LMFDB - Embedded newform 2352.4.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.a (trivial)

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:	yes
Analytic conductor:	$138.772492334$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 588)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +4.00000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +4.00000 q^{5} +9.00000 q^{9} +20.0000 q^{11} -4.00000 q^{13} +12.0000 q^{15} +24.0000 q^{17} -44.0000 q^{19} -72.0000 q^{23} -109.000 q^{25} +27.0000 q^{27} -38.0000 q^{29} -184.000 q^{31} +60.0000 q^{33} -30.0000 q^{37} -12.0000 q^{39} -216.000 q^{41} +164.000 q^{43} +36.0000 q^{45} -520.000 q^{47} +72.0000 q^{51} -146.000 q^{53} +80.0000 q^{55} -132.000 q^{57} -460.000 q^{59} +628.000 q^{61} -16.0000 q^{65} -556.000 q^{67} -216.000 q^{69} -592.000 q^{71} +1024.00 q^{73} -327.000 q^{75} +104.000 q^{79} +81.0000 q^{81} +324.000 q^{83} +96.0000 q^{85} -114.000 q^{87} +896.000 q^{89} -552.000 q^{93} -176.000 q^{95} -920.000 q^{97} +180.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	−4.00000	−0.0853385	−0.0426692	−	0.999089i	$-0.513586\pi$
$13$	−4.00000	−0.0853385	−0.0426692	+	0.999089i	$0.513586\pi$
$14$	0	0
$15$	12.0000	0.206559
$16$	0	0
$17$	24.0000	0.342403	0.171202	−	0.985236i	$-0.445235\pi$
$17$	24.0000	0.342403	0.171202	+	0.985236i	$0.445235\pi$
$18$	0	0
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−38.0000	−0.243325	−0.121662	−	0.992572i	$-0.538823\pi$
$29$	−38.0000	−0.243325	−0.121662	+	0.992572i	$0.538823\pi$
$30$	0	0
$31$	−184.000	−1.06604	−0.533022	−	0.846101i	$-0.678944\pi$
$31$	−184.000	−1.06604	−0.533022	+	0.846101i	$0.678944\pi$
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−30.0000	−0.133296	−0.0666482	−	0.997777i	$-0.521231\pi$
$37$	−30.0000	−0.133296	−0.0666482	+	0.997777i	$0.521231\pi$
$38$	0	0
$39$	−12.0000	−0.0492702
$40$	0	0
$41$	−216.000	−0.822769	−0.411385	−	0.911462i	$-0.634955\pi$
$41$	−216.000	−0.822769	−0.411385	+	0.911462i	$0.634955\pi$
$42$	0	0
$43$	164.000	0.581622	0.290811	−	0.956780i	$-0.406075\pi$
$43$	164.000	0.581622	0.290811	+	0.956780i	$0.406075\pi$
$44$	0	0
$45$	36.0000	0.119257
$46$	0	0
$47$	−520.000	−1.61383	−0.806913	−	0.590671i	$-0.798863\pi$
$47$	−520.000	−1.61383	−0.806913	+	0.590671i	$0.798863\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	72.0000	0.197687
$52$	0	0
$53$	−146.000	−0.378389	−0.189195	−	0.981940i	$-0.560588\pi$
$53$	−146.000	−0.378389	−0.189195	+	0.981940i	$0.560588\pi$
$54$	0	0
$55$	80.0000	0.196131
$56$	0	0
$57$	−132.000	−0.306734
$58$	0	0
$59$	−460.000	−1.01503	−0.507516	−	0.861642i	$-0.669436\pi$
$59$	−460.000	−1.01503	−0.507516	+	0.861642i	$0.669436\pi$
$60$	0	0
$61$	628.000	1.31815	0.659075	−	0.752077i	$-0.270948\pi$
$61$	628.000	1.31815	0.659075	+	0.752077i	$0.270948\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.0000	−0.0305316
$66$	0	0
$67$	−556.000	−1.01382	−0.506912	−	0.861998i	$-0.669213\pi$
$67$	−556.000	−1.01382	−0.506912	+	0.861998i	$0.669213\pi$
$68$	0	0
$69$	−216.000	−0.376860
$70$	0	0
$71$	−592.000	−0.989542	−0.494771	−	0.869023i	$-0.664748\pi$
$71$	−592.000	−0.989542	−0.494771	+	0.869023i	$0.664748\pi$
$72$	0	0
$73$	1024.00	1.64178	0.820891	−	0.571084i	$-0.193477\pi$
$73$	1024.00	1.64178	0.820891	+	0.571084i	$0.193477\pi$
$74$	0	0
$75$	−327.000	−0.503449
$76$	0	0
$77$	0	0
$78$	0	0
$79$	104.000	0.148113	0.0740564	−	0.997254i	$-0.476406\pi$
$79$	104.000	0.148113	0.0740564	+	0.997254i	$0.476406\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	324.000	0.428477	0.214239	−	0.976781i	$-0.431273\pi$
$83$	324.000	0.428477	0.214239	+	0.976781i	$0.431273\pi$
$84$	0	0
$85$	96.0000	0.122502
$86$	0	0
$87$	−114.000	−0.140484
$88$	0	0
$89$	896.000	1.06714	0.533572	−	0.845755i	$-0.320849\pi$
$89$	896.000	1.06714	0.533572	+	0.845755i	$0.320849\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−552.000	−0.615481
$94$	0	0
$95$	−176.000	−0.190076
$96$	0	0
$97$	−920.000	−0.963009	−0.481504	−	0.876444i	$-0.659909\pi$
$97$	−920.000	−0.963009	−0.481504	+	0.876444i	$0.659909\pi$
$98$	0	0
$99$	180.000	0.182734
$100$	0	0
$101$	1108.00	1.09159	0.545793	−	0.837920i	$-0.316229\pi$
$101$	1108.00	1.09159	0.545793	+	0.837920i	$0.316229\pi$
$102$	0	0
$103$	−1448.00	−1.38520	−0.692600	−	0.721321i	$-0.743535\pi$
