LMFDB - Embedded newform 3570.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3570,2,Mod(1,3570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3570, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3570.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3570.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:	$28.5065935216$
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	$\chi$	$=$	3570.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -1.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +1.00000 q^{68} +8.00000 q^{69} +1.00000 q^{70} +16.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -8.00000 q^{83} +1.00000 q^{84} +1.00000 q^{85} +6.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} -2.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} +4.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−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	$0.410544\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	−4.00000	−0.852803
$23$	−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
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.00000	0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−1.00000	−0.171499
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−1.00000	−0.154303
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	8.00000	1.17954
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	2.00000	0.277350
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	1.00000	0.133631
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	4.00000	0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	1.00000	0.121268
$69$	8.00000	0.963087
$70$	1.00000	0.119523
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	2.00000	0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	1.00000	0.109109
$85$	1.00000	0.108465
$86$	0	0
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−1.00000	−0.105409
$91$	−2.00000	−0.209657
$92$	−8.00000	−0.834058
$93$	8.00000	0.829561
$94$	4.00000	0.412568
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−1.00000	−0.101015
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	1.00000	0.0990148
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−2.00000	−0.196116
$105$	1.00000	0.0975900
$106$	14.0000	1.35980
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−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	$0.530535\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	−1.00000	−0.0916698
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	6.00000	0.543214
$123$	−2.00000	−0.180334
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−4.00000	−0.348155
$133$	4.00000	0.346844
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	−1.00000	−0.0857493
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−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	$0.445740\pi$
$140$	−1.00000	−0.0845154
$141$	4.00000	0.336861
$142$	−16.0000	−1.34269
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−2.00000	−0.165521
$147$	−1.00000	−0.0824786
$148$	6.00000	0.493197
$149$	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	$0.725663\pi$
$152$	4.00000	0.324443
$153$	1.00000	0.0808452
$154$	4.00000	0.322329
$155$	−8.00000	−0.642575
$156$	−2.00000	−0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	4.00000	0.318223
$159$	14.0000	1.11027
$160$	−1.00000	−0.0790569
$161$	8.00000	0.630488
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	2.00000	0.156174
$165$	−4.00000	−0.311400
$166$	8.00000	0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	−6.00000	−0.454859
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	−12.0000	−0.901975
$178$	−10.0000	−0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	1.00000	0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	2.00000	0.148250
$183$	6.00000	0.443533
$184$	8.00000	0.589768
$185$	6.00000	0.441129
$186$	−8.00000	−0.586588
$187$	4.00000	0.292509
$188$	−4.00000	−0.291730
$189$	1.00000	0.0727393
$190$	4.00000	0.290191
$191$	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	$0.275675\pi$
$194$	−10.0000	−0.717958
$195$	−2.00000	−0.143223
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	−1.00000	−0.0700140
$205$	2.00000	0.139686
$206$	16.0000	1.11477
$207$	−8.00000	−0.556038
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	−1.00000	−0.0690066
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−14.0000	−0.961524
$213$	−16.0000	−1.09630
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	2.00000	0.135457
$219$	−2.00000	−0.135147
$220$	4.00000	0.269680
$221$	2.00000	0.134535
$222$	6.00000	0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	8.00000	0.527504
$231$	4.00000	0.263181
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	−4.00000	−0.260931
$236$	12.0000	0.781133
$237$	4.00000	0.259828
