LMFDB - Embedded newform 930.2.a.a.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,-3] 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:	$0$
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)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} -3.00000 q^{22} +5.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} +4.00000 q^{29} -1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +4.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +3.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -5.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} +3.00000 q^{56} +3.00000 q^{57} -4.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -1.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{66} +2.00000 q^{67} -4.00000 q^{68} -5.00000 q^{69} -3.00000 q^{70} +7.00000 q^{71} -1.00000 q^{72} +5.00000 q^{73} -1.00000 q^{75} -3.00000 q^{76} -9.00000 q^{77} -2.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +12.0000 q^{83} +3.00000 q^{84} +4.00000 q^{85} -1.00000 q^{86} -4.00000 q^{87} -3.00000 q^{88} +1.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +5.00000 q^{92} -1.00000 q^{93} -10.0000 q^{94} +3.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.00000	0.801784
$15$	1.00000	0.258199
$16$	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
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	−3.00000	−0.639602
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	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
$33$	−3.00000	−0.522233
$34$	4.00000	0.685994
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	3.00000	0.486664
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	−3.00000	−0.462910
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	3.00000	0.452267
$45$	−1.00000	−0.149071
$46$	−5.00000	−0.737210
$47$	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
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	4.00000	0.560112
$52$	−2.00000	−0.277350
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	1.00000	0.136083
$55$	−3.00000	−0.404520
$56$	3.00000	0.400892
$57$	3.00000	0.397360
$58$	−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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−1.00000	−0.127000
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	3.00000	0.369274
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−4.00000	−0.485071
$69$	−5.00000	−0.601929
$70$	−3.00000	−0.358569
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	−1.00000	−0.117851
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	−3.00000	−0.344124
$77$	−9.00000	−1.02565
$78$	−2.00000	−0.226455
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	3.00000	0.327327
$85$	4.00000	0.433861
$86$	−1.00000	−0.107833
$87$	−4.00000	−0.428845
$88$	−3.00000	−0.319801
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	1.00000	0.105409
$91$	6.00000	0.628971
$92$	5.00000	0.521286
$93$	−1.00000	−0.103695
$94$	−10.0000	−1.03142
$95$	3.00000	0.307794
$96$	1.00000	0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−2.00000	−0.202031
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	−4.00000	−0.396059
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	0.196116
$105$	−3.00000	−0.292770
$106$	−3.00000	−0.291386
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	−1.00000	−0.0962250
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	−3.00000	−0.283473
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−3.00000	−0.280976
$115$	−5.00000	−0.466252
$116$	4.00000	0.371391
$117$	−2.00000	−0.184900
$118$	−6.00000	−0.552345
$119$	12.0000	1.10004