$103$	−1448.00	−1.38520	−0.692600	+	0.721321i	$0.743535\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1316.00	−1.18900	−0.594498	−	0.804097i	$-0.702649\pi$
$107$	−1316.00	−1.18900	−0.594498	+	0.804097i	$0.702649\pi$
$108$	0	0
$109$	−86.0000	−0.0755716	−0.0377858	−	0.999286i	$-0.512030\pi$
$109$	−86.0000	−0.0755716	−0.0377858	+	0.999286i	$0.512030\pi$
$110$	0	0
$111$	−90.0000	−0.0769588
$112$	0	0
$113$	1778.00	1.48018	0.740089	−	0.672509i	$-0.234783\pi$
$113$	1778.00	1.48018	0.740089	+	0.672509i	$0.234783\pi$
$114$	0	0
$115$	−288.000	−0.233532
$116$	0	0
$117$	−36.0000	−0.0284462
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−648.000	−0.475026
$124$	0	0
$125$	−936.000	−0.669747
$126$	0	0
$127$	928.000	0.648399	0.324200	−	0.945989i	$-0.394905\pi$
$127$	928.000	0.648399	0.324200	+	0.945989i	$0.394905\pi$
$128$	0	0
$129$	492.000	0.335800
$130$	0	0
$131$	−1404.00	−0.936397	−0.468199	−	0.883623i	$-0.655097\pi$
$131$	−1404.00	−0.936397	−0.468199	+	0.883623i	$0.655097\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	108.000	0.0688530
$136$	0	0
$137$	−1370.00	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−1370.00	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−516.000	−0.314867	−0.157434	−	0.987530i	$-0.550322\pi$
$139$	−516.000	−0.314867	−0.157434	+	0.987530i	$0.550322\pi$
$140$	0	0
$141$	−1560.00	−0.931743
$142$	0	0
$143$	−80.0000	−0.0467828
$144$	0	0
$145$	−152.000	−0.0870546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1390.00	0.764250	0.382125	−	0.924111i	$-0.375192\pi$
$149$	1390.00	0.764250	0.382125	+	0.924111i	$0.375192\pi$
$150$	0	0
$151$	−136.000	−0.0732949	−0.0366474	−	0.999328i	$-0.511668\pi$
$151$	−136.000	−0.0732949	−0.0366474	+	0.999328i	$0.511668\pi$
$152$	0	0
$153$	216.000	0.114134
$154$	0	0
$155$	−736.000	−0.381400
$156$	0	0
$157$	−148.000	−0.0752337	−0.0376168	−	0.999292i	$-0.511977\pi$
$157$	−148.000	−0.0752337	−0.0376168	+	0.999292i	$0.511977\pi$
$158$	0	0
$159$	−438.000	−0.218463
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1212.00	0.582400	0.291200	−	0.956662i	$-0.405946\pi$
$163$	1212.00	0.582400	0.291200	+	0.956662i	$0.405946\pi$
$164$	0	0
$165$	240.000	0.113236
$166$	0	0
$167$	−1976.00	−0.915614	−0.457807	−	0.889052i	$-0.651365\pi$
$167$	−1976.00	−0.915614	−0.457807	+	0.889052i	$0.651365\pi$
$168$	0	0
$169$	−2181.00	−0.992717
$170$	0	0
$171$	−396.000	−0.177093
$172$	0	0
$173$	−2692.00	−1.18306	−0.591529	−	0.806284i	$-0.701475\pi$
$173$	−2692.00	−1.18306	−0.591529	+	0.806284i	$0.701475\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1380.00	−0.586029
$178$	0	0
$179$	2580.00	1.07731	0.538654	−	0.842527i	$-0.318933\pi$
$179$	2580.00	1.07731	0.538654	+	0.842527i	$0.318933\pi$
$180$	0	0
$181$	−2036.00	−0.836103	−0.418052	−	0.908423i	$-0.637287\pi$
$181$	−2036.00	−0.836103	−0.418052	+	0.908423i	$0.637287\pi$
$182$	0	0
$183$	1884.00	0.761034
$184$	0	0
$185$	−120.000	−0.0476896
$186$	0	0
$187$	480.000	0.187706
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3960.00	−1.50019	−0.750093	−	0.661332i	$-0.769991\pi$
$191$	−3960.00	−1.50019	−0.750093	+	0.661332i	$0.769991\pi$
$192$	0	0
$193$	2.00000	0.000745923 0	0.000372962	−	1.00000i	$-0.499881\pi$
$193$	2.00000	0.000745923 0	0.000372962		1.00000i	$0.499881\pi$
$194$	0	0
$195$	−48.0000	−0.0176274
$196$	0	0
$197$	3774.00	1.36491	0.682453	−	0.730930i	$-0.260913\pi$
$197$	3774.00	1.36491	0.682453	+	0.730930i	$0.260913\pi$
$198$	0	0
$199$	3560.00	1.26815	0.634075	−	0.773272i	$-0.281381\pi$
$199$	3560.00	1.26815	0.634075	+	0.773272i	$0.281381\pi$
$200$	0	0
$201$	−1668.00	−0.585332
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−864.000	−0.294363
$206$	0	0
$207$	−648.000	−0.217580
$208$	0	0
$209$	−880.000	−0.291248
$210$	0	0
$211$	2692.00	0.878317	0.439159	−	0.898410i	$-0.355277\pi$
$211$	2692.00	0.878317	0.439159	+	0.898410i	$0.355277\pi$
$212$	0	0
$213$	−1776.00	−0.571312
$214$	0	0
$215$	656.000	0.208088
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3072.00	0.947884
$220$	0	0
$221$	−96.0000	−0.0292202
$222$	0	0
$223$	−4528.00	−1.35972	−0.679859	−	0.733342i	$-0.737959\pi$
$223$	−4528.00	−1.35972	−0.679859	+	0.733342i	$0.737959\pi$
$224$	0	0
$225$	−981.000	−0.290667
$226$	0	0
$227$	3652.00	1.06781	0.533903	−	0.845546i	$-0.320725\pi$
$227$	3652.00	1.06781	0.533903	+	0.845546i	$0.320725\pi$
$228$	0	0
