LMFDB - Embedded newform 5550.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5550,2,Mod(1,5550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5550, 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("5550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5550.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:	$44.3169731218$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5550.1

$q$ -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} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{21} +4.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -4.00000 q^{46} -10.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +2.00000 q^{57} +6.00000 q^{59} +8.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} -1.00000 q^{74} -2.00000 q^{76} -2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +2.00000 q^{84} -4.00000 q^{86} +6.00000 q^{89} +4.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} +10.0000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +3.00000 q^{98} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	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
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	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
$25$	0	0
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	2.00000	0.324443
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	−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	$0.543862\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	2.00000	0.264906
$58$	0	0
$59$	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
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−8.00000	−1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.00000	−0.242536
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	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
$91$	4.00000	0.419314
$92$	4.00000	0.417029
$93$	−8.00000	−0.829561
$94$	10.0000	1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	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
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	−2.00000	−0.188982
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	−6.00000	−0.552345
$119$	4.00000	0.366679
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	0	0
$126$	2.00000	0.178174
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	4.00000	0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	4.00000	0.340503
$139$	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
$141$	10.0000	0.842152
$142$	−8.00000	−0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.00000	−0.662085
$147$	3.00000	0.247436
$148$	1.00000	0.0821995
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	2.00000	0.162221
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	16.0000	1.27289
$159$	2.00000	0.158610
$160$	0	0
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−6.00000	−0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−4.00000	−0.296500
$183$	−8.00000	−0.591377
$184$	−4.00000	−0.294884
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	−10.0000	−0.729325
$189$	2.00000	0.145479
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	−8.00000	−0.557386
$207$	4.00000	0.278019
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−2.00000	−0.137361
$213$	−8.00000	−0.548151
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	−16.0000	−1.08615
$218$	12.0000	0.812743
$219$	−8.00000	−0.540590
$220$	0	0
$221$	4.00000	0.269069
$222$	1.00000	0.0671156
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	10.0000	0.665190
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	2.00000	0.132453
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	16.0000	1.03931
$238$	−4.00000	−0.259281
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	8.00000	0.512148
$245$	0	0
$246$	6.00000	0.382546
$247$	4.00000	0.254514
$248$	−8.00000	−0.508001
$249$	0	0
$250$	0	0
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	4.00000	0.249029
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−2.00000	−0.121268
$273$	−4.00000	−0.242091
$274$	−8.00000	−0.483298
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−12.0000	−0.719712
$279$	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	$0.596411\pi$
$282$	−10.0000	−0.595491
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	8.00000	0.468165
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−14.0000	−0.810998
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	10.0000	0.574485
$304$	−2.00000	−0.114708
$305$	0	0
$306$	2.00000	0.114332
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−2.00000	−0.113228
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	−2.00000	−0.112154
$319$	0	0
$320$	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
$326$	20.0000	1.10770
$327$	12.0000	0.663602
$328$	−6.00000	−0.331295
$329$	20.0000	1.10264
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	9.00000	0.489535
$339$	10.0000	0.543125
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−14.0000	−0.752645
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	6.00000	0.317999
$357$	−4.00000	−0.211702
$358$	−6.00000	−0.317110
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	6.00000	0.315353
$363$	11.0000	0.577350
$364$	4.00000	0.209657
$365$	0	0
$366$	8.00000	0.418167
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	4.00000	0.208514
$369$	6.00000	0.312348
$370$	0	0
$371$	4.00000	0.207670
$372$	−8.00000	−0.414781
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	10.0000	0.515711
$377$	0	0
$378$	−2.00000	−0.102869
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	−8.00000	−0.409316
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000	0.203331
