LMFDB - Embedded newform 930.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [930,2,Mod(1,930)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-1,1,1,1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.42608738798$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} -2.00000 q^{21} +4.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} -10.0000 q^{67} -6.00000 q^{68} -2.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -1.00000 q^{75} -8.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} -4.00000 q^{87} +4.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -1.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	1.00000	0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	−4.00000	−0.554700
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−4.00000	−0.525226
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−1.00000	−0.127000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−4.00000	−0.492366
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−4.00000	−0.464991
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−8.00000	−0.911685
$78$	−4.00000	−0.452911
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−2.00000	−0.218218
$85$	−6.00000	−0.650791
$86$	8.00000	0.862662
$87$	−4.00000	−0.428845
$88$	4.00000	0.426401
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−1.00000	−0.105409
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−1.00000	−0.103695
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	3.00000	0.303046
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−6.00000	−0.594089
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	4.00000	0.392232
$105$	−2.00000	−0.195180
$106$	2.00000	0.194257
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	−1.00000	−0.0962250
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	4.00000	0.381385
$111$	−4.00000	−0.379663
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	−4.00000	−0.369800
$118$	−6.00000	−0.552345
$119$	−12.0000	−1.10004
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	4.00000	0.350823
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	10.0000	0.863868
$135$	−1.00000	−0.0860663
$136$	6.00000	0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	2.00000	0.169031
$141$	12.0000	1.01058
$142$	14.0000	1.17485
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−4.00000	−0.331042
$147$	3.00000	0.247436
$148$	4.00000	0.328798
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	1.00000	0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	8.00000	0.644658
$155$	1.00000	0.0803219
$156$	4.00000	0.320256
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	8.00000	0.636446
$159$	2.00000	0.158610
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−6.00000	−0.468521
$165$	4.00000	0.311400
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	6.00000	0.460179
$171$	0	0
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	4.00000	0.303239
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	−6.00000	−0.450988
$178$	8.00000	0.599625
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	1.00000	0.0745356
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	8.00000	0.592999
$183$	−10.0000	−0.739221
$184$	0	0
$185$	4.00000	0.294086
$186$	1.00000	0.0733236
$187$	24.0000	1.75505
$188$	−12.0000	−0.875190
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	2.00000	0.143592
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	10.0000	0.705346
$202$	−14.0000	−0.985037
$203$	8.00000	0.561490
$204$	6.00000	0.420084
$205$	−6.00000	−0.419058
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−4.00000	−0.277350
$209$	0	0
$210$	2.00000	0.138013
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−2.00000	−0.137361
$213$	14.0000	0.959264
$214$	16.0000	1.09374
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	−18.0000	−1.21911
$219$	−4.00000	−0.270295
$220$	−4.00000	−0.269680
$221$	24.0000	1.61441
$222$	4.00000	0.268462
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	−4.00000	−0.262613
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	4.00000	0.261488
$235$	−12.0000	−0.782794
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	12.0000	0.777844
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−1.00000	−0.0645497
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	−3.00000	−0.191663
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	4.00000	0.253490
$250$	−1.00000	−0.0632456
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	12.0000	0.752947
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−8.00000	−0.498058
$259$	8.00000	0.497096
$260$	−4.00000	−0.248069
$261$	4.00000	0.247594
$262$	−14.0000	−0.864923