$229$	−4804.00	−1.38628	−0.693138	−	0.720805i	$-0.743772\pi$
$229$	−4804.00	−1.38628	−0.693138	+	0.720805i	$0.743772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2758.00	0.775462	0.387731	−	0.921773i	$-0.373259\pi$
$233$	2758.00	0.775462	0.387731	+	0.921773i	$0.373259\pi$
$234$	0	0
$235$	−2080.00	−0.577380
$236$	0	0
$237$	312.000	0.0855130
$238$	0	0
$239$	−6528.00	−1.76678	−0.883392	−	0.468635i	$-0.844746\pi$
$239$	−6528.00	−1.76678	−0.883392	+	0.468635i	$0.844746\pi$
$240$	0	0
$241$	−56.0000	−0.0149680	−0.00748398	−	0.999972i	$-0.502382\pi$
$241$	−56.0000	−0.0149680	−0.00748398	+	0.999972i	$0.502382\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	176.000	0.0453385
$248$	0	0
$249$	972.000	0.247382
$250$	0	0
$251$	−4900.00	−1.23221	−0.616106	−	0.787663i	$-0.711291\pi$
$251$	−4900.00	−1.23221	−0.616106	+	0.787663i	$0.711291\pi$
$252$	0	0
$253$	−1440.00	−0.357834
$254$	0	0
$255$	288.000	0.0707265
$256$	0	0
$257$	−6784.00	−1.64659	−0.823296	−	0.567612i	$-0.807867\pi$
$257$	−6784.00	−1.64659	−0.823296	+	0.567612i	$0.807867\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−342.000	−0.0811083
$262$	0	0
$263$	4544.00	1.06538	0.532690	−	0.846310i	$-0.321181\pi$
$263$	4544.00	1.06538	0.532690	+	0.846310i	$0.321181\pi$
$264$	0	0
$265$	−584.000	−0.135377
$266$	0	0
$267$	2688.00	0.616116
$268$	0	0
$269$	−4052.00	−0.918419	−0.459210	−	0.888328i	$-0.651867\pi$
$269$	−4052.00	−0.918419	−0.459210	+	0.888328i	$0.651867\pi$
$270$	0	0
$271$	2752.00	0.616871	0.308436	−	0.951245i	$-0.400195\pi$
$271$	2752.00	0.616871	0.308436	+	0.951245i	$0.400195\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2180.00	−0.478033
$276$	0	0
$277$	4366.00	0.947031	0.473515	−	0.880786i	$-0.342985\pi$
$277$	4366.00	0.947031	0.473515	+	0.880786i	$0.342985\pi$
$278$	0	0
$279$	−1656.00	−0.355348
$280$	0	0
$281$	7734.00	1.64189	0.820946	−	0.571006i	$-0.193447\pi$
$281$	7734.00	1.64189	0.820946	+	0.571006i	$0.193447\pi$
$282$	0	0
$283$	4052.00	0.851118	0.425559	−	0.904931i	$-0.360077\pi$
$283$	4052.00	0.851118	0.425559	+	0.904931i	$0.360077\pi$
$284$	0	0
$285$	−528.000	−0.109740
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4337.00	−0.882760
$290$	0	0
$291$	−2760.00	−0.555993
$292$	0	0
$293$	−3420.00	−0.681906	−0.340953	−	0.940080i	$-0.610750\pi$
$293$	−3420.00	−0.681906	−0.340953	+	0.940080i	$0.610750\pi$
$294$	0	0
$295$	−1840.00	−0.363149
$296$	0	0
$297$	540.000	0.105502
$298$	0	0
$299$	288.000	0.0557039
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3324.00	0.630227
$304$	0	0
$305$	2512.00	0.471596
$306$	0	0
$307$	−7324.00	−1.36157	−0.680786	−	0.732482i	$-0.738362\pi$
$307$	−7324.00	−1.36157	−0.680786	+	0.732482i	$0.738362\pi$
$308$	0	0
$309$	−4344.00	−0.799746
$310$	0	0
$311$	4192.00	0.764330	0.382165	−	0.924094i	$-0.375179\pi$
$311$	4192.00	0.764330	0.382165	+	0.924094i	$0.375179\pi$
$312$	0	0
$313$	−6840.00	−1.23521	−0.617603	−	0.786490i	$-0.711896\pi$
$313$	−6840.00	−1.23521	−0.617603	+	0.786490i	$0.711896\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6630.00	1.17469	0.587347	−	0.809335i	$-0.300172\pi$
$317$	6630.00	1.17469	0.587347	+	0.809335i	$0.300172\pi$
$318$	0	0
$319$	−760.000	−0.133391
$320$	0	0
$321$	−3948.00	−0.686467
$322$	0	0
$323$	−1056.00	−0.181911
$324$	0	0
$325$	436.000	0.0744152
$326$	0	0
$327$	−258.000	−0.0436313
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6868.00	1.14048	0.570241	−	0.821478i	$-0.306850\pi$
$331$	6868.00	1.14048	0.570241	+	0.821478i	$0.306850\pi$
$332$	0	0
$333$	−270.000	−0.0444322
$334$	0	0
$335$	−2224.00	−0.362717
$336$	0	0
$337$	−7378.00	−1.19260	−0.596299	−	0.802763i	$-0.703363\pi$
$337$	−7378.00	−1.19260	−0.596299	+	0.802763i	$0.703363\pi$
$338$	0	0
$339$	5334.00	0.854581
$340$	0	0
$341$	−3680.00	−0.584408
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−864.000	−0.134830
$346$	0	0
$347$	2676.00	0.413992	0.206996	−	0.978342i	$-0.433631\pi$
$347$	2676.00	0.413992	0.206996	+	0.978342i	$0.433631\pi$
$348$	0	0
$349$	−5124.00	−0.785907	−0.392953	−	0.919558i	$-0.628547\pi$
$349$	−5124.00	−0.785907	−0.392953	+	0.919558i	$0.628547\pi$
$350$	0	0
$351$	−108.000	−0.0164234
$352$	0	0
$353$	4560.00	0.687548	0.343774	−	0.939052i	$-0.388295\pi$
$353$	4560.00	0.687548	0.343774	+	0.939052i	$0.388295\pi$
$354$	0	0
$355$	−2368.00	−0.354029
