LMFDB - Embedded newform 510.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(1,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("510.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.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:	$4.07237050309$
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	$\chi$	$=$	510.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} +2.00000 q^{42} +2.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +2.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} -10.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -8.00000 q^{77} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -16.0000 q^{83} -2.00000 q^{84} +1.00000 q^{85} -2.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +4.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	2.00000	0.534522
$15$	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	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−4.00000	−0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−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
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	2.00000	0.308607
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	2.00000	0.267261
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	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	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000	1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	1.00000	0.121268
$69$	4.00000	0.481543
$70$	2.00000	0.239046
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	−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	$0.424791\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	−8.00000	−0.911685
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−8.00000	−0.883452
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−2.00000	−0.218218
$85$	1.00000	0.108465
$86$	−2.00000	−0.215666
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	4.00000	0.417029
$93$	−8.00000	−0.829561
$94$	8.00000	0.825137
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	3.00000	0.303046
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−1.00000	−0.0990148
$103$	−20.0000	−1.97066	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−20.0000	−1.97066	−0.985329	+	0.170664i	$0.945409\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	−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
$112$	−2.00000	−0.188982
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	−4.00000	−0.374634
$115$	4.00000	0.373002
$116$	6.00000	0.557086
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−2.00000	−0.183340
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	8.00000	0.721336
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.00000	0.348155
$133$	−8.00000	−0.693688
$134$	−2.00000	−0.172774
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−4.00000	−0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−2.00000	−0.169031
$141$	−8.00000	−0.673722
$142$	10.0000	0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	−4.00000	−0.331042
$147$	−3.00000	−0.247436
$148$	−6.00000	−0.493197
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	−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
$154$	8.00000	0.644658
$155$	−8.00000	−0.642575
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−4.00000	−0.318223
$159$	14.0000	1.11027
$160$	−1.00000	−0.0790569
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	8.00000	0.624695
$165$	4.00000	0.311400
$166$	16.0000	1.24184
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	2.00000	0.154303
$169$	−13.0000	−1.00000
$170$	−1.00000	−0.0766965
$171$	4.00000	0.305888
$172$	2.00000	0.152499
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−6.00000	−0.454859
$175$	−2.00000	−0.151186
$176$	4.00000	0.301511
$177$	6.00000	0.450988
$178$	−6.00000	−0.449719
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	1.00000	0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−4.00000	−0.294884
$185$	−6.00000	−0.441129
$186$	8.00000	0.586588
$187$	4.00000	0.292509
$188$	−8.00000	−0.583460
$189$	−2.00000	−0.145479
$190$	−4.00000	−0.290191