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−4.00000	−0.246183
$265$	−2.00000	−0.122859
$266$	0	0
$267$	8.00000	0.489592
$268$	−10.0000	−0.610847
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	1.00000	0.0608581
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	−6.00000	−0.363803
$273$	8.00000	0.484182
$274$	18.0000	1.08742
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	12.0000	0.719712
$279$	1.00000	0.0598684
$280$	−2.00000	−0.119523
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−12.0000	−0.714590
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	−16.0000	−0.946100
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	2.00000	0.117242
$292$	4.00000	0.234082
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−3.00000	−0.174964
$295$	6.00000	0.349334
$296$	−4.00000	−0.232495
$297$	4.00000	0.232104
$298$	−14.0000	−0.810998
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−16.0000	−0.922225
$302$	16.0000	0.920697
$303$	−14.0000	−0.804279
$304$	0	0
$305$	10.0000	0.572598
$306$	6.00000	0.342997
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	−8.00000	−0.455842
$309$	−14.0000	−0.796432
$310$	−1.00000	−0.0567962
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	−4.00000	−0.226455
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	6.00000	0.338600
$315$	2.00000	0.112687
$316$	−8.00000	−0.450035
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	−2.00000	−0.112154
$319$	−16.0000	−0.895828
$320$	1.00000	0.0559017
$321$	16.0000	0.893033
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−6.00000	−0.332309
$327$	−18.0000	−0.995402
$328$	6.00000	0.331295
$329$	−24.0000	−1.32316
$330$	−4.00000	−0.220193
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−4.00000	−0.219529
$333$	4.00000	0.219199
$334$	0	0
$335$	−10.0000	−0.546358
$336$	−2.00000	−0.109109
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−3.00000	−0.163178
$339$	6.00000	0.325875
$340$	−6.00000	−0.325396
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−20.0000	−1.07990
$344$	8.00000	0.431331
$345$	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	$0.465758\pi$
$348$	−4.00000	−0.214423
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−2.00000	−0.106904
$351$	4.00000	0.213504
$352$	4.00000	0.213201
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	6.00000	0.318896
$355$	−14.0000	−0.743043
$356$	−8.00000	−0.423999
$357$	12.0000	0.635107
$358$	4.00000	0.211407
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	−18.0000	−0.946059
$363$	−5.00000	−0.262432
$364$	−8.00000	−0.419314
$365$	4.00000	0.209370
$366$	10.0000	0.522708
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−4.00000	−0.207950
$371$	−4.00000	−0.207670
$372$	−1.00000	−0.0518476
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	−24.0000	−1.24101
$375$	−1.00000	−0.0516398
$376$	12.0000	0.618853
$377$	−16.0000	−0.824042
$378$	2.00000	0.102869
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	10.0000	0.511645
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	−8.00000	−0.407718
$386$	2.00000	0.101797
$387$	−8.00000	−0.406663
$388$	−2.00000	−0.101535
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	3.00000	0.151523
$393$	−14.0000	−0.706207
$394$	−2.00000	−0.100759
$395$	−8.00000	−0.402524
$396$	−4.00000	−0.201008
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	−10.0000	−0.498755
$403$	−4.00000	−0.199254
$404$	14.0000	0.696526
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	−16.0000	−0.793091
$408$	−6.00000	−0.297044
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	6.00000	0.296319
$411$	18.0000	0.887875
$412$	14.0000	0.689730
$413$	12.0000	0.590481
$414$	0	0
$415$	−4.00000	−0.196352
$416$	4.00000	0.196116
$417$	12.0000	0.587643
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−2.00000	−0.0975900
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	−16.0000	−0.778868
$423$	−12.0000	−0.583460
$424$	2.00000	0.0971286
$425$	−6.00000	−0.291043
$426$	−14.0000	−0.678302
$427$	20.0000	0.967868
$428$	−16.0000	−0.773389
$429$	−16.0000	−0.772487
$430$	8.00000	0.385794
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−1.00000	−0.0481125
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−2.00000	−0.0960031
$435$	−4.00000	−0.191785
$436$	18.0000	0.862044
$437$	0	0
$438$	4.00000	0.191127
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	4.00000	0.190693
$441$	−3.00000	−0.142857
$442$	−24.0000	−1.14156
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−4.00000	−0.189832
$445$	−8.00000	−0.379236
$446$	−16.0000	−0.757622
$447$	−14.0000	−0.662177
$448$	2.00000	0.0944911