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3656.00	−0.537483	−0.268741	−	0.963212i	$-0.586608\pi$
$359$	−3656.00	−0.537483	−0.268741	+	0.963212i	$0.586608\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	4096.00	0.587382
$366$	0	0
$367$	1616.00	0.229849	0.114924	−	0.993374i	$-0.463337\pi$
$367$	1616.00	0.229849	0.114924	+	0.993374i	$0.463337\pi$
$368$	0	0
$369$	−1944.00	−0.274256
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2734.00	0.379521	0.189760	−	0.981830i	$-0.439229\pi$
$373$	2734.00	0.379521	0.189760	+	0.981830i	$0.439229\pi$
$374$	0	0
$375$	−2808.00	−0.386679
$376$	0	0
$377$	152.000	0.0207650
$378$	0	0
$379$	1380.00	0.187034	0.0935169	−	0.995618i	$-0.470189\pi$
$379$	1380.00	0.187034	0.0935169	+	0.995618i	$0.470189\pi$
$380$	0	0
$381$	2784.00	0.374353
$382$	0	0
$383$	−6888.00	−0.918957	−0.459478	−	0.888189i	$-0.651964\pi$
$383$	−6888.00	−0.918957	−0.459478	+	0.888189i	$0.651964\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1476.00	0.193874
$388$	0	0
$389$	−2046.00	−0.266674	−0.133337	−	0.991071i	$-0.542569\pi$
$389$	−2046.00	−0.266674	−0.133337	+	0.991071i	$0.542569\pi$
$390$	0	0
$391$	−1728.00	−0.223501
$392$	0	0
$393$	−4212.00	−0.540629
$394$	0	0
$395$	416.000	0.0529905
$396$	0	0
$397$	3116.00	0.393923	0.196962	−	0.980411i	$-0.436893\pi$
$397$	3116.00	0.393923	0.196962	+	0.980411i	$0.436893\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2958.00	0.368368	0.184184	−	0.982892i	$-0.441036\pi$
$401$	2958.00	0.368368	0.184184	+	0.982892i	$0.441036\pi$
$402$	0	0
$403$	736.000	0.0909746
$404$	0	0
$405$	324.000	0.0397523
$406$	0	0
$407$	−600.000	−0.0730735
$408$	0	0
$409$	7944.00	0.960405	0.480202	−	0.877158i	$-0.340563\pi$
$409$	7944.00	0.960405	0.480202	+	0.877158i	$0.340563\pi$
$410$	0	0
$411$	−4110.00	−0.493264
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1296.00	0.153297
$416$	0	0
$417$	−1548.00	−0.181789
$418$	0	0
$419$	−4084.00	−0.476173	−0.238086	−	0.971244i	$-0.576520\pi$
$419$	−4084.00	−0.476173	−0.238086	+	0.971244i	$0.576520\pi$
$420$	0	0
$421$	−6306.00	−0.730013	−0.365007	−	0.931005i	$-0.618933\pi$
$421$	−6306.00	−0.730013	−0.365007	+	0.931005i	$0.618933\pi$
$422$	0	0
$423$	−4680.00	−0.537942
$424$	0	0
$425$	−2616.00	−0.298576
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−240.000	−0.0270100
$430$	0	0
$431$	11824.0	1.32144	0.660722	−	0.750631i	$-0.270250\pi$
$431$	11824.0	1.32144	0.660722	+	0.750631i	$0.270250\pi$
$432$	0	0
$433$	−4504.00	−0.499881	−0.249940	−	0.968261i	$-0.580411\pi$
$433$	−4504.00	−0.499881	−0.249940	+	0.968261i	$0.580411\pi$
$434$	0	0
$435$	−456.000	−0.0502610
$436$	0	0
$437$	3168.00	0.346787
$438$	0	0
$439$	−13056.0	−1.41943	−0.709714	−	0.704490i	$-0.751176\pi$
$439$	−13056.0	−1.41943	−0.709714	+	0.704490i	$0.751176\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−132.000	−0.0141569	−0.00707845	−	0.999975i	$-0.502253\pi$
$443$	−132.000	−0.0141569	−0.00707845	+	0.999975i	$0.502253\pi$
$444$	0	0
$445$	3584.00	0.381793
$446$	0	0
$447$	4170.00	0.441240
$448$	0	0
$449$	4866.00	0.511449	0.255725	−	0.966750i	$-0.417686\pi$
$449$	4866.00	0.511449	0.255725	+	0.966750i	$0.417686\pi$
$450$	0	0
$451$	−4320.00	−0.451044
$452$	0	0
$453$	−408.000	−0.0423168
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10106.0	1.03444	0.517220	−	0.855853i	$-0.326967\pi$
$457$	10106.0	1.03444	0.517220	+	0.855853i	$0.326967\pi$
$458$	0	0
$459$	648.000	0.0658955
$460$	0	0
$461$	−18036.0	−1.82217	−0.911085	−	0.412219i	$-0.864754\pi$
$461$	−18036.0	−1.82217	−0.911085	+	0.412219i	$0.864754\pi$
$462$	0	0
$463$	−5288.00	−0.530787	−0.265393	−	0.964140i	$-0.585502\pi$
$463$	−5288.00	−0.530787	−0.265393	+	0.964140i	$0.585502\pi$
$464$	0	0
$465$	−2208.00	−0.220201
$466$	0	0
$467$	−15164.0	−1.50258	−0.751291	−	0.659971i	$-0.770569\pi$
$467$	−15164.0	−1.50258	−0.751291	+	0.659971i	$0.770569\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−444.000	−0.0434362
$472$	0	0
$473$	3280.00	0.318847
$474$	0	0
$475$	4796.00	0.463275
$476$	0	0
$477$	−1314.00	−0.126130
$478$	0	0
$479$	−7896.00	−0.753189	−0.376594	−	0.926378i	$-0.622905\pi$
$479$	−7896.00	−0.753189	−0.376594	+	0.926378i	$0.622905\pi$
$480$	0	0
$481$	120.000	0.0113753
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3680.00	−0.344536
$486$	0	0
$487$	−2920.00	−0.271700	−0.135850	−	0.990729i	$-0.543376\pi$