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	1.00000	0.0721688
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−1.00000	−0.0707107
$201$	2.00000	0.141069
$202$	12.0000	0.844317
$203$	−12.0000	−0.842235
$204$	1.00000	0.0700140
$205$	8.00000	0.558744
$206$	20.0000	1.39347
$207$	4.00000	0.278019
$208$	0	0
$209$	16.0000	1.10674
$210$	2.00000	0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	14.0000	0.961524
$213$	−10.0000	−0.685189
$214$	4.00000	0.273434
$215$	2.00000	0.136399
$216$	−1.00000	−0.0680414
$217$	16.0000	1.08615
$218$	2.00000	0.135457
$219$	4.00000	0.270295
$220$	4.00000	0.269680
$221$	0	0
$222$	6.00000	0.402694
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	4.00000	0.264906
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−4.00000	−0.263752
$231$	−8.00000	−0.526361
$232$	−6.00000	−0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	6.00000	0.390567
$237$	4.00000	0.259828
$238$	2.00000	0.129641
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	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
$244$	2.00000	0.128037
$245$	−3.00000	−0.191663
$246$	−8.00000	−0.510061
$247$	0	0
$248$	8.00000	0.508001
$249$	−16.0000	−1.01396
$250$	−1.00000	−0.0632456
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−2.00000	−0.125988
$253$	16.0000	1.00591
$254$	−4.00000	−0.250982
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−2.00000	−0.124515
$259$	12.0000	0.745644
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−4.00000	−0.246183
$265$	14.0000	0.860013
$266$	8.00000	0.490511
$267$	6.00000	0.367194
$268$	2.00000	0.122169
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	−1.00000	−0.0608581
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	6.00000	0.362473
$275$	4.00000	0.241209
$276$	4.00000	0.240772
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	2.00000	0.119523
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	8.00000	0.476393
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−10.0000	−0.593391
$285$	4.00000	0.236940
$286$	0	0
$287$	−16.0000	−0.944450
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	−8.00000	−0.468968
$292$	4.00000	0.234082
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	3.00000	0.174964
$295$	6.00000	0.349334
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	−20.0000	−1.15857
$299$	0	0
$300$	1.00000	0.0577350
$301$	−4.00000	−0.230556
$302$	16.0000	0.920697
$303$	−12.0000	−0.689382
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	−1.00000	−0.0571662
$307$	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
$309$	−20.0000	−1.13776
$310$	8.00000	0.454369
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	4.00000	0.225018
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−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
$325$	0	0
$326$	−8.00000	−0.443079
$327$	−2.00000	−0.110600
$328$	−8.00000	−0.441726
$329$	16.0000	0.882109
$330$	−4.00000	−0.220193
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−16.0000	−0.878114
$333$	−6.00000	−0.328798
$334$	16.0000	0.875481
$335$	2.00000	0.109272
$336$	−2.00000	−0.109109
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	13.0000	0.707107
$339$	−18.0000	−0.977626
$340$	1.00000	0.0542326
$341$	−32.0000	−1.73290
$342$	−4.00000	−0.216295
$343$	20.0000	1.07990
$344$	−2.00000	−0.107833
$345$	4.00000	0.215353
$346$	−2.00000	−0.107521
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	6.00000	0.321634
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−4.00000	−0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−6.00000	−0.318896
$355$	−10.0000	−0.530745
$356$	6.00000	0.317999
$357$	−2.00000	−0.105851
$358$	−2.00000	−0.105703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	−14.0000	−0.735824
$363$	5.00000	0.262432
$364$	0	0
$365$	4.00000	0.209370
$366$	−2.00000	−0.104542
$367$	−38.0000	−1.98358	−0.991792	−	0.127862i	$-0.959188\pi$
$367$	−38.0000	−1.98358	−0.991792	+	0.127862i	$0.959188\pi$
$368$	4.00000	0.208514
$369$	8.00000	0.416463
$370$	6.00000	0.311925
$371$	−28.0000	−1.45369
$372$	−8.00000	−0.414781
$373$	36.0000	1.86401	0.932005	−	0.362446i	$-0.118058\pi$