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	2.00000	0.181071
$123$	−4.00000	−0.360668
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	3.00000	0.267261
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	−2.00000	−0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−3.00000	−0.261116
$133$	9.00000	0.780399
$134$	−2.00000	−0.172774
$135$	1.00000	0.0860663
$136$	4.00000	0.342997
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	5.00000	0.425628
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	3.00000	0.253546
$141$	−10.0000	−0.842152
$142$	−7.00000	−0.587427
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−5.00000	−0.413803
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	3.00000	0.243332
$153$	−4.00000	−0.323381
$154$	9.00000	0.725241
$155$	−1.00000	−0.0803219
$156$	2.00000	0.160128
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	1.00000	0.0795557
$159$	−3.00000	−0.237915
$160$	1.00000	0.0790569
$161$	−15.0000	−1.18217
$162$	−1.00000	−0.0785674
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	4.00000	0.312348
$165$	3.00000	0.233550
$166$	−12.0000	−0.931381
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	−3.00000	−0.229416
$172$	1.00000	0.0762493
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	4.00000	0.303239
$175$	−3.00000	−0.226779
$176$	3.00000	0.226134
$177$	−6.00000	−0.450988
$178$	−1.00000	−0.0749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.00000	−0.0745356
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−6.00000	−0.444750
$183$	2.00000	0.147844
$184$	−5.00000	−0.368605
$185$	0	0
$186$	1.00000	0.0733236
$187$	−12.0000	−0.877527
$188$	10.0000	0.729325
$189$	3.00000	0.218218
$190$	−3.00000	−0.217643
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	10.0000	0.717958
$195$	−2.00000	−0.143223
$196$	2.00000	0.142857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−3.00000	−0.213201
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	−1.00000	−0.0707107
$201$	−2.00000	−0.141069
$202$	1.00000	0.0703598
$203$	−12.0000	−0.842235
$204$	4.00000	0.280056
$205$	−4.00000	−0.279372
$206$	−16.0000	−1.11477
$207$	5.00000	0.347524
$208$	−2.00000	−0.138675
$209$	−9.00000	−0.622543
$210$	3.00000	0.207020
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	3.00000	0.206041
$213$	−7.00000	−0.479632
$214$	−9.00000	−0.615227
$215$	−1.00000	−0.0681994
$216$	1.00000	0.0680414
$217$	−3.00000	−0.203653
$218$	−20.0000	−1.35457
$219$	−5.00000	−0.337869
$220$	−3.00000	−0.202260
$221$	8.00000	0.538138
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	3.00000	0.200446
$225$	1.00000	0.0666667
$226$	−9.00000	−0.598671
$227$	−27.0000	−1.79205	−0.896026	−	0.444001i	$-0.853559\pi$
$227$	−27.0000	−1.79205	−0.896026	+	0.444001i	$0.853559\pi$
$228$	3.00000	0.198680
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	5.00000	0.329690
$231$	9.00000	0.592157
$232$	−4.00000	−0.262613
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	2.00000	0.130744
$235$	−10.0000	−0.652328
$236$	6.00000	0.390567
$237$	1.00000	0.0649570
$238$	−12.0000	−0.777844
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	1.00000	0.0645497
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	−2.00000	−0.127775
$246$	4.00000	0.255031
$247$	6.00000	0.381771
$248$	−1.00000	−0.0635001
$249$	−12.0000	−0.760469
$250$	1.00000	0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.00000	−0.188982
$253$	15.0000	0.943042
$254$	−8.00000	−0.501965
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−23.0000	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$257$	−23.0000	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$258$	1.00000	0.0622573