$487$	−2920.00	−0.271700	−0.135850	+	0.990729i	$0.543376\pi$
$488$	0	0
$489$	3636.00	0.336249
$490$	0	0
$491$	7932.00	0.729055	0.364528	−	0.931193i	$-0.381230\pi$
$491$	7932.00	0.729055	0.364528	+	0.931193i	$0.381230\pi$
$492$	0	0
$493$	−912.000	−0.0833152
$494$	0	0
$495$	720.000	0.0653770
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2004.00	0.179782	0.0898911	−	0.995952i	$-0.471348\pi$
$499$	2004.00	0.179782	0.0898911	+	0.995952i	$0.471348\pi$
$500$	0	0
$501$	−5928.00	−0.528630
$502$	0	0
$503$	−4496.00	−0.398542	−0.199271	−	0.979944i	$-0.563857\pi$
$503$	−4496.00	−0.398542	−0.199271	+	0.979944i	$0.563857\pi$
$504$	0	0
$505$	4432.00	0.390537
$506$	0	0
$507$	−6543.00	−0.573146
$508$	0	0
$509$	12620.0	1.09896	0.549481	−	0.835506i	$-0.314825\pi$
$509$	12620.0	1.09896	0.549481	+	0.835506i	$0.314825\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1188.00	−0.102245
$514$	0	0
$515$	−5792.00	−0.495584
$516$	0	0
$517$	−10400.0	−0.884703
$518$	0	0
$519$	−8076.00	−0.683039
$520$	0	0
$521$	18008.0	1.51429	0.757145	−	0.653247i	$-0.226594\pi$
$521$	18008.0	1.51429	0.757145	+	0.653247i	$0.226594\pi$
$522$	0	0
$523$	−13292.0	−1.11132	−0.555658	−	0.831411i	$-0.687534\pi$
$523$	−13292.0	−1.11132	−0.555658	+	0.831411i	$0.687534\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4416.00	−0.365017
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	−4140.00	−0.338344
$532$	0	0
$533$	864.000	0.0702139
$534$	0	0
$535$	−5264.00	−0.425388
$536$	0	0
$537$	7740.00	0.621984
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8570.00	−0.681059	−0.340530	−	0.940234i	$-0.610606\pi$
$541$	−8570.00	−0.681059	−0.340530	+	0.940234i	$0.610606\pi$
$542$	0	0
$543$	−6108.00	−0.482724
$544$	0	0
$545$	−344.000	−0.0270373
$546$	0	0
$547$	1916.00	0.149766	0.0748832	−	0.997192i	$-0.476142\pi$
$547$	1916.00	0.149766	0.0748832	+	0.997192i	$0.476142\pi$
$548$	0	0
$549$	5652.00	0.439383
$550$	0	0
$551$	1672.00	0.129273
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−360.000	−0.0275336
$556$	0	0
$557$	19926.0	1.51578	0.757892	−	0.652380i	$-0.226229\pi$
$557$	19926.0	1.51578	0.757892	+	0.652380i	$0.226229\pi$
$558$	0	0
$559$	−656.000	−0.0496348
$560$	0	0
$561$	1440.00	0.108372
$562$	0	0
$563$	−4244.00	−0.317697	−0.158848	−	0.987303i	$-0.550778\pi$
$563$	−4244.00	−0.317697	−0.158848	+	0.987303i	$0.550778\pi$
$564$	0	0
$565$	7112.00	0.529565
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22794.0	−1.67939	−0.839696	−	0.543057i	$-0.817267\pi$
$569$	−22794.0	−1.67939	−0.839696	+	0.543057i	$0.817267\pi$
$570$	0	0
$571$	−14028.0	−1.02811	−0.514057	−	0.857756i	$-0.671858\pi$
$571$	−14028.0	−1.02811	−0.514057	+	0.857756i	$0.671858\pi$
$572$	0	0
$573$	−11880.0	−0.866133
$574$	0	0
$575$	7848.00	0.569190
$576$	0	0
$577$	8368.00	0.603751	0.301876	−	0.953347i	$-0.402387\pi$
$577$	8368.00	0.603751	0.301876	+	0.953347i	$0.402387\pi$
$578$	0	0
$579$	6.00000	0.000430659 0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2920.00	−0.207434
$584$	0	0
$585$	−144.000	−0.0101772
$586$	0	0
$587$	−52.0000	−0.00365634	−0.00182817	−	0.999998i	$-0.500582\pi$
$587$	−52.0000	−0.00365634	−0.00182817	+	0.999998i	$0.500582\pi$
$588$	0	0
$589$	8096.00	0.566366
$590$	0	0
$591$	11322.0	0.788029
$592$	0	0
$593$	5808.00	0.402202	0.201101	−	0.979570i	$-0.435548\pi$
$593$	5808.00	0.402202	0.201101	+	0.979570i	$0.435548\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10680.0	0.732166
$598$	0	0
$599$	10464.0	0.713769	0.356884	−	0.934149i	$-0.383839\pi$
$599$	10464.0	0.713769	0.356884	+	0.934149i	$0.383839\pi$
$600$	0	0
$601$	1184.00	0.0803600	0.0401800	−	0.999192i	$-0.487207\pi$
$601$	1184.00	0.0803600	0.0401800	+	0.999192i	$0.487207\pi$
$602$	0	0
$603$	−5004.00	−0.337941
$604$	0	0
$605$	−3724.00	−0.250251
$606$	0	0
$607$	−13152.0	−0.879445	−0.439723	−	0.898134i	$-0.644923\pi$
$607$	−13152.0	−0.879445	−0.439723	+	0.898134i	$0.644923\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2080.00	0.137721
$612$	0	0
$613$	−18334.0	−1.20800	−0.603999	−	0.796985i	$-0.706427\pi$
$613$	−18334.0	−1.20800	−0.603999	+	0.796985i	$0.706427\pi$
$614$	0	0
$615$	−2592.00	−0.169950
$616$	0	0
$617$	−8122.00	−0.529950	−0.264975	−	0.964255i	$-0.585364\pi$
$617$	−8122.00	−0.529950	−0.264975	+	0.964255i	$0.585364\pi$
$618$	0	0
