LMFDB - Embedded newform 2550.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	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
$15$	0	0
$16$	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	$0.348268\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	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	$0.311919\pi$
$30$	0	0
$31$	−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
$35$	0	0
$36$	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
$39$	0	0
$40$	0	0
$41$	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	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−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
$55$	0	0
$56$	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	$0.372279\pi$
$60$	0	0
$61$	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	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−1.00000	−0.121268
$69$	4.00000	0.481543
$70$	0	0
$71$	−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	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	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	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.00000	0.883452
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−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	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−3.00000	−0.303046
$99$	4.00000	0.402015
$100$	0	0
$101$	−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	$-0.0545913\pi$
$103$	20.0000	1.97066	0.985329	+	0.170664i	$0.0545913\pi$
$104$	0	0
$105$	0	0
$106$	−14.0000	−1.35980
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	2.00000	0.188982
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	−8.00000	−0.721336
$124$	−8.00000	−0.718421
$125$	0	0
$126$	2.00000	0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	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
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	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	$0.554260\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−10.0000	−0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−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	$0.194400\pi$
$150$	0	0
$151$	−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
$155$	0	0
$156$	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
$160$	0	0
$161$	−8.00000	−0.630488
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	16.0000	1.24184
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−2.00000	−0.154303
$169$	−13.0000	−1.00000
$170$	0	0
$171$	4.00000	0.305888
$172$	−2.00000	−0.152499
$173$	−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
$175$	0	0
$176$	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	$0.476186\pi$
$180$	0	0
$181$	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
$185$	0	0
$186$	8.00000	0.586588
$187$	−4.00000	−0.292509
$188$	8.00000	0.583460
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−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	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	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	$0.591513\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	−12.0000	−0.844317
$203$	12.0000	0.842235
$204$	1.00000	0.0700140
$205$	0	0
$206$	20.0000	1.39347
$207$	−4.00000	−0.278019
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	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
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−16.0000	−1.08615
$218$	−2.00000	−0.135457
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	−6.00000	−0.402694
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	18.0000	1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−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	$0.828954\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	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	$0.5\pi$
$240$	0	0
$241$	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
$245$	0	0
$246$	−8.00000	−0.510061
$247$	0	0
$248$	−8.00000	−0.508001
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−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
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	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	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	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	$0.519420\pi$
$270$	0	0
$271$	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
$275$	0	0
$276$	4.00000	0.240772
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	−4.00000	−0.239904
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−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	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	0	0
$287$	16.0000	0.944450
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	−4.00000	−0.234082
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	20.0000	1.15857
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	−16.0000	−0.920697
$303$	12.0000	0.689382
$304$	4.00000	0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	8.00000	0.455842
$309$	−20.0000	−1.13776
$310$	0	0
$311$	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	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	4.00000	0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	14.0000	0.785081
$319$	24.0000	1.34374
$320$	0	0
$321$	−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
$330$	0	0
$331$	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
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−13.0000	−0.707107
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−2.00000	−0.107521
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−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	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	4.00000	0.213201
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	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	$0.533662\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	14.0000	0.735824
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	38.0000	1.98358	0.991792	−	0.127862i	$-0.0408116\pi$
$367$	38.0000	1.98358	0.991792	+	0.127862i	$0.0408116\pi$
$368$	−4.00000	−0.208514
$369$	8.00000	0.416463
$370$	0	0
$371$	−28.0000	−1.45369
$372$	8.00000	0.414781
