LMFDB - Embedded newform 7350.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7350.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -2.00000 q^{39} +2.00000 q^{41} +2.00000 q^{43} +2.00000 q^{44} +8.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +4.00000 q^{51} +2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{59} +10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} -4.00000 q^{68} +8.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +8.00000 q^{74} +2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +16.0000 q^{83} -2.00000 q^{86} -2.00000 q^{88} -14.0000 q^{89} -8.00000 q^{92} -2.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} +6.00000 q^{97} +2.00000 q^{99} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	4.00000	0.560112
$52$	2.00000	0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−4.00000	−0.485071
$69$	8.00000	0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	−8.00000	−0.834058
$93$	−2.00000	−0.207390
$94$	−10.0000	−1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−4.00000	−0.396059
$103$	−20.0000	−1.97066	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−20.0000	−1.97066	−0.985329	+	0.170664i	$0.945409\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−10.0000	−0.905357
$123$	−2.00000	−0.180334
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	4.00000	0.342997
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−8.00000	−0.681005
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	12.0000	1.00702
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−16.0000	−1.27289
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−16.0000	−1.24184
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	2.00000	0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	4.00000	0.300658
$178$	14.0000	1.04934
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	8.00000	0.589768
$185$	0	0
$186$	2.00000	0.146647
$187$	−8.00000	−0.585018
$188$	10.0000	0.729325
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−2.00000	−0.142134
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	−14.0000	−0.985037
$203$	0	0
$204$	4.00000	0.280056
$205$	0	0
$206$	20.0000	1.39347
$207$	−8.00000	−0.556038
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	2.00000	0.137361
$213$	12.0000	0.822226
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−8.00000	−0.538138
$222$	−8.00000	−0.536925
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	−2.00000	−0.127000
$249$	−16.0000	−1.01396
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	2.00000	0.124515
$259$	0	0
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	0	0
$267$	14.0000	0.856786
$268$	−2.00000	−0.122169
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	8.00000	0.481543
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	−4.00000	−0.239904
$279$	2.00000	0.119737
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	10.0000	0.595491
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	10.0000	0.585206
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	−2.00000	−0.116052
$298$	16.0000	0.926855
$299$	−16.0000	−0.925304
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	0	0
$306$	4.00000	0.228665
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	20.0000	1.13776
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	2.00000	0.113228
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	16.0000	0.900070
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	2.00000	0.112154
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−10.0000	−0.553849
$327$	2.00000	0.110600
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	16.0000	0.878114
$333$	−8.00000	−0.438397
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	−14.0000	−0.760376
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	6.00000	0.322562
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−2.00000	−0.106600
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	2.00000	0.105703
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−22.0000	−1.15629
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	10.0000	0.522708
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−8.00000	−0.417029
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	−2.00000	−0.103695
$373$	36.0000	1.86401	0.932005	−	0.362446i	$-0.118058\pi$
$373$	36.0000	1.86401	0.932005	+	0.362446i	$0.118058\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−10.0000	−0.515711
$377$	0	0
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	2.00000	0.101666
$388$	6.00000	0.304604
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	0	0
$393$	12.0000	0.605320
$394$	18.0000	0.906827
$395$	0	0
$396$	2.00000	0.100504
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	−2.00000	−0.0997509
$403$	4.00000	0.199254
$404$	14.0000	0.696526
$405$	0	0
$406$	0	0
$407$	−16.0000	−0.793091
$408$	−4.00000	−0.198030
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−20.0000	−0.985329
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000	0.194717
$423$	10.0000	0.486217
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−4.00000	−0.193122