$259$	0	0
$260$	2.00000	0.124035
$261$	4.00000	0.247594
$262$	−6.00000	−0.370681
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	3.00000	0.184637
$265$	−3.00000	−0.184289
$266$	−9.00000	−0.551825
$267$	−1.00000	−0.0611990
$268$	2.00000	0.122169
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	−1.00000	−0.0608581
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	−4.00000	−0.242536
$273$	−6.00000	−0.363137
$274$	10.0000	0.604122
$275$	3.00000	0.180907
$276$	−5.00000	−0.300965
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	14.0000	0.839664
$279$	1.00000	0.0598684
$280$	−3.00000	−0.179284
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	10.0000	0.595491
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	7.00000	0.415374
$285$	−3.00000	−0.177705
$286$	6.00000	0.354787
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	10.0000	0.586210
$292$	5.00000	0.292603
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	2.00000	0.116642
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−3.00000	−0.174078
$298$	11.0000	0.637213
$299$	−10.0000	−0.578315
$300$	−1.00000	−0.0577350
$301$	−3.00000	−0.172917
$302$	0	0
$303$	1.00000	0.0574485
$304$	−3.00000	−0.172062
$305$	2.00000	0.114520
$306$	4.00000	0.228665
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	−9.00000	−0.512823
$309$	−16.0000	−0.910208
$310$	1.00000	0.0567962
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−2.00000	−0.113228
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	5.00000	0.282166
$315$	3.00000	0.169031
$316$	−1.00000	−0.0562544
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	3.00000	0.168232
$319$	12.0000	0.671871
$320$	−1.00000	−0.0559017
$321$	−9.00000	−0.502331
$322$	15.0000	0.835917
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−14.0000	−0.775388
$327$	−20.0000	−1.10600
$328$	−4.00000	−0.220863
$329$	−30.0000	−1.65395
$330$	−3.00000	−0.165145
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	19.0000	1.03963
$335$	−2.00000	−0.109272
$336$	3.00000	0.163663
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	9.00000	0.489535
$339$	−9.00000	−0.488813
$340$	4.00000	0.216930
$341$	3.00000	0.162459
$342$	3.00000	0.162221
$343$	15.0000	0.809924
$344$	−1.00000	−0.0539164
$345$	5.00000	0.269191
$346$	22.0000	1.18273
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	−4.00000	−0.214423
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	3.00000	0.160357
$351$	2.00000	0.106752
$352$	−3.00000	−0.159901
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	6.00000	0.318896
$355$	−7.00000	−0.371521
$356$	1.00000	0.0529999
$357$	−12.0000	−0.635107
$358$	−4.00000	−0.211407
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	1.00000	0.0527046
$361$	−10.0000	−0.526316
$362$	−5.00000	−0.262794
$363$	2.00000	0.104973
$364$	6.00000	0.314485
$365$	−5.00000	−0.261712
$366$	−2.00000	−0.104542
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	5.00000	0.260643
$369$	4.00000	0.208232
$370$	0	0
$371$	−9.00000	−0.467257
$372$	−1.00000	−0.0518476
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	12.0000	0.620505
$375$	1.00000	0.0516398
$376$	−10.0000	−0.515711
$377$	−8.00000	−0.412021
$378$	−3.00000	−0.154303
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	3.00000	0.153897
$381$	−8.00000	−0.409852
$382$	−16.0000	−0.818631
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	9.00000	0.458682
$386$	6.00000	0.305392
$387$	1.00000	0.0508329
$388$	−10.0000	−0.507673
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	2.00000	0.101274
$391$	−20.0000	−1.01144
$392$	−2.00000	−0.101015