$373$	36.0000	1.86401	0.932005	+	0.362446i	$0.118058\pi$
$374$	−4.00000	−0.206835
$375$	1.00000	0.0516398
$376$	8.00000	0.412568
$377$	0	0
$378$	2.00000	0.102869
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	4.00000	0.205196
$381$	4.00000	0.204926
$382$	20.0000	1.02329
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−1.00000	−0.0510310
$385$	−8.00000	−0.407718
$386$	−16.0000	−0.814379
$387$	2.00000	0.101666
$388$	−8.00000	−0.406138
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	3.00000	0.151523
$393$	0	0
$394$	6.00000	0.302276
$395$	4.00000	0.201262
$396$	4.00000	0.201008
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	8.00000	0.401004
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−12.0000	−0.597022
$405$	1.00000	0.0496904
$406$	12.0000	0.595550
$407$	−24.0000	−1.18964
$408$	−1.00000	−0.0495074
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−8.00000	−0.395092
$411$	−6.00000	−0.295958
$412$	−20.0000	−0.985329
$413$	−12.0000	−0.590481
$414$	−4.00000	−0.196589
$415$	−16.0000	−0.785409
$416$	0	0
$417$	−4.00000	−0.195881
$418$	−16.0000	−0.782586
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	−2.00000	−0.0975900
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−4.00000	−0.194717
$423$	−8.00000	−0.388973
$424$	−14.0000	−0.679900
$425$	1.00000	0.0485071
$426$	10.0000	0.484502
$427$	−4.00000	−0.193574
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	1.00000	0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−16.0000	−0.768025
$435$	6.00000	0.287678
$436$	−2.00000	−0.0957826
$437$	16.0000	0.765384
$438$	−4.00000	−0.191127
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	−4.00000	−0.190693
$441$	−3.00000	−0.142857
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−6.00000	−0.284747
$445$	6.00000	0.284427
$446$	−4.00000	−0.189405
$447$	20.0000	0.945968
$448$	−2.00000	−0.0944911
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−1.00000	−0.0471405
$451$	32.0000	1.50682
$452$	−18.0000	−0.846649
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	26.0000	1.21490
$459$	1.00000	0.0466760
$460$	4.00000	0.186501
$461$	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
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	6.00000	0.278543
$465$	−8.00000	−0.370991
$466$	−10.0000	−0.463241
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	8.00000	0.369012
$471$	0	0
$472$	−6.00000	−0.276172
$473$	8.00000	0.367840
$474$	−4.00000	−0.183726
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	14.0000	0.641016
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−10.0000	−0.455488
$483$	−8.00000	−0.364013
$484$	5.00000	0.227273
$485$	−8.00000	−0.363261
$486$	−1.00000	−0.0453609
$487$	30.0000	1.35943	0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000	1.35943	0.679715	+	0.733476i	$0.262104\pi$
$488$	−2.00000	−0.0905357
$489$	8.00000	0.361773
$490$	3.00000	0.135526
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	8.00000	0.360668
$493$	6.00000	0.270226
$494$	0	0
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	20.0000	0.897123
$498$	16.0000	0.716977
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	1.00000	0.0447214
$501$	−16.0000	−0.714827
$502$	2.00000	0.0892644
$503$	44.0000	1.96186	0.980932	−	0.194354i	$-0.0622609\pi$
$503$	44.0000	1.96186	0.980932	+	0.194354i	$0.0622609\pi$
$504$	2.00000	0.0890871
$505$	−12.0000	−0.533993
$506$	−16.0000	−0.711287
$507$	−13.0000	−0.577350
$508$	4.00000	0.177471
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	−1.00000	−0.0442807
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−30.0000	−1.32324
$515$	−20.0000	−0.881305
$516$	2.00000	0.0880451
$517$	−32.0000	−1.40736
$518$	−12.0000	−0.527250
$519$	2.00000	0.0877903
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	−6.00000	−0.262613
$523$	30.0000	1.31181	0.655904	−	0.754844i	$-0.272288\pi$
$523$	30.0000	1.31181	0.655904	+	0.754844i	$0.272288\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	−16.0000	−0.697633
$527$	−8.00000	−0.348485
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	−14.0000	−0.608121
$531$	6.00000	0.260378
$532$	−8.00000	−0.346844