$238$	1.00000	0.0648204
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−1.00000	−0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	2.00000	0.127515
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	8.00000	0.506979
$250$	−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	$0.623635\pi$
$252$	−1.00000	−0.0629941
$253$	−32.0000	−2.01182
$254$	20.0000	1.25491
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	2.00000	0.124035
$261$	−6.00000	−0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	4.00000	0.246183
$265$	−14.0000	−0.860013
$266$	−4.00000	−0.245256
$267$	−10.0000	−0.611990
$268$	−8.00000	−0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	1.00000	0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	1.00000	0.0606339
$273$	2.00000	0.121046
$274$	−14.0000	−0.845771
$275$	4.00000	0.241209
$276$	8.00000	0.481543
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−4.00000	−0.239904
$279$	−8.00000	−0.478947
$280$	1.00000	0.0597614
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−4.00000	−0.238197
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	16.0000	0.949425
$285$	4.00000	0.236940
$286$	−8.00000	−0.473050
$287$	−2.00000	−0.118056
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	1.00000	0.0583212
$295$	12.0000	0.698667
$296$	−6.00000	−0.348743
$297$	−4.00000	−0.232104
$298$	−14.0000	−0.810998
$299$	−16.0000	−0.925304
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	−6.00000	−0.343559
$306$	−1.00000	−0.0571662
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	−4.00000	−0.227921
$309$	16.0000	0.910208
$310$	8.00000	0.454369
$311$	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	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	14.0000	0.790066
$315$	−1.00000	−0.0563436
$316$	−4.00000	−0.225018
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−14.0000	−0.785081
$319$	−24.0000	−1.34374
$320$	1.00000	0.0559017
$321$	4.00000	0.223258
$322$	−8.00000	−0.445823
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	20.0000	1.10770
$327$	2.00000	0.110600
$328$	−2.00000	−0.110432
$329$	4.00000	0.220527
$330$	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	$0.779499\pi$
$332$	−8.00000	−0.439057
$333$	6.00000	0.328798
$334$	0	0
$335$	−8.00000	−0.437087
$336$	1.00000	0.0545545
$337$	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
$340$	1.00000	0.0542326
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.00000	0.430706
$346$	2.00000	0.107521
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	6.00000	0.321634
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	1.00000	0.0534522
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	12.0000	0.637793
$355$	16.0000	0.849192
$356$	10.0000	0.529999
$357$	1.00000	0.0529256
$358$	−12.0000	−0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	−5.00000	−0.262432
$364$	−2.00000	−0.104828
$365$	2.00000	0.104685
$366$	−6.00000	−0.313625
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−8.00000	−0.417029
$369$	2.00000	0.104116
$370$	−6.00000	−0.311925
$371$	14.0000	0.726844
$372$	8.00000	0.414781
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−4.00000	−0.206835
$375$	−1.00000	−0.0516398
$376$	4.00000	0.206284
$377$	−12.0000	−0.618031
$378$	−1.00000	−0.0514344
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	−4.00000	−0.205196
$381$	20.0000	1.02463
$382$	0	0
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	1.00000	0.0510310
$385$	−4.00000	−0.203859
$386$	−18.0000	−0.916176
$387$	0	0
$388$	10.0000	0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	2.00000	0.101274
$391$	−8.00000	−0.404577
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	2.00000	0.100759
$395$	−4.00000	−0.201262
$396$	4.00000	0.201008
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−16.0000	−0.802008
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−8.00000	−0.399004
$403$	−16.0000	−0.797017
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	−6.00000	−0.297775
$407$	24.0000	1.18964
$408$	1.00000	0.0495074
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−2.00000	−0.0987730
$411$	−14.0000	−0.690569
$412$	−16.0000	−0.788263
$413$	−12.0000	−0.590481
$414$	8.00000	0.393179
$415$	−8.00000	−0.392705
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	16.0000	0.782586
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	1.00000	0.0487950
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−16.0000	−0.778868
$423$	−4.00000	−0.194487
$424$	14.0000	0.679900
$425$	1.00000	0.0485071
$426$	16.0000	0.775203
$427$	6.00000	0.290360
$428$	−4.00000	−0.193347
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−8.00000	−0.384012
$435$	6.00000	0.287678
$436$	−2.00000	−0.0957826
$437$	32.0000	1.53077
$438$	2.00000	0.0955637
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−4.00000	−0.190693