$393$	−6.00000	−0.302660
$394$	6.00000	0.302276
$395$	1.00000	0.0503155
$396$	3.00000	0.150756
$397$	35.0000	1.75660	0.878300	−	0.478110i	$-0.158678\pi$
$397$	35.0000	1.75660	0.878300	+	0.478110i	$0.158678\pi$
$398$	−7.00000	−0.350878
$399$	−9.00000	−0.450564
$400$	1.00000	0.0500000
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	2.00000	0.0997509
$403$	−2.00000	−0.0996271
$404$	−1.00000	−0.0497519
$405$	−1.00000	−0.0496904
$406$	12.0000	0.595550
$407$	0	0
$408$	−4.00000	−0.198030
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	4.00000	0.197546
$411$	10.0000	0.493264
$412$	16.0000	0.788263
$413$	−18.0000	−0.885722
$414$	−5.00000	−0.245737
$415$	−12.0000	−0.589057
$416$	2.00000	0.0980581
$417$	14.0000	0.685583
$418$	9.00000	0.440204
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−3.00000	−0.146385
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	19.0000	0.924906
$423$	10.0000	0.486217
$424$	−3.00000	−0.145693
$425$	−4.00000	−0.194029
$426$	7.00000	0.339151
$427$	6.00000	0.290360
$428$	9.00000	0.435031
$429$	6.00000	0.289683
$430$	1.00000	0.0482243
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	3.00000	0.144005
$435$	4.00000	0.191785
$436$	20.0000	0.957826
$437$	−15.0000	−0.717547
$438$	5.00000	0.238909
$439$	18.0000	0.859093	0.429547	−	0.903045i	$-0.358673\pi$
$439$	18.0000	0.859093	0.429547	+	0.903045i	$0.358673\pi$
$440$	3.00000	0.143019
$441$	2.00000	0.0952381
$442$	−8.00000	−0.380521
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	6.00000	0.284108
$447$	11.0000	0.520282
$448$	−3.00000	−0.141737
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	12.0000	0.565058
$452$	9.00000	0.423324
$453$	0	0
$454$	27.0000	1.26717
$455$	−6.00000	−0.281284
$456$	−3.00000	−0.140488
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−13.0000	−0.607450
$459$	4.00000	0.186704
$460$	−5.00000	−0.233126
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	−9.00000	−0.418718
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	4.00000	0.185695
$465$	1.00000	0.0463739
$466$	−15.0000	−0.694862
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−2.00000	−0.0924500
$469$	−6.00000	−0.277054
$470$	10.0000	0.461266
$471$	5.00000	0.230388
$472$	−6.00000	−0.276172
$473$	3.00000	0.137940
$474$	−1.00000	−0.0459315
$475$	−3.00000	−0.137649
$476$	12.0000	0.550019
$477$	3.00000	0.137361
$478$	−12.0000	−0.548867
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	18.0000	0.819878
$483$	15.0000	0.682524
$484$	−2.00000	−0.0909091
$485$	10.0000	0.454077
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	2.00000	0.0905357
$489$	−14.0000	−0.633102
$490$	2.00000	0.0903508
$491$	−37.0000	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$491$	−37.0000	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$492$	−4.00000	−0.180334
$493$	−16.0000	−0.720604
$494$	−6.00000	−0.269953
$495$	−3.00000	−0.134840
$496$	1.00000	0.0449013
$497$	−21.0000	−0.941979
$498$	12.0000	0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	19.0000	0.848857
$502$	12.0000	0.535586
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	3.00000	0.133631
$505$	1.00000	0.0444994
$506$	−15.0000	−0.666831
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	4.00000	0.177123
$511$	−15.0000	−0.663561
$512$	−1.00000	−0.0441942
$513$	3.00000	0.132453
$514$	23.0000	1.01449
$515$	−16.0000	−0.705044
$516$	−1.00000	−0.0440225
$517$	30.0000	1.31940
$518$	0	0
$519$	22.0000	0.965693
$520$	−2.00000	−0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−4.00000	−0.175075
$523$	−43.0000	−1.88026	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−43.0000	−1.88026	−0.940129	+	0.340818i	$0.889296\pi$