$533$	0	0
$534$	−6.00000	−0.259645
$535$	−4.00000	−0.172935
$536$	−2.00000	−0.0863868
$537$	2.00000	0.0863064
$538$	2.00000	0.0862261
$539$	−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	$0.723101\pi$
$542$	−16.0000	−0.687259
$543$	14.0000	0.600798
$544$	−1.00000	−0.0428746
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−6.00000	−0.256307
$549$	2.00000	0.0853579
$550$	−4.00000	−0.170561
$551$	24.0000	1.02243
$552$	−4.00000	−0.170251
$553$	−8.00000	−0.340195
$554$	−6.00000	−0.254916
$555$	−6.00000	−0.254686
$556$	−4.00000	−0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	4.00000	0.168880
$562$	10.0000	0.421825
$563$	40.0000	1.68580	0.842900	−	0.538071i	$-0.180847\pi$
$563$	40.0000	1.68580	0.842900	+	0.538071i	$0.180847\pi$
$564$	−8.00000	−0.336861
$565$	−18.0000	−0.757266
$566$	12.0000	0.504398
$567$	−2.00000	−0.0839921
$568$	10.0000	0.419591
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	−4.00000	−0.167542
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−20.0000	−0.835512
$574$	16.0000	0.667827
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−1.00000	−0.0415945
$579$	16.0000	0.664937
$580$	6.00000	0.249136
$581$	32.0000	1.32758
$582$	8.00000	0.331611
$583$	56.0000	2.31928
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	−3.00000	−0.123718
$589$	−32.0000	−1.31854
$590$	−6.00000	−0.247016
$591$	−6.00000	−0.246807
$592$	−6.00000	−0.246598
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	−4.00000	−0.164122
$595$	−2.00000	−0.0819920
$596$	20.0000	0.819232
$597$	−8.00000	−0.327418
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	−1.00000	−0.0408248
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	4.00000	0.163028
$603$	2.00000	0.0814463
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	12.0000	0.487467
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	−4.00000	−0.162221
$609$	−12.0000	−0.486265
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	1.00000	0.0404226
$613$	−36.0000	−1.45403	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−36.0000	−1.45403	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−2.00000	−0.0807134
$615$	8.00000	0.322591
$616$	8.00000	0.322329
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	20.0000	0.804518
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	−8.00000	−0.321288
$621$	4.00000	0.160514
$622$	−10.0000	−0.400963
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	16.0000	0.638978
$628$	0	0
$629$	−6.00000	−0.239236
$630$	2.00000	0.0796819
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	−4.00000	−0.159111
$633$	4.00000	0.158986
$634$	−18.0000	−0.714871
$635$	4.00000	0.158735
$636$	14.0000	0.555136
$637$	0	0
$638$	−24.0000	−0.950169
$639$	−10.0000	−0.395594
$640$	−1.00000	−0.0395285
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	4.00000	0.157867
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	−8.00000	−0.315244
$645$	2.00000	0.0787499
$646$	−4.00000	−0.157378
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−1.00000	−0.0392837
$649$	24.0000	0.942082
$650$	0	0
$651$	16.0000	0.627089
$652$	8.00000	0.313304
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	8.00000	0.312348
$657$	4.00000	0.156055
$658$	−16.0000	−0.623745
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	4.00000	0.155700
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	0	0
$663$	0	0
$664$	16.0000	0.620920
$665$	−8.00000	−0.310227
$666$	6.00000	0.232495
$667$	24.0000	0.929284
$668$	−16.0000	−0.619059
$669$	4.00000	0.154649
$670$	−2.00000	−0.0772667
$671$	8.00000	0.308837
$672$	2.00000	0.0771517
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	−20.0000	−0.770371
$675$	1.00000	0.0384900
$676$	−13.0000	−0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	18.0000	0.691286
$679$	16.0000	0.614024
$680$	−1.00000	−0.0383482
$681$	−12.0000	−0.459841
$682$	32.0000	1.22534
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	4.00000	0.152944
$685$	−6.00000	−0.229248
$686$	−20.0000	−0.763604
$687$	−26.0000	−0.991962
$688$	2.00000	0.0762493
$689$	0	0
$690$	−4.00000	−0.152277
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	2.00000	0.0760286