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	−6.00000	−0.282216
$453$	16.0000	0.751746
$454$	−20.0000	−0.938647
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	26.0000	1.21490
$459$	6.00000	0.280056
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	−8.00000	−0.372194
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	4.00000	0.185695
$465$	−1.00000	−0.0463739
$466$	−6.00000	−0.277945
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−4.00000	−0.184900
$469$	−20.0000	−0.923514
$470$	12.0000	0.553519
$471$	6.00000	0.276465
$472$	−6.00000	−0.276172
$473$	32.0000	1.47136
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	−2.00000	−0.0915737
$478$	−20.0000	−0.914779
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	1.00000	0.0456435
$481$	−16.0000	−0.729537
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	−2.00000	−0.0908153
$486$	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
$489$	−6.00000	−0.271329
$490$	3.00000	0.135526
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000	0.270501
$493$	−24.0000	−1.08091
$494$	0	0
$495$	−4.00000	−0.179787
$496$	1.00000	0.0449013
$497$	−28.0000	−1.25597
$498$	−4.00000	−0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	28.0000	1.24970
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−2.00000	−0.0890871
$505$	14.0000	0.622992
$506$	0	0
$507$	−3.00000	−0.133235
$508$	−12.0000	−0.532414
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−6.00000	−0.265684
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	14.0000	0.616914
$516$	8.00000	0.352180
$517$	48.0000	2.11104
$518$	−8.00000	−0.351500
$519$	−6.00000	−0.263371
$520$	4.00000	0.175412
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−4.00000	−0.175075
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	14.0000	0.611593
$525$	−2.00000	−0.0872872
$526$	−24.0000	−1.04645
$527$	−6.00000	−0.261364
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	2.00000	0.0868744
$531$	6.00000	0.260378
$532$	0	0
$533$	24.0000	1.03956
$534$	−8.00000	−0.346194
$535$	−16.0000	−0.691740
$536$	10.0000	0.431934
$537$	4.00000	0.172613
$538$	12.0000	0.517357
$539$	12.0000	0.516877
$540$	−1.00000	−0.0430331
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	24.0000	1.03089
$543$	−18.0000	−0.772454
$544$	6.00000	0.257248
$545$	18.0000	0.771035
$546$	−8.00000	−0.342368
$547$	46.0000	1.96682	0.983409	−	0.181402i	$-0.0580636\pi$
$547$	46.0000	1.96682	0.983409	+	0.181402i	$0.0580636\pi$
$548$	−18.0000	−0.768922
$549$	10.0000	0.426790
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	8.00000	0.339887
$555$	−4.00000	−0.169791
$556$	−12.0000	−0.508913
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−1.00000	−0.0423334
$559$	32.0000	1.35346
$560$	2.00000	0.0845154
$561$	−24.0000	−1.01328
$562$	−6.00000	−0.253095
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	12.0000	0.505291
$565$	−6.00000	−0.252422
$566$	−14.0000	−0.588464
$567$	2.00000	0.0839921
$568$	14.0000	0.587427
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	16.0000	0.668994
$573$	10.0000	0.417756
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−19.0000	−0.790296
$579$	2.00000	0.0831172
$580$	4.00000	0.166091
$581$	−8.00000	−0.331896
$582$	−2.00000	−0.0829027
$583$	8.00000	0.331326
$584$	−4.00000	−0.165521
$585$	−4.00000	−0.165380
$586$	−26.0000	−1.07405
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	−6.00000	−0.247016
$591$	−2.00000	−0.0822690
$592$	4.00000	0.164399
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	−4.00000	−0.164122
$595$	−12.0000	−0.491952
$596$	14.0000	0.573462
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	1.00000	0.0408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	16.0000	0.652111
$603$	−10.0000	−0.407231
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	14.0000	0.568711
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	−10.0000	−0.404888
$611$	48.0000	1.94187
$612$	−6.00000	−0.242536
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	−2.00000	−0.0807134
$615$	6.00000	0.241943
$616$	8.00000	0.322329
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	14.0000	0.563163
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	−6.00000	−0.240578
$623$	−16.0000	−0.641026
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	4.00000	0.159872
$627$	0	0
$628$	−6.00000	−0.239426
$629$	−24.0000	−0.956943
$630$	−2.00000	−0.0796819
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	8.00000	0.318223
$633$	−16.0000	−0.635943
$634$	−26.0000	−1.03259
$635$	−12.0000	−0.476205
$636$	2.00000	0.0793052
$637$	12.0000	0.475457
$638$	16.0000	0.633446
$639$	−14.0000	−0.553831
$640$	−1.00000	−0.0395285