$524$	6.00000	0.262111
$525$	3.00000	0.130931
$526$	−24.0000	−1.04645
$527$	−4.00000	−0.174243
$528$	−3.00000	−0.130558
$529$	2.00000	0.0869565
$530$	3.00000	0.130312
$531$	6.00000	0.260378
$532$	9.00000	0.390199
$533$	−8.00000	−0.346518
$534$	1.00000	0.0432742
$535$	−9.00000	−0.389104
$536$	−2.00000	−0.0863868
$537$	−4.00000	−0.172613
$538$	−2.00000	−0.0862261
$539$	6.00000	0.258438
$540$	1.00000	0.0430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−13.0000	−0.558398
$543$	−5.00000	−0.214571
$544$	4.00000	0.171499
$545$	−20.0000	−0.856706
$546$	6.00000	0.256776
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−10.0000	−0.427179
$549$	−2.00000	−0.0853579
$550$	−3.00000	−0.127920
$551$	−12.0000	−0.511217
$552$	5.00000	0.212814
$553$	3.00000	0.127573
$554$	−12.0000	−0.509831
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−1.00000	−0.0423714	−0.0211857	−	0.999776i	$-0.506744\pi$
$557$	−1.00000	−0.0423714	−0.0211857	+	0.999776i	$0.506744\pi$
$558$	−1.00000	−0.0423334
$559$	−2.00000	−0.0845910
$560$	3.00000	0.126773
$561$	12.0000	0.506640
$562$	−16.0000	−0.674919
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−10.0000	−0.421076
$565$	−9.00000	−0.378633
$566$	24.0000	1.00880
$567$	−3.00000	−0.125988
$568$	−7.00000	−0.293713
$569$	−23.0000	−0.964210	−0.482105	−	0.876113i	$-0.660128\pi$
$569$	−23.0000	−0.964210	−0.482105	+	0.876113i	$0.660128\pi$
$570$	3.00000	0.125656
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	−6.00000	−0.250873
$573$	−16.0000	−0.668410
$574$	12.0000	0.500870
$575$	5.00000	0.208514
$576$	1.00000	0.0416667
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	1.00000	0.0415945
$579$	6.00000	0.249351
$580$	−4.00000	−0.166091
$581$	−36.0000	−1.49353
$582$	−10.0000	−0.414513
$583$	9.00000	0.372742
$584$	−5.00000	−0.206901
$585$	2.00000	0.0826898
$586$	−30.0000	−1.23929
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	−2.00000	−0.0824786
$589$	−3.00000	−0.123613
$590$	6.00000	0.247016
$591$	6.00000	0.246807
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	3.00000	0.123091
$595$	−12.0000	−0.491952
$596$	−11.0000	−0.450578
$597$	−7.00000	−0.286491
$598$	10.0000	0.408930
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	1.00000	0.0408248
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	3.00000	0.122271
$603$	2.00000	0.0814463
$604$	0	0
$605$	2.00000	0.0813116
$606$	−1.00000	−0.0406222
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	3.00000	0.121666
$609$	12.0000	0.486265
$610$	−2.00000	−0.0809776
$611$	−20.0000	−0.809113
$612$	−4.00000	−0.161690
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−16.0000	−0.645707
$615$	4.00000	0.161296
$616$	9.00000	0.362620
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	16.0000	0.643614
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	−1.00000	−0.0401610
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−3.00000	−0.120192
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−30.0000	−1.19904
$627$	9.00000	0.359425
$628$	−5.00000	−0.199522
$629$	0	0
$630$	−3.00000	−0.119523
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	1.00000	0.0397779
$633$	19.0000	0.755182
$634$	8.00000	0.317721
$635$	−8.00000	−0.317470
$636$	−3.00000	−0.118958
$637$	−4.00000	−0.158486
$638$	−12.0000	−0.475085
$639$	7.00000	0.276916
$640$	1.00000	0.0395285
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	9.00000	0.355202
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	−15.0000	−0.591083
$645$	1.00000	0.0393750
$646$	−12.0000	−0.472134
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	2.00000	0.0784465