$693$	−8.00000	−0.303895
$694$	20.0000	0.759190
$695$	−4.00000	−0.151729
$696$	−6.00000	−0.227429
$697$	8.00000	0.303022
$698$	14.0000	0.529908
$699$	10.0000	0.378235
$700$	−2.00000	−0.0755929
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	4.00000	0.150756
$705$	−8.00000	−0.301297
$706$	−6.00000	−0.225813
$707$	24.0000	0.902613
$708$	6.00000	0.225494
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	10.0000	0.375293
$711$	4.00000	0.150012
$712$	−6.00000	−0.224860
$713$	−32.0000	−1.19841
$714$	2.00000	0.0748481
$715$	0	0
$716$	2.00000	0.0747435
$717$	0	0
$718$	4.00000	0.149279
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	1.00000	0.0372678
$721$	40.0000	1.48968
$722$	3.00000	0.111648
$723$	10.0000	0.371904
$724$	14.0000	0.520306
$725$	6.00000	0.222834
$726$	−5.00000	−0.185567
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	2.00000	0.0739727
$732$	2.00000	0.0739221
$733$	−44.0000	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$734$	38.0000	1.40261
$735$	−3.00000	−0.110657
$736$	−4.00000	−0.147442
$737$	8.00000	0.294684
$738$	−8.00000	−0.294484
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	28.0000	1.02791
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	8.00000	0.293294
$745$	20.0000	0.732743
$746$	−36.0000	−1.31805
$747$	−16.0000	−0.585409
$748$	4.00000	0.146254
$749$	8.00000	0.292314
$750$	−1.00000	−0.0365148
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	−8.00000	−0.291730
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	−16.0000	−0.582300
$756$	−2.00000	−0.0727393
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	28.0000	1.01701
$759$	16.0000	0.580763
$760$	−4.00000	−0.145095
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	−4.00000	−0.144905
$763$	4.00000	0.144810
$764$	−20.0000	−0.723575
$765$	1.00000	0.0361551
$766$	16.0000	0.578103
$767$	0	0
$768$	1.00000	0.0360844
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	8.00000	0.288300
$771$	30.0000	1.08042
$772$	16.0000	0.575853
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−2.00000	−0.0718885
$775$	−8.00000	−0.287368
$776$	8.00000	0.287183
$777$	12.0000	0.430498
$778$	36.0000	1.29066
$779$	32.0000	1.14652
$780$	0	0
$781$	−40.0000	−1.43131
$782$	−4.00000	−0.143040
$783$	6.00000	0.214423
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−6.00000	−0.213741
$789$	16.0000	0.569615
$790$	−4.00000	−0.142314
$791$	36.0000	1.28001
$792$	−4.00000	−0.142134
$793$	0	0
$794$	−34.0000	−1.20661
$795$	14.0000	0.496529
$796$	−8.00000	−0.283552
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	8.00000	0.283197
$799$	−8.00000	−0.283020
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	−20.0000	−0.706225
$803$	16.0000	0.564628
$804$	2.00000	0.0705346
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	12.0000	0.422159
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	−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	$0.522373\pi$
$812$	−12.0000	−0.421117
$813$	16.0000	0.561144
$814$	24.0000	0.841200
$815$	8.00000	0.280228
$816$	1.00000	0.0350070
$817$	8.00000	0.279885
$818$	26.0000	0.909069
$819$	0	0
$820$	8.00000	0.279372
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	6.00000	0.209274
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	20.0000	0.696733
$825$	4.00000	0.139262
$826$	12.0000	0.417533
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	4.00000	0.139010
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	16.0000	0.555368
$831$	6.00000	0.208138
$832$	0	0
$833$	−3.00000	−0.103944
$834$	4.00000	0.138509
$835$	−16.0000	−0.553703
$836$	16.0000	0.553372
$837$	−8.00000	−0.276520
$838$	−32.0000	−1.10542
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	−10.0000	−0.344418
$844$	4.00000	0.137686
$845$	−13.0000	−0.447214
$846$	8.00000	0.275046
$847$	−10.0000	−0.343604
$848$	14.0000	0.480762
$849$	−12.0000	−0.411839
$850$	−1.00000	−0.0342997
$851$	−24.0000	−0.822709
$852$	−10.0000	−0.342594
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	4.00000	0.136877
$855$	4.00000	0.136797
$856$	4.00000	0.136717
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	2.00000	0.0681994