$441$	1.00000	0.0476190
$442$	−2.00000	−0.0951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−6.00000	−0.284747
$445$	10.0000	0.474045
$446$	0	0
$447$	−14.0000	−0.662177
$448$	−1.00000	−0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−1.00000	−0.0471405
$451$	8.00000	0.376705
$452$	−14.0000	−0.658505
$453$	16.0000	0.751746
$454$	−28.0000	−1.31411
$455$	−2.00000	−0.0937614
$456$	−4.00000	−0.187317
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	10.0000	0.467269
$459$	−1.00000	−0.0466760
$460$	−8.00000	−0.373002
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	−4.00000	−0.186097
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	6.00000	0.277945
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	2.00000	0.0924500
$469$	8.00000	0.369406
$470$	4.00000	0.184506
$471$	14.0000	0.645086
$472$	−12.0000	−0.552345
$473$	0	0
$474$	−4.00000	−0.183726
$475$	−4.00000	−0.183533
$476$	−1.00000	−0.0458349
$477$	−14.0000	−0.641016
$478$	24.0000	1.09773
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	1.00000	0.0456435
$481$	12.0000	0.547153
$482$	10.0000	0.455488
$483$	−8.00000	−0.364013
$484$	5.00000	0.227273
$485$	10.0000	0.454077
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	20.0000	0.904431
$490$	−1.00000	−0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−2.00000	−0.0901670
$493$	−6.00000	−0.270226
$494$	8.00000	0.359937
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	−16.0000	−0.717698
$498$	−8.00000	−0.358489
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	32.0000	1.42257
$507$	9.00000	0.399704
$508$	−20.0000	−0.887357
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	1.00000	0.0442807
$511$	−2.00000	−0.0884748
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	6.00000	0.264649
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−16.0000	−0.703679
$518$	6.00000	0.263625
$519$	2.00000	0.0877903
$520$	−2.00000	−0.0877058
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−8.00000	−0.348485
$528$	−4.00000	−0.174078
$529$	41.0000	1.78261
$530$	14.0000	0.608121
$531$	12.0000	0.520756
$532$	4.00000	0.173422
$533$	4.00000	0.173259
$534$	10.0000	0.432742
$535$	−4.00000	−0.172935
$536$	8.00000	0.345547
$537$	−12.0000	−0.517838
$538$	18.0000	0.776035
$539$	4.00000	0.172292
$540$	−1.00000	−0.0430331
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	8.00000	0.343629
$543$	22.0000	0.944110
$544$	−1.00000	−0.0428746
$545$	−2.00000	−0.0856706
$546$	−2.00000	−0.0855921
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	14.0000	0.598050
$549$	−6.00000	−0.256074
$550$	−4.00000	−0.170561
$551$	24.0000	1.02243
$552$	−8.00000	−0.340503
$553$	4.00000	0.170097
$554$	26.0000	1.10463
$555$	−6.00000	−0.254686
$556$	4.00000	0.169638
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	−4.00000	−0.168880
$562$	6.00000	0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	4.00000	0.168430
$565$	−14.0000	−0.588984
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	−16.0000	−0.671345
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−4.00000	−0.167542
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	2.00000	0.0834784
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−1.00000	−0.0415945
$579$	−18.0000	−0.748054
$580$	−6.00000	−0.249136
$581$	8.00000	0.331896
$582$	10.0000	0.414513
$583$	−56.0000	−2.31928
$584$	−2.00000	−0.0827606
$585$	2.00000	0.0826898
$586$	−14.0000	−0.578335
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	−1.00000	−0.0412393
$589$	32.0000	1.31854
$590$	−12.0000	−0.494032
$591$	2.00000	0.0822690
$592$	6.00000	0.246598
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	4.00000	0.164122
$595$	−1.00000	−0.0409960
$596$	14.0000	0.573462
$597$	−16.0000	−0.654836
$598$	16.0000	0.654289
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000	0.0408248
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	6.00000	0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	4.00000	0.162221
$609$	−6.00000	−0.243132
$610$	6.00000	0.242933
$611$	−8.00000	−0.323645
$612$	1.00000	0.0404226
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−12.0000	−0.484281
$615$	−2.00000	−0.0806478
$616$	4.00000	0.161165
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−16.0000	−0.643614
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−8.00000	−0.321288
$621$	8.00000	0.321029
$622$	−20.0000	−0.801927
$623$	−10.0000	−0.400642
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−26.0000	−1.03917
$627$	16.0000	0.638978
$628$	−14.0000	−0.558661
$629$	6.00000	0.239236
$630$	1.00000	0.0398410
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	4.00000	0.159111
$633$	−16.0000	−0.635943
$634$	2.00000	0.0794301
$635$	−20.0000	−0.793676
$636$	14.0000	0.555136
$637$	2.00000	0.0792429