$651$	3.00000	0.117579
$652$	14.0000	0.548282
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	20.0000	0.782062
$655$	−6.00000	−0.234439
$656$	4.00000	0.156174
$657$	5.00000	0.195069
$658$	30.0000	1.16952
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	3.00000	0.116775
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−12.0000	−0.466393
$663$	−8.00000	−0.310694
$664$	−12.0000	−0.465690
$665$	−9.00000	−0.349005
$666$	0	0
$667$	20.0000	0.774403
$668$	−19.0000	−0.735132
$669$	6.00000	0.231973
$670$	2.00000	0.0772667
$671$	−6.00000	−0.231627
$672$	−3.00000	−0.115728
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	2.00000	0.0770371
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	9.00000	0.345643
$679$	30.0000	1.15129
$680$	−4.00000	−0.153393
$681$	27.0000	1.03464
$682$	−3.00000	−0.114876
$683$	−19.0000	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$683$	−19.0000	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$684$	−3.00000	−0.114708
$685$	10.0000	0.382080
$686$	−15.0000	−0.572703
$687$	−13.0000	−0.495981
$688$	1.00000	0.0381246
$689$	−6.00000	−0.228582
$690$	−5.00000	−0.190347
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	−22.0000	−0.836315
$693$	−9.00000	−0.341882
$694$	32.0000	1.21470
$695$	14.0000	0.531050
$696$	4.00000	0.151620
$697$	−16.0000	−0.606043
$698$	24.0000	0.908413
$699$	−15.0000	−0.567352
$700$	−3.00000	−0.113389
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	3.00000	0.113067
$705$	10.0000	0.376622
$706$	−18.0000	−0.677439
$707$	3.00000	0.112827
$708$	−6.00000	−0.225494
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	7.00000	0.262705
$711$	−1.00000	−0.0375029
$712$	−1.00000	−0.0374766
$713$	5.00000	0.187251
$714$	12.0000	0.449089
$715$	6.00000	0.224387
$716$	4.00000	0.149487
$717$	−12.0000	−0.448148
$718$	−15.0000	−0.559795
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	−1.00000	−0.0372678
$721$	−48.0000	−1.78761
$722$	10.0000	0.372161
$723$	18.0000	0.669427
$724$	5.00000	0.185824
$725$	4.00000	0.148556
$726$	−2.00000	−0.0742270
$727$	45.0000	1.66896	0.834479	−	0.551040i	$-0.185769\pi$
$727$	45.0000	1.66896	0.834479	+	0.551040i	$0.185769\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	5.00000	0.185058
$731$	−4.00000	−0.147945
$732$	2.00000	0.0739221
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	−2.00000	−0.0738213
$735$	2.00000	0.0737711
$736$	−5.00000	−0.184302
$737$	6.00000	0.221013
$738$	−4.00000	−0.147242
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	9.00000	0.330400
$743$	−37.0000	−1.35740	−0.678699	−	0.734416i	$-0.737456\pi$
$743$	−37.0000	−1.35740	−0.678699	+	0.734416i	$0.737456\pi$
$744$	1.00000	0.0366618
$745$	11.0000	0.403009
$746$	9.00000	0.329513
$747$	12.0000	0.439057
$748$	−12.0000	−0.438763
$749$	−27.0000	−0.986559
$750$	−1.00000	−0.0365148
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	10.0000	0.364662
$753$	12.0000	0.437304
$754$	8.00000	0.291343
$755$	0	0
$756$	3.00000	0.109109
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	−15.0000	−0.544825
$759$	−15.0000	−0.544466
$760$	−3.00000	−0.108821
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	8.00000	0.289809
$763$	−60.0000	−2.17215
$764$	16.0000	0.578860
$765$	4.00000	0.144620
$766$	−12.0000	−0.433578
$767$	−12.0000	−0.433295
$768$	−1.00000	−0.0360844
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	−9.00000	−0.324337
$771$	23.0000	0.828325
$772$	−6.00000	−0.215945
$773$	−31.0000	−1.11499	−0.557496	−	0.830179i	$-0.688238\pi$
$773$	−31.0000	−1.11499	−0.557496	+	0.830179i	$0.688238\pi$
$774$	−1.00000	−0.0359443