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	10.0000	0.477818
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	8.00000	0.380521
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−16.0000	−0.757622
$447$	16.0000	0.756774
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	14.0000	0.658505
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	42.0000	1.96468	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	42.0000	1.96468	0.982339	+	0.187112i	$0.0599128\pi$
$458$	10.0000	0.467269
$459$	4.00000	0.186704
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	−14.0000	−0.648537
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	4.00000	0.184115
$473$	4.00000	0.183920
$474$	16.0000	0.734904
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	8.00000	0.365911
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	−20.0000	−0.910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	−10.0000	−0.452679
$489$	−10.0000	−0.452216
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	0	0
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	16.0000	0.716977
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	−20.0000	−0.892644
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	0	0
$506$	16.0000	0.711287
$507$	9.00000	0.399704
$508$	12.0000	0.532414
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	20.0000	0.879599
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	−2.00000	−0.0870388
$529$	41.0000	1.78261
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	−14.0000	−0.605839
$535$	0	0
$536$	2.00000	0.0863868
$537$	2.00000	0.0863064
$538$	−10.0000	−0.431131
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−14.0000	−0.601351
$543$	−22.0000	−0.944110
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	2.00000	0.0854358
$549$	10.0000	0.426790
$550$	0	0
$551$	0	0
$552$	−8.00000	−0.340503
$553$	0	0
$554$	−28.0000	−1.18961
$555$	0	0
$556$	4.00000	0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−2.00000	−0.0846668
$559$	4.00000	0.169182
$560$	0	0
$561$	8.00000	0.337760
$562$	−14.0000	−0.590554
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−10.0000	−0.421076
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	12.0000	0.503509
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	0.0415945
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	6.00000	0.248708
$583$	4.00000	0.165663
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−30.0000	−1.23929
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	−8.00000	−0.328798
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−16.0000	−0.655386
$597$	10.0000	0.409273
$598$	16.0000	0.654289
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	0	0
$606$	14.0000	0.568711
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.0000	0.809113
$612$	−4.00000	−0.161690
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−20.0000	−0.804518
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	−20.0000	−0.801927
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	−10.0000	−0.399043
$629$	32.0000	1.27592
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−16.0000	−0.636446
$633$	4.00000	0.158986
$634$	6.00000	0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	−12.0000	−0.473602
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	−1.00000	−0.0392837
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	10.0000	0.391630
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	2.00000	0.0780869
$657$	10.0000	0.390137
$658$	0	0
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	4.00000	0.155464
$663$	8.00000	0.310694
$664$	−16.0000	−0.620920
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	18.0000	0.696441
$669$	−16.0000	−0.618596
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	14.0000	0.537667
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	−4.00000	−0.153168
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.0000	0.381524
$688$	2.00000	0.0762493
$689$	4.00000	0.152388
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	−8.00000	−0.303022
$698$	10.0000	0.378506
$699$	−14.0000	−0.529529
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	2.00000	0.0753778
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	4.00000	0.150329
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	14.0000	0.524672
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	8.00000	0.298765
$718$	20.0000	0.746393
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	0	0
$722$	19.0000	0.707107
$723$	−20.0000	−0.743808
$724$	22.0000	0.817624
$725$	0	0
$726$	−7.00000	−0.259794
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	−10.0000	−0.369611
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	8.00000	0.294884
$737$	−4.00000	−0.147342
$738$	−2.00000	−0.0736210
$739$	48.0000	1.76571	0.882854	−	0.469647i	$-0.155619\pi$
$739$	48.0000	1.76571	0.882854	+	0.469647i	$0.155619\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−36.0000	−1.31805
$747$	16.0000	0.585409
$748$	−8.00000	−0.292509
$749$	0	0