$388$	−14.0000	−0.710742
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	3.00000	0.151523
$393$	6.00000	0.302660
$394$	−10.0000	−0.503793
$395$	0	0
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−4.00000	−0.199502
$403$	−16.0000	−0.797017
$404$	−10.0000	−0.497519
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−2.00000	−0.0990148
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	8.00000	0.394132
$413$	−12.0000	−0.590481
$414$	−4.00000	−0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	8.00000	0.389434
$423$	−10.0000	−0.486217
$424$	2.00000	0.0971286
$425$	0	0
$426$	8.00000	0.387601
$427$	−16.0000	−0.774294
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−12.0000	−0.574696
$437$	−8.00000	−0.382692
$438$	8.00000	0.382255
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−4.00000	−0.190261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−1.00000	−0.0474579
$445$	0	0
$446$	26.0000	1.23114
$447$	−14.0000	−0.662177
$448$	−2.00000	−0.0944911
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	0	0
$452$	−10.0000	−0.470360
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−2.00000	−0.0934539
$459$	2.00000	0.0933520
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	8.00000	0.369406
$470$	0	0
$471$	6.00000	0.276465
$472$	−6.00000	−0.276172
$473$	0	0
$474$	−16.0000	−0.734904
$475$	0	0
$476$	4.00000	0.183340
$477$	−2.00000	−0.0915737
$478$	8.00000	0.365911
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	22.0000	1.00207
$483$	8.00000	0.364013
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−8.00000	−0.362143
$489$	20.0000	0.904431
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	0	0
$502$	−30.0000	−1.33897
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	6.00000	0.266207
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	26.0000	1.14681
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	2.00000	0.0878750
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−6.00000	−0.261612
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	6.00000	0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	−6.00000	−0.258919
$538$	−6.00000	−0.258678
$539$	0	0
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	2.00000	0.0857493
$545$	0	0
$546$	4.00000	0.171184
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	8.00000	0.341743
$549$	8.00000	0.341432
$550$	0	0
$551$	0	0
$552$	4.00000	0.170251
$553$	32.0000	1.36078
$554$	14.0000	0.594803
$555$	0	0
$556$	12.0000	0.508913
$557$	−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
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	10.0000	0.421076
$565$	0	0
$566$	12.0000	0.504398
$567$	−2.00000	−0.0839921
$568$	−8.00000	−0.335673
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	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
$573$	−8.00000	−0.334205
$574$	12.0000	0.500870
$575$	0	0
$576$	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	$0.566745\pi$
$578$	13.0000	0.540729
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	−14.0000	−0.580319
$583$	0	0
$584$	−8.00000	−0.331042
$585$	0	0
$586$	6.00000	0.247858
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	3.00000	0.123718
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−10.0000	−0.411345
$592$	1.00000	0.0410997
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	0	0
$595$	0	0
$596$	14.0000	0.573462
$597$	16.0000	0.654836
$598$	8.00000	0.327144
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	0	0
$605$	0	0
$606$	−10.0000	−0.406222
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	0	0
$611$	20.0000	0.809113
$612$	−2.00000	−0.0808452
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	8.00000	0.321807
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	4.00000	0.160385
$623$	−12.0000	−0.480770
$624$	2.00000	0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	−6.00000	−0.239426
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	16.0000	0.636446
$633$	8.00000	0.317971
$634$	26.0000	1.03259
$635$	0	0
$636$	2.00000	0.0793052
$637$	6.00000	0.237729
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	4.00000	0.157867
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−4.00000	−0.157378
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	−20.0000	−0.783260
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	6.00000	0.234261
$657$	8.00000	0.312110
$658$	−20.0000	−0.779681
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−10.0000	−0.388661
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	0	0
$668$	0	0
$669$	26.0000	1.00522
$670$	0	0
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−10.0000	−0.384048
$679$	28.0000	1.07454
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−20.0000	−0.763604
$687$	−2.00000	−0.0763048
$688$	4.00000	0.152499
$689$	4.00000	0.152388
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	26.0000	0.984115
$699$	0	0
$700$	0	0
$701$	−4.00000	−0.151078	−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000	−0.151078	−0.0755390	+	0.997143i	$0.524068\pi$
$702$	−2.00000	−0.0754851
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	20.0000	0.752177
$708$	−6.00000	−0.225494
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	−6.00000	−0.224860
$713$	32.0000	1.19841
$714$	4.00000	0.149696
$715$	0	0
$716$	6.00000	0.224231
$717$	8.00000	0.298765
$718$	16.0000	0.597115
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	15.0000	0.558242
$723$	22.0000	0.818189
$724$	−6.00000	−0.222988
$725$	0	0
$726$	−11.0000	−0.408248
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	−8.00000	−0.295689
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	−4.00000	−0.146845
