.601678	−0.300839	−	0.953675i	$-0.597267\pi$
$991$	−5.85578e8	−0.601678	−0.300839	+	0.953675i	$0.597267\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.56293e9	1.58661
$996$	0	0
$997$	− 8.06481e8i	− 0.813783i	−0.913477	−	0.406891i	$-0.866613\pi$
$997$	− 8.06481e8i	− 0.813783i	0.913477	−	0.406891i	$-0.133387\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.h.b.161.4	yes	32
3.2	odd	2	inner	576.7.h.b.161.31	yes	32
4.3	odd	2	inner	576.7.h.b.161.2	yes	32
8.3	odd	2	inner	576.7.h.b.161.29	yes	32
8.5	even	2	inner	576.7.h.b.161.30	yes	32
12.11	even	2	inner	576.7.h.b.161.32	yes	32
24.5	odd	2	inner	576.7.h.b.161.1	✓	32
24.11	even	2	inner	576.7.h.b.161.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.7.h.b.161.1	✓	32	24.5	odd	2	inner
576.7.h.b.161.2	yes	32	4.3	odd	2	inner
576.7.h.b.161.3	yes	32	24.11	even	2	inner
576.7.h.b.161.4	yes	32	1.1	even	1	trivial
576.7.h.b.161.29	yes	32	8.3	odd	2	inner
576.7.h.b.161.30	yes	32	8.5	even	2	inner
576.7.h.b.161.31	yes	32	3.2	odd	2	inner
576.7.h.b.161.32	yes	32	12.11	even	2	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-238.767 q^{5} +590.190 q^{7} +O(q^{10})\)	\(q-238.767 q^{5} +590.190 q^{7} -646.721 q^{11} +1796.55i q^{13} +4420.15i q^{17} -169.523i q^{19} -15636.4i q^{23} +41384.6 q^{25} -14520.1 q^{29} +8109.94 q^{31} -140918. q^{35} +81346.0i q^{37} -30885.6i q^{41} -92332.6i q^{43} -142425. i q^{47} +230675. q^{49} +183794. q^{53} +154416. q^{55} -18321.5 q^{59} +211001. i q^{61} -428958. i q^{65} -478057. i q^{67} +582966. i q^{71} -351679. q^{73} -381688. q^{77} +444742. q^{79} +870207. q^{83} -1.05539e6i q^{85} -57220.8i q^{89} +1.06031e6i q^{91} +40476.5i q^{95} -1.53212e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) \) 32 * q + 169888 * q^25 + 829792 * q^49 - 1493888 * q^73 - 15893248 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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