$619$	−5980.00	−0.388298	−0.194149	−	0.980972i	$-0.562195\pi$
$619$	−5980.00	−0.388298	−0.194149	+	0.980972i	$0.562195\pi$
$620$	0	0
$621$	−1944.00	−0.125620
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	−2640.00	−0.168152
$628$	0	0
$629$	−720.000	−0.0456411
$630$	0	0
$631$	−12528.0	−0.790383	−0.395192	−	0.918599i	$-0.629322\pi$
$631$	−12528.0	−0.790383	−0.395192	+	0.918599i	$0.629322\pi$
$632$	0	0
$633$	8076.00	0.507097
$634$	0	0
$635$	3712.00	0.231978
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5328.00	−0.329847
$640$	0	0
$641$	20798.0	1.28155	0.640773	−	0.767730i	$-0.278614\pi$
$641$	20798.0	1.28155	0.640773	+	0.767730i	$0.278614\pi$
$642$	0	0
$643$	1932.00	0.118492	0.0592462	−	0.998243i	$-0.481130\pi$
$643$	1932.00	0.118492	0.0592462	+	0.998243i	$0.481130\pi$
$644$	0	0
$645$	1968.00	0.120139
$646$	0	0
$647$	8424.00	0.511873	0.255936	−	0.966694i	$-0.417616\pi$
$647$	8424.00	0.511873	0.255936	+	0.966694i	$0.417616\pi$
$648$	0	0
$649$	−9200.00	−0.556443
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17750.0	−1.06372	−0.531862	−	0.846831i	$-0.678507\pi$
$653$	−17750.0	−1.06372	−0.531862	+	0.846831i	$0.678507\pi$
$654$	0	0
$655$	−5616.00	−0.335016
$656$	0	0
$657$	9216.00	0.547261
$658$	0	0
$659$	27580.0	1.63029	0.815147	−	0.579254i	$-0.196656\pi$
$659$	27580.0	1.63029	0.815147	+	0.579254i	$0.196656\pi$
$660$	0	0
$661$	−9292.00	−0.546773	−0.273386	−	0.961904i	$-0.588144\pi$
$661$	−9292.00	−0.546773	−0.273386	+	0.961904i	$0.588144\pi$
$662$	0	0
$663$	−288.000	−0.0168703
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2736.00	0.158828
$668$	0	0
$669$	−13584.0	−0.785034
$670$	0	0
$671$	12560.0	0.722613
$672$	0	0
$673$	11486.0	0.657879	0.328940	−	0.944351i	$-0.393309\pi$
$673$	11486.0	0.657879	0.328940	+	0.944351i	$0.393309\pi$
$674$	0	0
$675$	−2943.00	−0.167816
$676$	0	0
$677$	7116.00	0.403974	0.201987	−	0.979388i	$-0.435260\pi$
$677$	7116.00	0.403974	0.201987	+	0.979388i	$0.435260\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10956.0	0.616498
$682$	0	0
$683$	7612.00	0.426450	0.213225	−	0.977003i	$-0.431603\pi$
$683$	7612.00	0.426450	0.213225	+	0.977003i	$0.431603\pi$
$684$	0	0
$685$	−5480.00	−0.305664
$686$	0	0
$687$	−14412.0	−0.800367
$688$	0	0
$689$	584.000	0.0322912
$690$	0	0
$691$	21572.0	1.18761	0.593804	−	0.804609i	$-0.297625\pi$
$691$	21572.0	1.18761	0.593804	+	0.804609i	$0.297625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2064.00	−0.112650
$696$	0	0
$697$	−5184.00	−0.281719
$698$	0	0
$699$	8274.00	0.447713
$700$	0	0
$701$	−1702.00	−0.0917028	−0.0458514	−	0.998948i	$-0.514600\pi$
$701$	−1702.00	−0.0917028	−0.0458514	+	0.998948i	$0.514600\pi$
$702$	0	0
$703$	1320.00	0.0708176
$704$	0	0
$705$	−6240.00	−0.333350
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6370.00	0.337419	0.168710	−	0.985666i	$-0.446040\pi$
$709$	6370.00	0.337419	0.168710	+	0.985666i	$0.446040\pi$
$710$	0	0
$711$	936.000	0.0493709
$712$	0	0
$713$	13248.0	0.695851
$714$	0	0
$715$	−320.000	−0.0167375
$716$	0	0
$717$	−19584.0	−1.02005
$718$	0	0
$719$	8808.00	0.456861	0.228430	−	0.973560i	$-0.426641\pi$
$719$	8808.00	0.456861	0.228430	+	0.973560i	$0.426641\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−168.000	−0.00864175
$724$	0	0
$725$	4142.00	0.212179
$726$	0	0
$727$	−17768.0	−0.906436	−0.453218	−	0.891400i	$-0.649724\pi$
$727$	−17768.0	−0.906436	−0.453218	+	0.891400i	$0.649724\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	3936.00	0.199149
$732$	0	0
$733$	−5564.00	−0.280370	−0.140185	−	0.990125i	$-0.544770\pi$
$733$	−5564.00	−0.280370	−0.140185	+	0.990125i	$0.544770\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11120.0	−0.555781
$738$	0	0
$739$	17564.0	0.874293	0.437146	−	0.899390i	$-0.355989\pi$
$739$	17564.0	0.874293	0.437146	+	0.899390i	$0.355989\pi$
$740$	0	0
$741$	528.000	0.0261762
$742$	0	0
$743$	38280.0	1.89012	0.945059	−	0.326901i	$-0.106004\pi$
$743$	38280.0	1.89012	0.945059	+	0.326901i	$0.106004\pi$
$744$	0	0
$745$	5560.00	0.273426
$746$	0	0
$747$	2916.00	0.142826
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36192.0	−1.75854	−0.879271	−	0.476322i	$-0.841970\pi$
$751$	−36192.0	−1.75854	−0.879271	+	0.476322i	$0.841970\pi$
$752$	0	0
$753$	−14700.0	−0.711418
$754$	0	0
$755$	−544.000	−0.0262228
$756$	0	0
$757$	−14.0000	−0.000672178 0	−0.000336089	−	1.00000i	$-0.500107\pi$