$775$	1.00000	0.0359211
$776$	10.0000	0.358979
$777$	0	0
$778$	26.0000	0.932145
$779$	−12.0000	−0.429945
$780$	−2.00000	−0.0716115
$781$	21.0000	0.751439
$782$	20.0000	0.715199
$783$	−4.00000	−0.142948
$784$	2.00000	0.0714286
$785$	5.00000	0.178458
$786$	6.00000	0.214013
$787$	−35.0000	−1.24762	−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000	−1.24762	−0.623808	+	0.781578i	$0.714415\pi$
$788$	−6.00000	−0.213741
$789$	−24.0000	−0.854423
$790$	−1.00000	−0.0355784
$791$	−27.0000	−0.960009
$792$	−3.00000	−0.106600
$793$	4.00000	0.142044
$794$	−35.0000	−1.24210
$795$	3.00000	0.106399
$796$	7.00000	0.248108
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	9.00000	0.318597
$799$	−40.0000	−1.41510
$800$	−1.00000	−0.0353553
$801$	1.00000	0.0353333
$802$	−25.0000	−0.882781
$803$	15.0000	0.529339
$804$	−2.00000	−0.0705346
$805$	15.0000	0.528681
$806$	2.00000	0.0704470
$807$	−2.00000	−0.0704033
$808$	1.00000	0.0351799
$809$	17.0000	0.597688	0.298844	−	0.954302i	$-0.403399\pi$
$809$	17.0000	0.597688	0.298844	+	0.954302i	$0.403399\pi$
$810$	1.00000	0.0351364
$811$	−27.0000	−0.948098	−0.474049	−	0.880498i	$-0.657208\pi$
$811$	−27.0000	−0.948098	−0.474049	+	0.880498i	$0.657208\pi$
$812$	−12.0000	−0.421117
$813$	−13.0000	−0.455930
$814$	0	0
$815$	−14.0000	−0.490399
$816$	4.00000	0.140028
$817$	−3.00000	−0.104957
$818$	−8.00000	−0.279713
$819$	6.00000	0.209657
$820$	−4.00000	−0.139686
$821$	40.0000	1.39601	0.698005	−	0.716093i	$-0.254071\pi$
$821$	40.0000	1.39601	0.698005	+	0.716093i	$0.254071\pi$
$822$	−10.0000	−0.348790
$823$	−18.0000	−0.627441	−0.313720	−	0.949515i	$-0.601575\pi$
$823$	−18.0000	−0.627441	−0.313720	+	0.949515i	$0.601575\pi$
$824$	−16.0000	−0.557386
$825$	−3.00000	−0.104447
$826$	18.0000	0.626300
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	5.00000	0.173762
$829$	−35.0000	−1.21560	−0.607800	−	0.794090i	$-0.707948\pi$
$829$	−35.0000	−1.21560	−0.607800	+	0.794090i	$0.707948\pi$
$830$	12.0000	0.416526
$831$	−12.0000	−0.416275
$832$	−2.00000	−0.0693375
$833$	−8.00000	−0.277184
$834$	−14.0000	−0.484780
$835$	19.0000	0.657522
$836$	−9.00000	−0.311272
$837$	−1.00000	−0.0345651
$838$	−12.0000	−0.414533
$839$	51.0000	1.76072	0.880358	−	0.474310i	$-0.157302\pi$
$839$	51.0000	1.76072	0.880358	+	0.474310i	$0.157302\pi$
$840$	3.00000	0.103510
$841$	−13.0000	−0.448276
$842$	22.0000	0.758170
$843$	−16.0000	−0.551069
$844$	−19.0000	−0.654007
$845$	9.00000	0.309609
$846$	−10.0000	−0.343807
$847$	6.00000	0.206162
$848$	3.00000	0.103020
$849$	24.0000	0.823678
$850$	4.00000	0.137199
$851$	0	0
$852$	−7.00000	−0.239816
$853$	−41.0000	−1.40381	−0.701907	−	0.712269i	$-0.747668\pi$
$853$	−41.0000	−1.40381	−0.701907	+	0.712269i	$0.747668\pi$
$854$	−6.00000	−0.205316
$855$	3.00000	0.102598
$856$	−9.00000	−0.307614
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−6.00000	−0.204837
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	−1.00000	−0.0340997
$861$	12.0000	0.408959
$862$	−16.0000	−0.544962
$863$	43.0000	1.46374	0.731869	−	0.681446i	$-0.238649\pi$
$863$	43.0000	1.46374	0.731869	+	0.681446i	$0.238649\pi$
$864$	1.00000	0.0340207
$865$	22.0000	0.748022
$866$	1.00000	0.0339814
$867$	1.00000	0.0339618
$868$	−3.00000	−0.101827
$869$	−3.00000	−0.101768
$870$	−4.00000	−0.135613
$871$	−4.00000	−0.135535
$872$	−20.0000	−0.677285
$873$	−10.0000	−0.338449
$874$	15.0000	0.507383
$875$	3.00000	0.101419
$876$	−5.00000	−0.168934
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	−18.0000	−0.607471
$879$	−30.0000	−1.01187
$880$	−3.00000	−0.101130
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	−2.00000	−0.0673435