$638$	24.0000	0.950169
$639$	16.0000	0.632950
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−4.00000	−0.157867
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	4.00000	0.157378
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	−1.00000	−0.0392837
$649$	48.0000	1.88416
$650$	−2.00000	−0.0784465
$651$	−8.00000	−0.313545
$652$	−20.0000	−0.783260
$653$	−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
$657$	2.00000	0.0780274
$658$	−4.00000	−0.155936
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	−4.00000	−0.155700
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	28.0000	1.08825
$663$	−2.00000	−0.0776736
$664$	8.00000	0.310460
$665$	4.00000	0.155113
$666$	−6.00000	−0.232495
$667$	48.0000	1.85857
$668$	0	0
$669$	0	0
$670$	8.00000	0.309067
$671$	−24.0000	−0.926510
$672$	−1.00000	−0.0385758
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−2.00000	−0.0770371
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−14.0000	−0.537667
$679$	−10.0000	−0.383765
$680$	−1.00000	−0.0383482
$681$	−28.0000	−1.07296
$682$	32.0000	1.22534
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	14.0000	0.534913
$686$	1.00000	0.0381802
$687$	10.0000	0.381524
$688$	0	0
$689$	−28.0000	−1.06672
$690$	−8.00000	−0.304555
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−2.00000	−0.0760286
$693$	−4.00000	−0.151947
$694$	28.0000	1.06287
$695$	4.00000	0.151729
$696$	−6.00000	−0.227429
$697$	2.00000	0.0757554
$698$	10.0000	0.378506
$699$	6.00000	0.226941
$700$	−1.00000	−0.0377964
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	2.00000	0.0754851
$703$	−24.0000	−0.905177
$704$	4.00000	0.150756
$705$	4.00000	0.150649
$706$	−18.0000	−0.677439
$707$	−6.00000	−0.225653
$708$	−12.0000	−0.450988
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−16.0000	−0.600469
$711$	−4.00000	−0.150012
$712$	−10.0000	−0.374766
$713$	64.0000	2.39682
$714$	−1.00000	−0.0374241
$715$	8.00000	0.299183
$716$	12.0000	0.448461
$717$	24.0000	0.896296
$718$	−24.0000	−0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	1.00000	0.0372678
$721$	16.0000	0.595871
$722$	3.00000	0.111648
$723$	10.0000	0.371904
$724$	−22.0000	−0.817624
$725$	−6.00000	−0.222834
$726$	5.00000	0.185567
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	0	0
$732$	6.00000	0.221766
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	24.0000	0.885856
$735$	−1.00000	−0.0368856
$736$	8.00000	0.294884
$737$	−32.0000	−1.17874
$738$	−2.00000	−0.0736210
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	6.00000	0.220564
$741$	8.00000	0.293887
$742$	−14.0000	−0.513956
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−8.00000	−0.293294
$745$	14.0000	0.512920
$746$	26.0000	0.951928
$747$	−8.00000	−0.292705
$748$	4.00000	0.146254
$749$	4.00000	0.146157
$750$	1.00000	0.0365148
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−4.00000	−0.145865
$753$	12.0000	0.437304
$754$	12.0000	0.437014
$755$	−16.0000	−0.582300
$756$	1.00000	0.0363696
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	32.0000	1.16229
$759$	32.0000	1.16153
$760$	4.00000	0.145095
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	−20.0000	−0.724524
$763$	2.00000	0.0724049
$764$	0	0
$765$	1.00000	0.0361551
$766$	20.0000	0.722629
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	4.00000	0.144150
$771$	6.00000	0.216085
$772$	18.0000	0.647834
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−10.0000	−0.358979
$777$	6.00000	0.215249
$778$	18.0000	0.645331
$779$	−8.00000	−0.286630
$780$	−2.00000	−0.0716115
$781$	64.0000	2.29010
$782$	8.00000	0.286079
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	−14.0000	−0.499681
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	4.00000	0.142314
$791$	14.0000	0.497783
$792$	−4.00000	−0.142134
$793$	−12.0000	−0.426132
$794$	−14.0000	−0.496841
$795$	14.0000	0.496529
$796$	16.0000	0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	4.00000	0.141598
$799$	−4.00000	−0.141510
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	18.0000	0.635602
$803$	8.00000	0.282314
$804$	8.00000	0.282138
$805$	8.00000	0.281963
$806$	16.0000	0.563576
$807$	18.0000	0.633630
$808$	−6.00000	−0.211079
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−1.00000	−0.0351364
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	6.00000	0.210559
$813$	8.00000	0.280572
$814$	−24.0000	−0.841200
$815$	−20.0000	−0.700569
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−10.0000	−0.349642
$819$	−2.00000	−0.0698857
$820$	2.00000	0.0698430
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	14.0000	0.488306
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	16.0000	0.557386
$825$	−4.00000	−0.139262
$826$	12.0000	0.417533
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−8.00000	−0.278019