$373$	−36.0000	−1.86401	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000	−1.86401	−0.932005	+	0.362446i	$0.881942\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	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	$0.755460\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	−20.0000	−1.02329
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−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
$395$	0	0
$396$	4.00000	0.201008
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−8.00000	−0.401004
$399$	−8.00000	−0.400501
$400$	0	0
$401$	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
$405$	0	0
$406$	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	$0.722231\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	20.0000	0.985329
$413$	12.0000	0.590481
$414$	−4.00000	−0.196589
$415$	0	0
$416$	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	$0.214378\pi$
$420$	0	0
$421$	−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
$425$	0	0
$426$	10.0000	0.484502
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	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	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−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	$0.732928\pi$
$440$	0	0
$441$	−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
$445$	0	0
$446$	−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	$0.591382\pi$
$450$	0	0
$451$	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	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−26.0000	−1.21490
$459$	1.00000	0.0466760
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	−8.00000	−0.372194
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	−8.00000	−0.367840
$474$	−4.00000	−0.183726
$475$	0	0
$476$	−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	$0.259642\pi$
$480$	0	0
$481$	0	0
$482$	10.0000	0.455488
$483$	8.00000	0.364013
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−30.0000	−1.35943	−0.679715	−	0.733476i	$-0.737896\pi$
$487$	−30.0000	−1.35943	−0.679715	+	0.733476i	$0.737896\pi$
$488$	2.00000	0.0905357
$489$	8.00000	0.361773
$490$	0	0
$491$	−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
$495$	0	0
$496$	−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	$0.201741\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	−2.00000	−0.0892644
$503$	−44.0000	−1.96186	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000	−1.96186	−0.980932	+	0.194354i	$0.937739\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	−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	$0.794022\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−30.0000	−1.32324
$515$	0	0
$516$	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	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	0	0
$525$	0	0
$526$	−16.0000	−0.697633
$527$	8.00000	0.348485
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	8.00000	0.346844
$533$	0	0
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	−2.00000	−0.0863064
$538$	−2.00000	−0.0862261
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−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
$545$	0	0
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	6.00000	0.256307
$549$	2.00000	0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000	0.170251
$553$	8.00000	0.340195
$554$	−6.00000	−0.254916
$555$	0	0
$556$	−4.00000	−0.169638
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	0	0
$561$	4.00000	0.168880
$562$	−10.0000	−0.421825
$563$	−40.0000	−1.68580	−0.842900	−	0.538071i	$-0.819153\pi$
$563$	−40.0000	−1.68580	−0.842900	+	0.538071i	$0.819153\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	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	$0.247483\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	20.0000	0.835512
$574$	16.0000	0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	1.00000	0.0415945
$579$	16.0000	0.664937
$580$	0	0
$581$	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	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	3.00000	0.123718
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	6.00000	0.246598
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	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	$0.236966\pi$
$600$	0	0
$601$	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
$605$	0	0
$606$	12.0000	0.487467
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	4.00000	0.162221
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	36.0000	1.45403	0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000	1.45403	0.727013	+	0.686624i	$0.240908\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	8.00000	0.322329
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	−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	$0.577528\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	10.0000	0.400963
$623$	12.0000	0.480770
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	−16.0000	−0.638978
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−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
$635$	0	0
$636$	14.0000	0.555136
$637$	0	0
$638$	24.0000	0.950169
$639$	−10.0000	−0.395594
$640$	0	0
$641$	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	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−4.00000	−0.157378
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	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	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	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	$0.698638\pi$
$660$	0	0
$661$	−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
$665$	0	0
$666$	6.00000	0.232495
$667$	−24.0000	−0.929284
$668$	16.0000	0.619059
$669$	4.00000	0.154649
$670$	0	0
$671$	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
$675$	0	0
$676$	−13.0000	−0.500000
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−18.0000	−0.691286
$679$	16.0000	0.614024
$680$	0	0
$681$	−12.0000	−0.459841
$682$	−32.0000	−1.22534
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−20.0000	−0.763604
$687$	26.0000	0.991962
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−2.00000	−0.0760286
$693$	8.00000	0.303895
$694$	20.0000	0.759190
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−8.00000	−0.303022