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	8.00000	0.269069
$885$	6.00000	0.201688
$886$	−15.0000	−0.503935
$887$	−46.0000	−1.54453	−0.772264	−	0.635301i	$-0.780876\pi$
$887$	−46.0000	−1.54453	−0.772264	+	0.635301i	$0.780876\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	1.00000	0.0335201
$891$	3.00000	0.100504
$892$	−6.00000	−0.200895
$893$	−30.0000	−1.00391
$894$	−11.0000	−0.367895
$895$	−4.00000	−0.133705
$896$	3.00000	0.100223
$897$	10.0000	0.333890
$898$	−2.00000	−0.0667409
$899$	4.00000	0.133407
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	−12.0000	−0.399556
$903$	3.00000	0.0998337
$904$	−9.00000	−0.299336
$905$	−5.00000	−0.166206
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	−27.0000	−0.896026
$909$	−1.00000	−0.0331679
$910$	6.00000	0.198898
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	3.00000	0.0993399
$913$	36.0000	1.19143
$914$	−10.0000	−0.330771
$915$	−2.00000	−0.0661180
$916$	13.0000	0.429532
$917$	−18.0000	−0.594412
$918$	−4.00000	−0.132020
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	5.00000	0.164845
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−14.0000	−0.460816
$924$	9.00000	0.296078
$925$	0	0
$926$	−4.00000	−0.131448
$927$	16.0000	0.525509
$928$	−4.00000	−0.131306
$929$	−41.0000	−1.34517	−0.672583	−	0.740022i	$-0.734815\pi$
$929$	−41.0000	−1.34517	−0.672583	+	0.740022i	$0.734815\pi$
$930$	−1.00000	−0.0327913
$931$	−6.00000	−0.196642
$932$	15.0000	0.491341
$933$	0	0
$934$	−24.0000	−0.785304
$935$	12.0000	0.392442
$936$	2.00000	0.0653720
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	6.00000	0.195907
$939$	−30.0000	−0.979013
$940$	−10.0000	−0.326164
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	−5.00000	−0.162909
$943$	20.0000	0.651290
$944$	6.00000	0.195283
$945$	−3.00000	−0.0975900
$946$	−3.00000	−0.0975384
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	1.00000	0.0324785
$949$	−10.0000	−0.324614
$950$	3.00000	0.0973329
$951$	8.00000	0.259418
$952$	−12.0000	−0.388922
$953$	−52.0000	−1.68445	−0.842223	−	0.539130i	$-0.818753\pi$
$953$	−52.0000	−1.68445	−0.842223	+	0.539130i	$0.818753\pi$
$954$	−3.00000	−0.0971286
$955$	−16.0000	−0.517748
$956$	12.0000	0.388108
$957$	−12.0000	−0.387905
$958$	−3.00000	−0.0969256
$959$	30.0000	0.968751
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	0	0
$963$	9.00000	0.290021
$964$	−18.0000	−0.579741
$965$	6.00000	0.193147
$966$	−15.0000	−0.482617
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	2.00000	0.0642824
$969$	−12.0000	−0.385496
$970$	−10.0000	−0.321081
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	−1.00000	−0.0320750
$973$	42.0000	1.34646
$974$	−32.0000	−1.02535
$975$	2.00000	0.0640513
$976$	−2.00000	−0.0640184
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	14.0000	0.447671
$979$	3.00000	0.0958804
$980$	−2.00000	−0.0638877
$981$	20.0000	0.638551
$982$	37.0000	1.18072
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	4.00000	0.127515
$985$	6.00000	0.191176
$986$	16.0000	0.509544
$987$	30.0000	0.954911
$988$	6.00000	0.190885
$989$	5.00000	0.158991
$990$	3.00000	0.0953463
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	−1.00000	−0.0317500
$993$	−12.0000	−0.380808
$994$	21.0000	0.666080
$995$	−7.00000	−0.221915
$996$	−12.0000	−0.380235
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.a.1.1	✓	1	1.1	even	1	trivial
2790.2.a.y.1.1		1	3.2	odd	2
4650.2.a.bv.1.1		1	5.4	even	2
4650.2.d.y.3349.1		2	5.2	odd	4
4650.2.d.y.3349.2		2	5.3	odd	4
7440.2.a.v.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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