LMFDB - Embedded newform 55.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [55,2,Mod(1,55)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("55.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(55, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 55 = 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	55.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$0.439177211117$
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	$\chi$	$=$	55.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -3.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -3.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{55} +6.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +2.00000 q^{65} -16.0000 q^{67} -6.00000 q^{68} +8.00000 q^{71} +9.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +6.00000 q^{85} +4.00000 q^{86} +3.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} -4.00000 q^{92} -12.0000 q^{94} -4.00000 q^{95} +10.0000 q^{97} -7.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	−3.00000	−1.00000
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−3.00000	−0.707107
$19$	−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
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	0	0
$29$	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	$0.755135\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	3.00000	0.500000
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−3.00000	−0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	−3.00000	−0.447214
$46$	4.00000	0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	9.00000	1.06066
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	4.00000	0.431331
$87$	0	0
$88$	3.00000	0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−4.00000	−0.410391
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−7.00000	−0.707107
$99$	3.00000	0.301511
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−6.00000	−0.557086
$117$	−6.00000	−0.554700
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	0	0
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	2.00000	0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−16.0000	−1.38219
$135$	0	0
$136$	−18.0000	−1.54349
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	−2.00000	−0.167248
$144$	3.00000	0.250000
$145$	6.00000	0.498273
$146$	14.0000	1.15865
$147$	0	0
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	12.0000	0.973329
$153$	−18.0000	−1.45521
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	5.00000	0.395285
$161$	0	0
$162$	9.00000	0.707107
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	12.0000	0.917663
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	10.0000	0.749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	3.00000	0.223607
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	−12.0000	−0.884652
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−6.00000	−0.438763
$188$	12.0000	0.875190
$189$	0	0
$190$	−4.00000	−0.290191
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	7.00000	0.500000
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	3.00000	0.213201
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−10.0000	−0.703598
$203$	0	0
$204$	0	0
$205$	2.00000	0.139686
$206$	−4.00000	−0.278693
$207$	−12.0000	−0.834058
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	12.0000	0.820303
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	−18.0000	−1.21911
$219$	0	0
$220$	1.00000	0.0674200
$221$	12.0000	0.807207
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	−18.0000	−1.18176
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−6.00000	−0.392232
$235$	−12.0000	−0.782794
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	10.0000	0.640184
$245$	−7.00000	−0.447214
$246$	0	0
$247$	−8.00000	−0.509028
$248$	24.0000	1.52400
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	0	0
$260$	−2.00000	−0.124035
$261$	−18.0000	−1.11417
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	0	0
$268$	16.0000	0.977356
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	18.0000	1.08742
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	12.0000	0.719712
$279$	24.0000	1.43684
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	−15.0000	−0.883883
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	0	0
$292$	−14.0000	−0.819288
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	6.00000	0.348743
$297$	0	0
$298$	−10.0000	−0.579284
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	0	0
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	−18.0000	−1.02899
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	7.00000	0.391312
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	−9.00000	−0.500000
$325$	2.00000	0.110940
$326$	16.0000	0.886158
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	−8.00000	−0.437741
$335$	−16.0000	−0.874173
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−6.00000	−0.325396
$341$	8.00000	0.433224
$342$	12.0000	0.648886
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	−5.00000	−0.266501
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−10.0000	−0.529999
$357$	0	0
$358$	4.00000	0.211407
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	9.00000	0.474342
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	0	0
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−4.00000	−0.208514
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	0	0
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	36.0000	1.85656
$377$	12.0000	0.618031
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	8.00000	0.409316