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	−16.0000	−0.631470
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−1.00000	−0.0392837
$649$	−24.0000	−0.942082
$650$	4.00000	0.156893
$651$	−2.00000	−0.0783862
$652$	6.00000	0.234978
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	18.0000	0.703856
$655$	14.0000	0.547025
$656$	−6.00000	−0.234261
$657$	4.00000	0.156055
$658$	24.0000	0.935617
$659$	−22.0000	−0.856998	−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	4.00000	0.155700
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	28.0000	1.08825
$663$	−24.0000	−0.932083
$664$	4.00000	0.155230
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	10.0000	0.386334
$671$	−40.0000	−1.54418
$672$	2.00000	0.0771517
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	−8.00000	−0.308148
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	−6.00000	−0.230429
$679$	−4.00000	−0.153506
$680$	6.00000	0.230089
$681$	−20.0000	−0.766402
$682$	4.00000	0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	20.0000	0.763604
$687$	26.0000	0.991962
$688$	−8.00000	−0.304997
$689$	8.00000	0.304776
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	6.00000	0.228086
$693$	−8.00000	−0.303895
$694$	−4.00000	−0.151838
$695$	−12.0000	−0.455186
$696$	4.00000	0.151620
$697$	36.0000	1.36360
$698$	−2.00000	−0.0757011
$699$	−6.00000	−0.226941
$700$	2.00000	0.0755929
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	−4.00000	−0.150756
$705$	12.0000	0.451946
$706$	−10.0000	−0.376355
$707$	28.0000	1.05305
$708$	−6.00000	−0.225494
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	14.0000	0.525411
$711$	−8.00000	−0.300023
$712$	8.00000	0.299813
$713$	0	0
$714$	−12.0000	−0.449089
$715$	16.0000	0.598366
$716$	−4.00000	−0.149487
$717$	−20.0000	−0.746914
$718$	−30.0000	−1.11959
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	1.00000	0.0372678
$721$	28.0000	1.04277
$722$	19.0000	0.707107
$723$	−10.0000	−0.371904
$724$	18.0000	0.668965
$725$	4.00000	0.148556
$726$	5.00000	0.185567
$727$	−30.0000	−1.11264	−0.556319	−	0.830969i	$-0.687787\pi$
$727$	−30.0000	−1.11264	−0.556319	+	0.830969i	$0.687787\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	48.0000	1.77534
$732$	−10.0000	−0.369611
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	40.0000	1.47342
$738$	6.00000	0.220863
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	4.00000	0.146845
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	1.00000	0.0366618
$745$	14.0000	0.512920
$746$	−18.0000	−0.659027
$747$	−4.00000	−0.146352
$748$	24.0000	0.877527
$749$	−32.0000	−1.16925
$750$	1.00000	0.0365148
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	−12.0000	−0.437595
$753$	28.0000	1.02038
$754$	16.0000	0.582686
$755$	−16.0000	−0.582300
$756$	−2.00000	−0.0727393
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	−12.0000	−0.434714
$763$	36.0000	1.30329
$764$	−10.0000	−0.361787
$765$	−6.00000	−0.216930
$766$	−24.0000	−0.867155
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	8.00000	0.288300
$771$	2.00000	0.0720282
$772$	−2.00000	−0.0719816
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	8.00000	0.287554
$775$	1.00000	0.0359211
$776$	2.00000	0.0717958
$777$	−8.00000	−0.286998
$778$	−36.0000	−1.29066
$779$	0	0
$780$	4.00000	0.143223
$781$	56.0000	2.00384
$782$	0	0
$783$	−4.00000	−0.142948
$784$	−3.00000	−0.107143
$785$	−6.00000	−0.214149
$786$	14.0000	0.499363
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	2.00000	0.0712470
$789$	−24.0000	−0.854423
$790$	8.00000	0.284627
$791$	−12.0000	−0.426671
$792$	4.00000	0.142134
$793$	−40.0000	−1.42044
$794$	18.0000	0.638796
$795$	2.00000	0.0709327
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	−1.00000	−0.0353553
$801$	−8.00000	−0.282666
$802$	4.00000	0.141245
$803$	−16.0000	−0.564628
$804$	10.0000	0.352673
$805$	0	0
$806$	4.00000	0.140894
$807$	12.0000	0.422420
$808$	−14.0000	−0.492518
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	−1.00000	−0.0351364
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	8.00000	0.280745
$813$	24.0000	0.841717
$814$	16.0000	0.560800
$815$	6.00000	0.210171
$816$	6.00000	0.210042
$817$	0	0
$818$	−6.00000	−0.209785
$819$	−8.00000	−0.279543
$820$	−6.00000	−0.209529
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	−18.0000	−0.627822
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−14.0000	−0.487713
$825$	4.00000	0.139262
$826$	−12.0000	−0.417533
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	4.00000	0.138842
$831$	8.00000	0.277517
$832$	−4.00000	−0.138675
$833$	18.0000	0.623663
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	30.0000	1.03633