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	10.0000	0.364662
$753$	−20.0000	−0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	−8.00000	−0.290573
$759$	16.0000	0.580763
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	12.0000	0.434714
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−14.0000	−0.505841
$767$	−8.00000	−0.288863
$768$	−1.00000	−0.0360844
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	18.0000	0.647834
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	24.0000	0.860442
$779$	0	0
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−32.0000	−1.14432
$783$	0	0
$784$	0	0
$785$	0	0
$786$	−12.0000	−0.428026
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	20.0000	0.710221
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	−14.0000	−0.494666
$802$	14.0000	0.494357
$803$	20.0000	0.705785
$804$	2.00000	0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−10.0000	−0.352017
$808$	−14.0000	−0.492518
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−14.0000	−0.491001
$814$	16.0000	0.560800
$815$	0	0
$816$	4.00000	0.140028
$817$	0	0
$818$	−32.0000	−1.11885
$819$	0	0
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	2.00000	0.0697580
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	20.0000	0.696733
$825$	0	0
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	−8.00000	−0.278019
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	2.00000	0.0693375
$833$	0	0
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	36.0000	1.24360
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−38.0000	−1.30957
$843$	−14.0000	−0.482186
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−10.0000	−0.343807
$847$	0	0
$848$	2.00000	0.0686803
$849$	−4.00000	−0.137280
$850$	0	0
$851$	64.0000	2.19389
$852$	12.0000	0.411113
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	4.00000	0.136558
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−38.0000	−1.29129
$867$	1.00000	0.0339618
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−4.00000	−0.135535
$872$	2.00000	0.0677285
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−12.0000	−0.405211	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000	−0.405211	−0.202606	+	0.979260i	$0.564941\pi$
$878$	−26.0000	−0.877457
$879$	−30.0000	−1.01187
$880$	0	0
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	28.0000	0.940678
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	16.0000	0.535720
$893$	0	0
$894$	−16.0000	−0.535120
$895$	0	0
$896$	0	0
$897$	16.0000	0.534224
$898$	−6.00000	−0.200223
$899$	0	0
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−4.00000	−0.133185
$903$	0	0
$904$	−14.0000	−0.465633
$905$	0	0
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−12.0000	−0.398234
$909$	14.0000	0.464351
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	−42.0000	−1.38924
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−4.00000	−0.132020
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−18.0000	−0.592798
$923$	−24.0000	−0.789970
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	−20.0000	−0.656886
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	14.0000	0.458585
$933$	−20.0000	−0.654771
$934$	−8.00000	−0.261768
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	−10.0000	−0.325818
$943$	−16.0000	−0.521032
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−4.00000	−0.130051
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	−16.0000	−0.519656
$949$	20.0000	0.649227
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	4.00000	0.129234
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	16.0000	0.515861
$963$	−12.0000	−0.386695
$964$	20.0000	0.644157
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	28.0000	0.897178
$975$	0	0
$976$	10.0000	0.320092
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	10.0000	0.319765
$979$	−28.0000	−0.894884
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−6.00000	−0.191468
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−2.00000	−0.0635001
$993$	4.00000	0.126936
$994$	0	0
$995$	0	0
$996$	−16.0000	−0.506979
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−40.0000	−1.26618
$999$	8.00000	0.253109

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.o.1.1		1
5.4	even	2		1470.2.a.p.1.1	yes	1
7.6	odd	2		7350.2.a.bh.1.1		1
15.14	odd	2		4410.2.a.n.1.1		1
35.4	even	6		1470.2.i.c.961.1		2
35.9	even	6		1470.2.i.c.361.1		2
35.19	odd	6		1470.2.i.g.361.1		2
35.24	odd	6		1470.2.i.g.961.1		2
35.34	odd	2		1470.2.a.n.1.1	✓	1
105.104	even	2		4410.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.n.1.1	✓	1	35.34	odd	2
1470.2.a.p.1.1	yes	1	5.4	even	2
1470.2.i.c.361.1		2	35.9	even	6
1470.2.i.c.961.1		2	35.4	even	6
1470.2.i.g.361.1		2	35.19	odd	6
1470.2.i.g.961.1		2	35.24	odd	6
4410.2.a.e.1.1		1	105.104	even	2
4410.2.a.n.1.1		1	15.14	odd	2
7350.2.a.o.1.1		1	1.1	even	1	trivial
7350.2.a.bh.1.1		1	7.6	odd	2

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7350.2.a.o.1.1

Level	$7350$
Weight	$2$
Character	7350.1
Self dual	yes
Analytic conductor	$58.690$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$