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	0	0
$386$	−26.0000	−1.32337
$387$	−12.0000	−0.609994
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	21.0000	1.06066
$393$	0	0
$394$	2.00000	0.100759
$395$	8.00000	0.402524
$396$	−3.00000	−0.150756
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	10.0000	0.497519
$405$	9.00000	0.447214
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	4.00000	0.197066
$413$	0	0
$414$	−12.0000	−0.589768
$415$	−4.00000	−0.196352
$416$	10.0000	0.490290
$417$	0	0
$418$	4.00000	0.195646
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000	0.194717
$423$	36.0000	1.75038
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	3.00000	0.143019
$441$	21.0000	1.00000
$442$	12.0000	0.570782
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	−4.00000	−0.189405
$447$	0	0
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−3.00000	−0.141421
$451$	−2.00000	−0.0941763
$452$	6.00000	0.282216
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	−12.0000	−0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	10.0000	0.455488
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	10.0000	0.454077
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	−7.00000	−0.316228
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	−8.00000	−0.359937
$495$	3.00000	0.134840
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−16.0000	−0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	18.0000	0.793946
$515$	−4.00000	−0.176261
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−18.0000	−0.787839
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	−48.0000	−2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	−12.0000	−0.520756
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	12.0000	0.518805
$536$	48.0000	2.07328
$537$	0	0
$538$	−18.0000	−0.776035
$539$	7.00000	0.301511
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	30.0000	1.28624
$545$	−18.0000	−0.771035
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−18.0000	−0.768922
$549$	30.0000	1.28037
$550$	−1.00000	−0.0426401
$551$	−24.0000	−1.02243
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	24.0000	1.01600
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	4.00000	0.168133
$567$	0	0
$568$	−24.0000	−1.00702
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	−21.0000	−0.875000
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−6.00000	−0.249136
$581$	0	0
$582$	0	0
$583$	2.00000	0.0828315
$584$	−42.0000	−1.73797
$585$	−6.00000	−0.248069
$586$	10.0000	0.413096
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	4.00000	0.164677
$591$	0	0
$592$	2.00000	0.0821995
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	8.00000	0.327144
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	48.0000	1.95471
$604$	−8.00000	−0.325515
$605$	1.00000	0.0406558
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	−10.0000	−0.404888
$611$	−24.0000	−0.970936
$612$	18.0000	0.727607
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	2.00000	0.0798087
$629$	−12.0000	−0.478471
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	−24.0000	−0.954669
$633$	0	0
$634$	−18.0000	−0.714871
$635$	16.0000	0.634941
$636$	0	0
$637$	−14.0000	−0.554700
$638$	−6.00000	−0.237542
$639$	−24.0000	−0.949425
$640$	−3.00000	−0.118585
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	−27.0000	−1.06066
$649$	−4.00000	−0.157014
$650$	2.00000	0.0784465
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	−2.00000	−0.0780869
$657$	−42.0000	−1.63858
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	24.0000	0.929284
$668$	8.00000	0.309529
$669$	0	0
$670$	−16.0000	−0.618134
$671$	10.0000	0.386046
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	0	0
$680$	−18.0000	−0.690268
$681$	0	0
$682$	8.00000	0.306336
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	−12.0000	−0.458831
$685$	18.0000	0.687745
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−4.00000	−0.151838
$695$	12.0000	0.455186
$696$	0	0
$697$	12.0000	0.454532
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	8.00000	0.300235
$711$	−24.0000	−0.900070
$712$	−30.0000	−1.12430
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−32.0000	−1.19423
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−3.00000	−0.111648
$723$	0	0
$724$	10.0000	0.371647
$725$	6.00000	0.222834
$726$	0	0
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	14.0000	0.518163
$731$	24.0000	0.887672
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	20.0000	0.737210
$737$	16.0000	0.589368
$738$	−6.00000	−0.220863
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	18.0000	0.659027
$747$	12.0000	0.439057
$748$	6.00000	0.219382
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	12.0000	0.437014
$755$	8.00000	0.291150
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	12.0000	0.435286
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	−18.0000	−0.650791
$766$	−12.0000	−0.433578
$767$	8.00000	0.288863
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	26.0000	0.935760
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−12.0000	−0.431331
$775$	−8.00000	−0.287368
$776$	−30.0000	−1.07694
$777$	0	0
$778$	6.00000	0.215110
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−8.00000	−0.286263
$782$	24.0000	0.858238
$783$	0	0
$784$	7.00000	0.250000
$785$	−2.00000	−0.0713831
$786$	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
$790$	8.00000	0.284627
$791$	0	0
$792$	−9.00000	−0.319801
$793$	−20.0000	−0.710221
$794$	30.0000	1.06466
$795$	0	0