$743$	−38.0000	−1.39408	−0.697042	−	0.717030i	$-0.745501\pi$
$743$	−38.0000	−1.39408	−0.697042	+	0.717030i	$0.745501\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−10.0000	−0.364662
$753$	−30.0000	−1.09326
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	6.00000	0.217357
$763$	24.0000	0.868858
$764$	8.00000	0.289430
$765$	0	0
$766$	−4.00000	−0.144526
$767$	−12.0000	−0.433295
$768$	−1.00000	−0.0360844
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	26.0000	0.936367
$772$	14.0000	0.503871
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	14.0000	0.502571
$777$	2.00000	0.0717496
$778$	16.0000	0.573628
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−6.00000	−0.214013
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	10.0000	0.356235
$789$	−6.00000	−0.213606
$790$	0	0
$791$	20.0000	0.711118
$792$	0	0
$793$	−16.0000	−0.568177
$794$	34.0000	1.20661
$795$	0	0
$796$	−16.0000	−0.567105
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	4.00000	0.141598
$799$	20.0000	0.707549
$800$	0	0
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	16.0000	0.563576
$807$	−6.00000	−0.211210
$808$	10.0000	0.351799
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	2.00000	0.0700140
$817$	−8.00000	−0.279885
$818$	−14.0000	−0.489499
$819$	4.00000	0.139771
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	8.00000	0.279032
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	12.0000	0.417533
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	4.00000	0.139010
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	−2.00000	−0.0693375
$833$	6.00000	0.207888
$834$	12.0000	0.415526
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	20.0000	0.690889
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	20.0000	0.689246
$843$	10.0000	0.344418
$844$	−8.00000	−0.275371
$845$	0	0
$846$	10.0000	0.343807
$847$	22.0000	0.755929
$848$	−2.00000	−0.0686803
$849$	12.0000	0.411839
$850$	0	0
$851$	4.00000	0.137118
$852$	−8.00000	−0.274075
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	−4.00000	−0.136717
$857$	34.0000	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$857$	34.0000	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	−16.0000	−0.544962
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	16.0000	0.543702
$867$	13.0000	0.441503
$868$	−16.0000	−0.543075
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	12.0000	0.406371
$873$	−14.0000	−0.473828
$874$	8.00000	0.270604
$875$	0	0
$876$	−8.00000	−0.270295
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	8.00000	0.269987
$879$	6.00000	0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	3.00000	0.101015
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−12.0000	−0.403148
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	1.00000	0.0335578
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	−26.0000	−0.870544
$893$	20.0000	0.669274
$894$	14.0000	0.468230
$895$	0	0
$896$	2.00000	0.0668153
$897$	8.00000	0.267112
$898$	2.00000	0.0667409
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	8.00000	0.266223
$904$	10.0000	0.332595
$905$	0	0
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−12.0000	−0.398234
$909$	−10.0000	−0.331679
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	14.0000	0.463079
$915$	0	0
$916$	2.00000	0.0660819
$917$	12.0000	0.396275
$918$	−2.00000	−0.0660098
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	8.00000	0.262754
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	4.00000	0.130954
$934$	12.0000	0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	−8.00000	−0.261209
$939$	10.0000	0.326338
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	−6.00000	−0.195491
$943$	24.0000	0.781548
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	16.0000	0.519656
$949$	−16.0000	−0.519382
$950$	0	0
$951$	26.0000	0.843108
$952$	−4.00000	−0.129641
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	24.0000	0.775405
$959$	−16.0000	−0.516667
$960$	0	0
$961$	33.0000	1.06452
$962$	2.00000	0.0644826
$963$	4.00000	0.128898
$964$	−22.0000	−0.708572
$965$	0	0
$966$	−8.00000	−0.257396
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	11.0000	0.353553
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	−1.00000	−0.0320750
$973$	−24.0000	−0.769405
$974$	0	0
$975$	0	0
$976$	8.00000	0.256074
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−20.0000	−0.639529
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	−8.00000	−0.255290
$983$	10.0000	0.318950	0.159475	−	0.987202i	$-0.449020\pi$
$983$	10.0000	0.318950	0.159475	+	0.987202i	$0.449020\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	−20.0000	−0.636607
$988$	4.00000	0.127257
$989$	16.0000	0.508770
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	−8.00000	−0.254000
$993$	−10.0000	−0.317340
$994$	16.0000	0.507489
$995$	0	0
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	6.00000	0.189927
$999$	−1.00000	−0.0316386

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.c.1.1		1
5.2	odd	4		1110.2.d.a.889.1	✓	2
5.3	odd	4		1110.2.d.a.889.2	yes	2
5.4	even	2		5550.2.a.bp.1.1		1
15.2	even	4		3330.2.d.g.1999.2		2
15.8	even	4		3330.2.d.g.1999.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.d.a.889.1	✓	2	5.2	odd	4
1110.2.d.a.889.2	yes	2	5.3	odd	4
3330.2.d.g.1999.1		2	15.8	even	4
3330.2.d.g.1999.2		2	15.2	even	4
5550.2.a.c.1.1		1	1.1	even	1	trivial
5550.2.a.bp.1.1		1	5.4	even	2

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5550.2.a.c.1.1

Level	$5550$
Weight	$2$
Character	5550.1
Self dual	yes
Analytic conductor	$44.317$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$