$698$	−14.0000	−0.529908
$699$	10.0000	0.378235
$700$	0	0
$701$	−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
$705$	0	0
$706$	−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	$0.488043\pi$
$710$	0	0
$711$	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	$0.415929\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	−3.00000	−0.111648
$723$	−10.0000	−0.371904
$724$	14.0000	0.520306
$725$	0	0
$726$	−5.00000	−0.185567
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	−2.00000	−0.0739221
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	38.0000	1.40261
$735$	0	0
$736$	−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	$0.700309\pi$
$740$	0	0
$741$	0	0
$742$	−28.0000	−1.02791
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−36.0000	−1.31805
$747$	16.0000	0.585409
$748$	−4.00000	−0.146254
$749$	8.00000	0.292314
$750$	0	0
$751$	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
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	−28.0000	−1.01701
$759$	16.0000	0.580763
$760$	0	0
$761$	−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
$765$	0	0
$766$	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	$0.857550\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	−16.0000	−0.575853
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	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	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	6.00000	0.213741
$789$	16.0000	0.569615
$790$	0	0
$791$	36.0000	1.28001
$792$	4.00000	0.142134
$793$	0	0
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−8.00000	−0.283552
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−8.00000	−0.283197
$799$	−8.00000	−0.283020
$800$	0	0
$801$	6.00000	0.212000
$802$	20.0000	0.706225
$803$	−16.0000	−0.564628
$804$	2.00000	0.0705346
$805$	0	0
$806$	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	$0.5\pi$
$810$	0	0
$811$	−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
$815$	0	0
$816$	1.00000	0.0350070
$817$	−8.00000	−0.279885
$818$	−26.0000	−0.909069
$819$	0	0
$820$	0	0
$821$	−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	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	20.0000	0.696733
$825$	0	0
$826$	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	$0.566904\pi$
$828$	−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	$0.729400\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	3.00000	0.103944
$834$	4.00000	0.138509
$835$	0	0
$836$	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	$0.533027\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	10.0000	0.344418
$844$	4.00000	0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	10.0000	0.343604
$848$	−14.0000	−0.480762
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−24.0000	−0.822709
$852$	10.0000	0.342594
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	4.00000	0.136717
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	10.0000	0.340601
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−6.00000	−0.203888
$867$	−1.00000	−0.0339618
$868$	−16.0000	−0.543075
$869$	16.0000	0.542763
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	8.00000	0.270759
$874$	−16.0000	−0.541208
$875$	0	0
$876$	4.00000	0.135147
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−28.0000	−0.944954
$879$	18.0000	0.607125
$880$	0	0
$881$	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	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−6.00000	−0.201347
$889$	−8.00000	−0.268311
$890$	0	0
$891$	4.00000	0.134005
$892$	−4.00000	−0.133930
$893$	32.0000	1.07084
$894$	−20.0000	−0.668900
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−48.0000	−1.60089
$900$	0	0
$901$	14.0000	0.466408
$902$	32.0000	1.06548
$903$	4.00000	0.133112
$904$	18.0000	0.598671
$905$	0	0
$906$	16.0000	0.531564
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	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
$915$	0	0
$916$	−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	$0.414993\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	24.0000	0.790398
$923$	0	0
$924$	−8.00000	−0.263181
$925$	0	0
$926$	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	$0.675913\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	−10.0000	−0.327561
$933$	−10.0000	−0.327385
$934$	28.0000	0.916188
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−4.00000	−0.130605
$939$	−8.00000	−0.261070
$940$	0	0
$941$	−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
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	−2.00000	−0.0648204
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	−14.0000	−0.453267
$955$	0	0
$956$	0	0
$957$	−24.0000	−0.775810
$958$	30.0000	0.969256
$959$	12.0000	0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	4.00000	0.128898
$964$	10.0000	0.322078
$965$	0	0
$966$	8.00000	0.257396
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	5.00000	0.160706
$969$	4.00000	0.128499
$970$	0	0
$971$	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	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	8.00000	0.255812
$979$	24.0000	0.767043
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−6.00000	−0.191468
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−8.00000	−0.255031
$985$	0	0
$986$	−6.00000	−0.191079
$987$	−16.0000	−0.509286
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	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
$995$	0	0
$996$	−16.0000	−0.506979
$997$	62.0000	1.96356	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000	1.96356	0.981780	+	0.190022i	$0.0608559\pi$
$998$	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	2550.2.a.y.1.1		1
3.2	odd	2		7650.2.a.y.1.1		1
5.2	odd	4		2550.2.d.j.2449.2		2
5.3	odd	4		2550.2.d.j.2449.1		2
5.4	even	2		510.2.a.b.1.1	✓	1
15.14	odd	2		1530.2.a.i.1.1		1
20.19	odd	2		4080.2.a.n.1.1		1
85.84	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	5.4	even	2
1530.2.a.i.1.1		1	15.14	odd	2
2550.2.a.y.1.1		1	1.1	even	1	trivial
2550.2.d.j.2449.1		2	5.3	odd	4
2550.2.d.j.2449.2		2	5.2	odd	4
4080.2.a.n.1.1		1	20.19	odd	2
7650.2.a.y.1.1		1	3.2	odd	2
8670.2.a.c.1.1		1	85.84	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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