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	5.00000	0.176777
$801$	−30.0000	−1.06000
$802$	2.00000	0.0706225
$803$	−14.0000	−0.494049
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	0	0
$808$	30.0000	1.05540
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	9.00000	0.316228
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	16.0000	0.560456
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−6.00000	−0.209785
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−36.0000	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	12.0000	0.417029
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	14.0000	0.485363
$833$	−42.0000	−1.45521
$834$	0	0
$835$	−8.00000	−0.276851
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−28.0000	−0.967244
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	6.00000	0.206774
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−9.00000	−0.309609
$846$	36.0000	1.23771
$847$	0	0
$848$	2.00000	0.0686803
$849$	0	0
$850$	6.00000	0.205798
$851$	−8.00000	−0.274236
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	−36.0000	−1.23045
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−24.0000	−0.817443
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	−22.0000	−0.747590
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−32.0000	−1.08428
$872$	54.0000	1.82867
$873$	−30.0000	−1.01535
$874$	−16.0000	−0.541208
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	1.00000	0.0337100
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	21.0000	0.707107
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	8.00000	0.268765
$887$	56.0000	1.88030	0.940148	−	0.340766i	$-0.110687\pi$
$887$	56.0000	1.88030	0.940148	+	0.340766i	$0.110687\pi$
$888$	0	0
$889$	0	0
$890$	10.0000	0.335201
$891$	−9.00000	−0.301511
$892$	4.00000	0.133930
$893$	48.0000	1.60626
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	0	0
$898$	2.00000	0.0667409
$899$	−48.0000	−1.60089
$900$	3.00000	0.100000
$901$	−12.0000	−0.399778
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	18.0000	0.598671
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	20.0000	0.663723
$909$	30.0000	0.995037
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	−26.0000	−0.860004
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	−34.0000	−1.11973
$923$	16.0000	0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−36.0000	−1.18303
$927$	12.0000	0.394132
$928$	30.0000	0.984798
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	−6.00000	−0.196537
$933$	0	0
$934$	0	0
$935$	−6.00000	−0.196221
$936$	18.0000	0.588348
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	0	0
$940$	12.0000	0.391397
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−4.00000	−0.130051
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	6.00000	0.194257
$955$	8.00000	0.258874
$956$	−8.00000	−0.258738
$957$	0	0
$958$	24.0000	0.775405
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−4.00000	−0.128965
$963$	−36.0000	−1.16008
$964$	−10.0000	−0.322078
$965$	−26.0000	−0.836970
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−3.00000	−0.0964237
$969$	0	0
$970$	10.0000	0.321081
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	0	0
$974$	28.0000	0.897178
$975$	0	0
$976$	10.0000	0.320092
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	7.00000	0.223607
$981$	54.0000	1.72409
$982$	−28.0000	−0.893516
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	36.0000	1.14647
$987$	0	0
$988$	8.00000	0.254514
$989$	16.0000	0.508770
$990$	3.00000	0.0953463
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.2.a.a.1.1	✓	1
3.2	odd	2		495.2.a.a.1.1		1
4.3	odd	2		880.2.a.h.1.1		1
5.2	odd	4		275.2.b.b.199.2		2
5.3	odd	4		275.2.b.b.199.1		2
5.4	even	2		275.2.a.a.1.1		1
7.6	odd	2		2695.2.a.c.1.1		1
8.3	odd	2		3520.2.a.n.1.1		1
8.5	even	2		3520.2.a.p.1.1		1
11.2	odd	10		605.2.g.c.81.1		4
11.3	even	5		605.2.g.a.251.1		4
11.4	even	5		605.2.g.a.511.1		4
11.5	even	5		605.2.g.a.366.1		4
11.6	odd	10		605.2.g.c.366.1		4
11.7	odd	10		605.2.g.c.511.1		4
11.8	odd	10		605.2.g.c.251.1		4
11.9	even	5		605.2.g.a.81.1		4
11.10	odd	2		605.2.a.b.1.1		1
12.11	even	2		7920.2.a.i.1.1		1
13.12	even	2		9295.2.a.b.1.1		1
15.2	even	4		2475.2.c.f.199.1		2
15.8	even	4		2475.2.c.f.199.2		2
15.14	odd	2		2475.2.a.i.1.1		1
20.3	even	4		4400.2.b.n.4049.2		2
20.7	even	4		4400.2.b.n.4049.1		2
20.19	odd	2		4400.2.a.p.1.1		1
33.32	even	2		5445.2.a.i.1.1		1
44.43	even	2		9680.2.a.r.1.1		1
55.54	odd	2		3025.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.a.1.1	✓	1	1.1	even	1	trivial
275.2.a.a.1.1		1	5.4	even	2
275.2.b.b.199.1		2	5.3	odd	4
275.2.b.b.199.2		2	5.2	odd	4
495.2.a.a.1.1		1	3.2	odd	2
605.2.a.b.1.1		1	11.10	odd	2
605.2.g.a.81.1		4	11.9	even	5
605.2.g.a.251.1		4	11.3	even	5
605.2.g.a.366.1		4	11.5	even	5
605.2.g.a.511.1		4	11.4	even	5
605.2.g.c.81.1		4	11.2	odd	10
605.2.g.c.251.1		4	11.8	odd	10
605.2.g.c.366.1		4	11.6	odd	10
605.2.g.c.511.1		4	11.7	odd	10
880.2.a.h.1.1		1	4.3	odd	2
2475.2.a.i.1.1		1	15.14	odd	2
2475.2.c.f.199.1		2	15.2	even	4
2475.2.c.f.199.2		2	15.8	even	4
2695.2.a.c.1.1		1	7.6	odd	2
3025.2.a.f.1.1		1	55.54	odd	2
3520.2.a.n.1.1		1	8.3	odd	2
3520.2.a.p.1.1		1	8.5	even	2
4400.2.a.p.1.1		1	20.19	odd	2
4400.2.b.n.4049.1		2	20.7	even	4
4400.2.b.n.4049.2		2	20.3	even	4
5445.2.a.i.1.1		1	33.32	even	2
7920.2.a.i.1.1		1	12.11	even	2
9295.2.a.b.1.1		1	13.12	even	2
9680.2.a.r.1.1		1	44.43	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	55.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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