$757$	−14.0000	−0.000672178 0	−0.000336089		1.00000i	$0.500107\pi$
$758$	0	0
$759$	−4320.00	−0.206596
$760$	0	0
$761$	−26504.0	−1.26251	−0.631254	−	0.775576i	$-0.717460\pi$
$761$	−26504.0	−1.26251	−0.631254	+	0.775576i	$0.717460\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	864.000	0.0408340
$766$	0	0
$767$	1840.00	0.0866213
$768$	0	0
$769$	40184.0	1.88436	0.942180	−	0.335109i	$-0.108773\pi$
$769$	40184.0	1.88436	0.942180	+	0.335109i	$0.108773\pi$
$770$	0	0
$771$	−20352.0	−0.950661
$772$	0	0
$773$	−35340.0	−1.64436	−0.822181	−	0.569226i	$-0.807243\pi$
$773$	−35340.0	−1.64436	−0.822181	+	0.569226i	$0.807243\pi$
$774$	0	0
$775$	20056.0	0.929591
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9504.00	0.437120
$780$	0	0
$781$	−11840.0	−0.542469
$782$	0	0
$783$	−1026.00	−0.0468279
$784$	0	0
$785$	−592.000	−0.0269164
$786$	0	0
$787$	14852.0	0.672702	0.336351	−	0.941737i	$-0.390807\pi$
$787$	14852.0	0.672702	0.336351	+	0.941737i	$0.390807\pi$
$788$	0	0
$789$	13632.0	0.615098
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2512.00	−0.112489
$794$	0	0
$795$	−1752.00	−0.0781598
$796$	0	0
$797$	19788.0	0.879457	0.439728	−	0.898131i	$-0.355075\pi$
$797$	19788.0	0.879457	0.439728	+	0.898131i	$0.355075\pi$
$798$	0	0
$799$	−12480.0	−0.552579
$800$	0	0
$801$	8064.00	0.355715
$802$	0	0
$803$	20480.0	0.900029
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12156.0	−0.530250
$808$	0	0
$809$	16986.0	0.738190	0.369095	−	0.929392i	$-0.379668\pi$
$809$	16986.0	0.738190	0.369095	+	0.929392i	$0.379668\pi$
$810$	0	0
$811$	26596.0	1.15156	0.575778	−	0.817606i	$-0.304699\pi$
$811$	26596.0	1.15156	0.575778	+	0.817606i	$0.304699\pi$
$812$	0	0
$813$	8256.00	0.356151
$814$	0	0
$815$	4848.00	0.208366
$816$	0	0
$817$	−7216.00	−0.309004
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34898.0	−1.48349	−0.741747	−	0.670680i	$-0.766002\pi$
$821$	−34898.0	−1.48349	−0.741747	+	0.670680i	$0.766002\pi$
$822$	0	0
$823$	−12928.0	−0.547560	−0.273780	−	0.961792i	$-0.588274\pi$
$823$	−12928.0	−0.547560	−0.273780	+	0.961792i	$0.588274\pi$
$824$	0	0
$825$	−6540.00	−0.275992
$826$	0	0
$827$	43164.0	1.81494	0.907472	−	0.420112i	$-0.138009\pi$
$827$	43164.0	1.81494	0.907472	+	0.420112i	$0.138009\pi$
$828$	0	0
$829$	−41228.0	−1.72727	−0.863635	−	0.504117i	$-0.831818\pi$
$829$	−41228.0	−1.72727	−0.863635	+	0.504117i	$0.831818\pi$
$830$	0	0
$831$	13098.0	0.546768
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7904.00	−0.327580
$836$	0	0
$837$	−4968.00	−0.205160
$838$	0	0
$839$	1368.00	0.0562915	0.0281458	−	0.999604i	$-0.491040\pi$
$839$	1368.00	0.0562915	0.0281458	+	0.999604i	$0.491040\pi$
$840$	0	0
$841$	−22945.0	−0.940793
$842$	0	0
$843$	23202.0	0.947947
$844$	0	0
$845$	−8724.00	−0.355165
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12156.0	0.491393
$850$	0	0
$851$	2160.00	0.0870080
$852$	0	0
$853$	5276.00	0.211778	0.105889	−	0.994378i	$-0.466231\pi$
$853$	5276.00	0.211778	0.105889	+	0.994378i	$0.466231\pi$
$854$	0	0
$855$	−1584.00	−0.0633587
$856$	0	0
$857$	840.000	0.0334817	0.0167409	−	0.999860i	$-0.494671\pi$
$857$	840.000	0.0334817	0.0167409	+	0.999860i	$0.494671\pi$
$858$	0	0
$859$	26028.0	1.03383	0.516917	−	0.856035i	$-0.327079\pi$
$859$	26028.0	1.03383	0.516917	+	0.856035i	$0.327079\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29448.0	1.16155	0.580777	−	0.814063i	$-0.302749\pi$
$863$	29448.0	1.16155	0.580777	+	0.814063i	$0.302749\pi$
$864$	0	0
$865$	−10768.0	−0.423264
$866$	0	0
$867$	−13011.0	−0.509662
$868$	0	0
$869$	2080.00	0.0811958
$870$	0	0
$871$	2224.00	0.0865182
$872$	0	0
$873$	−8280.00	−0.321003
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25866.0	0.995932	0.497966	−	0.867196i	$-0.334080\pi$
$877$	25866.0	0.995932	0.497966	+	0.867196i	$0.334080\pi$
$878$	0	0
$879$	−10260.0	−0.393699
$880$	0	0
$881$	9472.00	0.362225	0.181112	−	0.983462i	$-0.442030\pi$
$881$	9472.00	0.362225	0.181112	+	0.983462i	$0.442030\pi$
$882$	0	0
$883$	−49372.0	−1.88165	−0.940827	−	0.338888i	$-0.889949\pi$
$883$	−49372.0	−1.88165	−0.940827	+	0.338888i	$0.889949\pi$
$884$	0	0
$885$	−5520.00	−0.209664
$886$	0	0
$887$	11160.0	0.422453	0.211227	−	0.977437i	$-0.432254\pi$
$887$	11160.0	0.422453	0.211227	+	0.977437i	$0.432254\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1620.00	0.0609114