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	2.00000	0.0690066
$841$	−13.0000	−0.448276
$842$	30.0000	1.03387
$843$	−6.00000	−0.206651
$844$	16.0000	0.550743
$845$	3.00000	0.103203
$846$	12.0000	0.412568
$847$	10.0000	0.343604
$848$	−2.00000	−0.0686803
$849$	−14.0000	−0.480479
$850$	6.00000	0.205798
$851$	0	0
$852$	14.0000	0.479632
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	16.0000	0.546869
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	16.0000	0.546231
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	−8.00000	−0.272798
$861$	12.0000	0.408959
$862$	10.0000	0.340601
$863$	−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
$865$	6.00000	0.204006
$866$	4.00000	0.135926
$867$	−19.0000	−0.645274
$868$	2.00000	0.0678844
$869$	32.0000	1.08553
$870$	4.00000	0.135613
$871$	40.0000	1.35535
$872$	−18.0000	−0.609557
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	2.00000	0.0676123
$876$	−4.00000	−0.135147
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−24.0000	−0.809961
$879$	−26.0000	−0.876958
$880$	−4.00000	−0.134840
$881$	−56.0000	−1.88669	−0.943344	−	0.331816i	$-0.892339\pi$
$881$	−56.0000	−1.88669	−0.943344	+	0.331816i	$0.892339\pi$
$882$	3.00000	0.101015
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	24.0000	0.807207
$885$	−6.00000	−0.201688
$886$	12.0000	0.403148
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	4.00000	0.134231
$889$	−24.0000	−0.804934
$890$	8.00000	0.268161
$891$	−4.00000	−0.134005
$892$	16.0000	0.535720
$893$	0	0
$894$	14.0000	0.468230
$895$	−4.00000	−0.133705
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−20.0000	−0.667409
$899$	4.00000	0.133407
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	−24.0000	−0.799113
$903$	16.0000	0.532447
$904$	6.00000	0.199557
$905$	18.0000	0.598340
$906$	−16.0000	−0.531564
$907$	58.0000	1.92586	0.962929	−	0.269754i	$-0.0869425\pi$
$907$	58.0000	1.92586	0.962929	+	0.269754i	$0.0869425\pi$
$908$	20.0000	0.663723
$909$	14.0000	0.464351
$910$	8.00000	0.265197
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	4.00000	0.132308
$915$	−10.0000	−0.330590
$916$	−26.0000	−0.859064
$917$	28.0000	0.924641
$918$	−6.00000	−0.198030
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	−24.0000	−0.790398
$923$	56.0000	1.84326
$924$	8.00000	0.263181
$925$	4.00000	0.131519
$926$	−12.0000	−0.394344
$927$	14.0000	0.459820
$928$	−4.00000	−0.131306
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	1.00000	0.0327913
$931$	0	0
$932$	6.00000	0.196537
$933$	−6.00000	−0.196431
$934$	24.0000	0.785304
$935$	24.0000	0.784884
$936$	4.00000	0.130744
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	20.0000	0.653023
$939$	4.00000	0.130535
$940$	−12.0000	−0.391397
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	−6.00000	−0.195491
$943$	0	0
$944$	6.00000	0.195283
$945$	−2.00000	−0.0650600
$946$	−32.0000	−1.04041
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	8.00000	0.259828
$949$	−16.0000	−0.519382
$950$	0	0
$951$	−26.0000	−0.843108
$952$	12.0000	0.388922
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	2.00000	0.0647524
$955$	−10.0000	−0.323592
$956$	20.0000	0.646846
$957$	16.0000	0.517207
$958$	14.0000	0.452319
$959$	−36.0000	−1.16250
$960$	−1.00000	−0.0322749
$961$	1.00000	0.0322581
$962$	16.0000	0.515861
$963$	−16.0000	−0.515593
$964$	10.0000	0.322078
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	2.00000	0.0642161
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	−1.00000	−0.0320750
$973$	−24.0000	−0.769405
$974$	28.0000	0.897178
$975$	4.00000	0.128103
$976$	10.0000	0.320092
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	6.00000	0.191859
$979$	32.0000	1.02272
$980$	−3.00000	−0.0958315
$981$	18.0000	0.574696
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−6.00000	−0.191273
$985$	2.00000	0.0637253
$986$	24.0000	0.764316
$987$	24.0000	0.763928
$988$	0	0
$989$	0	0
$990$	4.00000	0.127128
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−1.00000	−0.0317500
$993$	28.0000	0.888553
$994$	28.0000	0.888106
$995$	0	0
$996$	4.00000	0.126745
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	4.00000	0.126618
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.d.1.1	✓	1
3.2	odd	2		2790.2.a.t.1.1		1
4.3	odd	2		7440.2.a.y.1.1		1
5.2	odd	4		4650.2.d.p.3349.1		2
5.3	odd	4		4650.2.d.p.3349.2		2
5.4	even	2		4650.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.d.1.1	✓	1	1.1	even	1	trivial
2790.2.a.t.1.1		1	3.2	odd	2
4650.2.a.bl.1.1		1	5.4	even	2
4650.2.d.p.3349.1		2	5.2	odd	4
4650.2.d.p.3349.2		2	5.3	odd	4
7440.2.a.y.1.1		1	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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