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	8.00000	0.277684
$831$	26.0000	0.901930
$832$	2.00000	0.0693375
$833$	1.00000	0.0346479
$834$	4.00000	0.138509
$835$	0	0
$836$	−16.0000	−0.553372
$837$	8.00000	0.276520
$838$	16.0000	0.552711
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	−1.00000	−0.0345033
$841$	7.00000	0.241379
$842$	−22.0000	−0.758170
$843$	6.00000	0.206651
$844$	16.0000	0.550743
$845$	−9.00000	−0.309609
$846$	4.00000	0.137523
$847$	−5.00000	−0.171802
$848$	−14.0000	−0.480762
$849$	4.00000	0.137280
$850$	−1.00000	−0.0342997
$851$	−48.0000	−1.64542
$852$	−16.0000	−0.548151
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	−6.00000	−0.205316
$855$	−4.00000	−0.136797
$856$	4.00000	0.136717
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	8.00000	0.273115
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	8.00000	0.272481
$863$	−56.0000	−1.90626	−0.953131	−	0.302558i	$-0.902160\pi$
$863$	−56.0000	−1.90626	−0.953131	+	0.302558i	$0.902160\pi$
$864$	1.00000	0.0340207
$865$	−2.00000	−0.0680020
$866$	34.0000	1.15537
$867$	−1.00000	−0.0339618
$868$	8.00000	0.271538
$869$	−16.0000	−0.542763
$870$	−6.00000	−0.203419
$871$	−16.0000	−0.542139
$872$	2.00000	0.0677285
$873$	10.0000	0.338449
$874$	−32.0000	−1.08242
$875$	−1.00000	−0.0338062
$876$	−2.00000	−0.0675737
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	8.00000	0.269987
$879$	−14.0000	−0.472208
$880$	4.00000	0.134840
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−1.00000	−0.0336718
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	2.00000	0.0672673
$885$	−12.0000	−0.403376
$886$	36.0000	1.20944
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	6.00000	0.201347
$889$	20.0000	0.670778
$890$	−10.0000	−0.335201
$891$	4.00000	0.134005
$892$	0	0
$893$	16.0000	0.535420
$894$	14.0000	0.468230
$895$	12.0000	0.401116
$896$	1.00000	0.0334077
$897$	16.0000	0.534224
$898$	−6.00000	−0.200223
$899$	48.0000	1.60089
$900$	1.00000	0.0333333
$901$	−14.0000	−0.466408
$902$	−8.00000	−0.266371
$903$	0	0
$904$	14.0000	0.465633
$905$	−22.0000	−0.731305
$906$	−16.0000	−0.531564
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	28.0000	0.929213
$909$	6.00000	0.199007
$910$	2.00000	0.0662994
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	4.00000	0.132453
$913$	−32.0000	−1.05905
$914$	−18.0000	−0.595387
$915$	6.00000	0.198354
$916$	−10.0000	−0.330409
$917$	0	0
$918$	1.00000	0.0330049
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	8.00000	0.263752
$921$	−12.0000	−0.395413
$922$	−14.0000	−0.461065
$923$	32.0000	1.05329
$924$	4.00000	0.131590
$925$	6.00000	0.197279
$926$	−20.0000	−0.657241
$927$	−16.0000	−0.525509
$928$	6.00000	0.196960
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	−8.00000	−0.262330
$931$	−4.00000	−0.131095
$932$	−6.00000	−0.196537
$933$	−20.0000	−0.654771
$934$	32.0000	1.04707
$935$	4.00000	0.130814
$936$	−2.00000	−0.0653720
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	−8.00000	−0.261209
$939$	−26.0000	−0.848478
$940$	−4.00000	−0.130466
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	−14.0000	−0.456145
$943$	−16.0000	−0.521032
$944$	12.0000	0.390567
$945$	1.00000	0.0325300
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	4.00000	0.129914
$949$	4.00000	0.129845
$950$	4.00000	0.129777
$951$	2.00000	0.0648544
$952$	1.00000	0.0324102
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	−24.0000	−0.776215
$957$	24.0000	0.775810
$958$	−12.0000	−0.387702
$959$	−14.0000	−0.452084
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	−4.00000	−0.128898
$964$	−10.0000	−0.322078
$965$	18.0000	0.579441
$966$	8.00000	0.257396
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	−5.00000	−0.160706
$969$	4.00000	0.128499
$970$	−10.0000	−0.321081
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	−8.00000	−0.256337
$975$	−2.00000	−0.0640513
$976$	−6.00000	−0.192055
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−20.0000	−0.639529
$979$	40.0000	1.27841
$980$	1.00000	0.0319438
$981$	−2.00000	−0.0638551
$982$	−12.0000	−0.382935
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	2.00000	0.0637577
$985$	−2.00000	−0.0637253
$986$	6.00000	0.191079
$987$	−4.00000	−0.127321
$988$	−8.00000	−0.254514
$989$	0	0
$990$	−4.00000	−0.127128
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	8.00000	0.254000
$993$	28.0000	0.888553
$994$	16.0000	0.507489
$995$	16.0000	0.507234
$996$	8.00000	0.253490
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	24.0000	0.759707
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3570.2.a.g.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3570.2.a.g.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3570.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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