$861$	−16.0000	−0.545279
$862$	−10.0000	−0.340601
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	−1.00000	−0.0340207
$865$	2.00000	0.0680020
$866$	−6.00000	−0.203888
$867$	1.00000	0.0339618
$868$	16.0000	0.543075
$869$	16.0000	0.542763
$870$	−6.00000	−0.203419
$871$	0	0
$872$	2.00000	0.0677285
$873$	−8.00000	−0.270759
$874$	−16.0000	−0.541208
$875$	−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	$0.598292\pi$
$878$	28.0000	0.944954
$879$	18.0000	0.607125
$880$	4.00000	0.134840
$881$	44.0000	1.48240	0.741199	−	0.671286i	$-0.234258\pi$
$881$	44.0000	1.48240	0.741199	+	0.671286i	$0.234258\pi$
$882$	3.00000	0.101015
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	6.00000	0.201347
$889$	−8.00000	−0.268311
$890$	−6.00000	−0.201120
$891$	4.00000	0.134005
$892$	4.00000	0.133930
$893$	−32.0000	−1.07084
$894$	−20.0000	−0.668900
$895$	2.00000	0.0668526
$896$	2.00000	0.0668153
$897$	0	0
$898$	12.0000	0.400445
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	14.0000	0.466408
$902$	−32.0000	−1.06548
$903$	−4.00000	−0.133112
$904$	18.0000	0.598671
$905$	14.0000	0.465376
$906$	16.0000	0.531564
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	−12.0000	−0.398234
$909$	−12.0000	−0.398015
$910$	0	0
$911$	38.0000	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$912$	4.00000	0.132453
$913$	−64.0000	−2.11809
$914$	18.0000	0.595387
$915$	2.00000	0.0661180
$916$	−26.0000	−0.859064
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−4.00000	−0.131876
$921$	2.00000	0.0659022
$922$	−24.0000	−0.790398
$923$	0	0
$924$	−8.00000	−0.263181
$925$	−6.00000	−0.197279
$926$	4.00000	0.131448
$927$	−20.0000	−0.656886
$928$	−6.00000	−0.196960
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	8.00000	0.262330
$931$	−12.0000	−0.393284
$932$	10.0000	0.327561
$933$	10.0000	0.327385
$934$	28.0000	0.916188
$935$	4.00000	0.130814
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	4.00000	0.130605
$939$	−8.00000	−0.261070
$940$	−8.00000	−0.260931
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	6.00000	0.195283
$945$	−2.00000	−0.0650600
$946$	−8.00000	−0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	4.00000	0.129914
$949$	0	0
$950$	−4.00000	−0.129777
$951$	18.0000	0.583690
$952$	2.00000	0.0648204
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−14.0000	−0.453267
$955$	−20.0000	−0.647185
$956$	0	0
$957$	24.0000	0.775810
$958$	−30.0000	−0.969256
$959$	12.0000	0.387500
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	−4.00000	−0.128898
$964$	10.0000	0.322078
$965$	16.0000	0.515058
$966$	8.00000	0.257396
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−5.00000	−0.160706
$969$	4.00000	0.128499
$970$	8.00000	0.256865
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	1.00000	0.0320750
$973$	8.00000	0.256468
$974$	−30.0000	−0.961262
$975$	0	0
$976$	2.00000	0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−8.00000	−0.255812
$979$	24.0000	0.767043
$980$	−3.00000	−0.0958315
$981$	−2.00000	−0.0638551
$982$	6.00000	0.191468
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	−8.00000	−0.255031
$985$	−6.00000	−0.191176
$986$	−6.00000	−0.191079
$987$	16.0000	0.509286
$988$	0	0
$989$	8.00000	0.254385
$990$	−4.00000	−0.127128
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	−20.0000	−0.634361
$995$	−8.00000	−0.253617
$996$	−16.0000	−0.506979
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	−36.0000	−1.13956
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	510.2.a.b.1.1	✓	1
3.2	odd	2		1530.2.a.i.1.1		1
4.3	odd	2		4080.2.a.n.1.1		1
5.2	odd	4		2550.2.d.j.2449.1		2
5.3	odd	4		2550.2.d.j.2449.2		2
5.4	even	2		2550.2.a.y.1.1		1
15.14	odd	2		7650.2.a.y.1.1		1
17.16	even	2		8670.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.b.1.1	✓	1	1.1	even	1	trivial
1530.2.a.i.1.1		1	3.2	odd	2
2550.2.a.y.1.1		1	5.4	even	2
2550.2.d.j.2449.1		2	5.2	odd	4
2550.2.d.j.2449.2		2	5.3	odd	4
4080.2.a.n.1.1		1	4.3	odd	2
7650.2.a.y.1.1		1	15.14	odd	2
8670.2.a.c.1.1		1	17.16	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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