$892$	0	0
$893$	22880.0	0.857391
$894$	0	0
$895$	10320.0	0.385430
$896$	0	0
$897$	864.000	0.0321607
$898$	0	0
$899$	6992.00	0.259395
$900$	0	0
$901$	−3504.00	−0.129562
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8144.00	−0.299133
$906$	0	0
$907$	−22708.0	−0.831319	−0.415660	−	0.909520i	$-0.636449\pi$
$907$	−22708.0	−0.831319	−0.415660	+	0.909520i	$0.636449\pi$
$908$	0	0
$909$	9972.00	0.363862
$910$	0	0
$911$	16192.0	0.588875	0.294437	−	0.955671i	$-0.404868\pi$
$911$	16192.0	0.588875	0.294437	+	0.955671i	$0.404868\pi$
$912$	0	0
$913$	6480.00	0.234892
$914$	0	0
$915$	7536.00	0.272276
$916$	0	0
$917$	0	0
$918$	0	0
$919$	46320.0	1.66263	0.831314	−	0.555802i	$-0.187589\pi$
$919$	46320.0	1.66263	0.831314	+	0.555802i	$0.187589\pi$
$920$	0	0
$921$	−21972.0	−0.786104
$922$	0	0
$923$	2368.00	0.0844460
$924$	0	0
$925$	3270.00	0.116235
$926$	0	0
$927$	−13032.0	−0.461734
$928$	0	0
$929$	−2280.00	−0.0805214	−0.0402607	−	0.999189i	$-0.512819\pi$
$929$	−2280.00	−0.0805214	−0.0402607	+	0.999189i	$0.512819\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12576.0	0.441286
$934$	0	0
$935$	1920.00	0.0671558
$936$	0	0
$937$	49056.0	1.71034	0.855171	−	0.518347i	$-0.173452\pi$
$937$	49056.0	1.71034	0.855171	+	0.518347i	$0.173452\pi$
$938$	0	0
$939$	−20520.0	−0.713147
$940$	0	0
$941$	−24876.0	−0.861779	−0.430890	−	0.902405i	$-0.641800\pi$
$941$	−24876.0	−0.861779	−0.430890	+	0.902405i	$0.641800\pi$
$942$	0	0
$943$	15552.0	0.537055
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23428.0	−0.803915	−0.401958	−	0.915658i	$-0.631670\pi$
$947$	−23428.0	−0.803915	−0.401958	+	0.915658i	$0.631670\pi$
$948$	0	0
$949$	−4096.00	−0.140107
$950$	0	0
$951$	19890.0	0.678210
$952$	0	0
$953$	−6678.00	−0.226990	−0.113495	−	0.993539i	$-0.536205\pi$
$953$	−6678.00	−0.226990	−0.113495	+	0.993539i	$0.536205\pi$
$954$	0	0
$955$	−15840.0	−0.536723
$956$	0	0
$957$	−2280.00	−0.0770135
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4065.00	0.136451
$962$	0	0
$963$	−11844.0	−0.396332
$964$	0	0
$965$	8.00000	0.000266870 0
$966$	0	0
$967$	−15544.0	−0.516920	−0.258460	−	0.966022i	$-0.583215\pi$
$967$	−15544.0	−0.516920	−0.258460	+	0.966022i	$0.583215\pi$
$968$	0	0
$969$	−3168.00	−0.105027
$970$	0	0
$971$	31124.0	1.02865	0.514324	−	0.857596i	$-0.328043\pi$
$971$	31124.0	1.02865	0.514324	+	0.857596i	$0.328043\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1308.00	0.0429636
$976$	0	0
$977$	50062.0	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$977$	50062.0	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$978$	0	0
$979$	17920.0	0.585011
$980$	0	0
$981$	−774.000	−0.0251905
$982$	0	0
$983$	−328.000	−0.0106425	−0.00532125	−	0.999986i	$-0.501694\pi$
$983$	−328.000	−0.0106425	−0.00532125	+	0.999986i	$0.501694\pi$
$984$	0	0
$985$	15096.0	0.488323
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11808.0	−0.379649
$990$	0	0
$991$	20872.0	0.669042	0.334521	−	0.942388i	$-0.391425\pi$
$991$	20872.0	0.669042	0.334521	+	0.942388i	$0.391425\pi$
$992$	0	0
$993$	20604.0	0.658457
$994$	0	0
$995$	14240.0	0.453707
$996$	0	0
$997$	−46924.0	−1.49057	−0.745285	−	0.666746i	$-0.767687\pi$
$997$	−46924.0	−1.49057	−0.745285	+	0.666746i	$0.767687\pi$
$998$	0	0
$999$	−810.000	−0.0256529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.be.1.1		1
4.3	odd	2		588.4.a.b.1.1	✓	1
7.6	odd	2		2352.4.a.g.1.1		1
12.11	even	2		1764.4.a.d.1.1		1
28.3	even	6		588.4.i.b.373.1		2
28.11	odd	6		588.4.i.g.373.1		2
28.19	even	6		588.4.i.b.361.1		2
28.23	odd	6		588.4.i.g.361.1		2
28.27	even	2		588.4.a.e.1.1	yes	1
84.11	even	6		1764.4.k.j.1549.1		2
84.23	even	6		1764.4.k.j.361.1		2
84.47	odd	6		1764.4.k.g.361.1		2
84.59	odd	6		1764.4.k.g.1549.1		2
84.83	odd	2		1764.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.b.1.1	✓	1	4.3	odd	2
588.4.a.e.1.1	yes	1	28.27	even	2
588.4.i.b.361.1		2	28.19	even	6
588.4.i.b.373.1		2	28.3	even	6
588.4.i.g.361.1		2	28.23	odd	6
588.4.i.g.373.1		2	28.11	odd	6
1764.4.a.d.1.1		1	12.11	even	2
1764.4.a.i.1.1		1	84.83	odd	2
1764.4.k.g.361.1		2	84.47	odd	6
1764.4.k.g.1549.1		2	84.59	odd	6
1764.4.k.j.361.1		2	84.23	even	6
1764.4.k.j.1549.1		2	84.11	even	6
2352.4.a.g.1.1		1	7.6	odd	2
2352.4.a.be.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

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