LMFDB - Embedded newform 5520.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} +5.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -5.00000 q^{29} -3.00000 q^{31} -2.00000 q^{33} -3.00000 q^{35} -7.00000 q^{37} -2.00000 q^{39} -11.0000 q^{41} +8.00000 q^{43} +1.00000 q^{45} -8.00000 q^{47} +2.00000 q^{49} +5.00000 q^{51} +5.00000 q^{53} -2.00000 q^{55} +2.00000 q^{57} +1.00000 q^{59} -8.00000 q^{61} -3.00000 q^{63} -2.00000 q^{65} +9.00000 q^{67} +1.00000 q^{69} -1.00000 q^{71} +10.0000 q^{73} +1.00000 q^{75} +6.00000 q^{77} +1.00000 q^{81} -15.0000 q^{83} +5.00000 q^{85} -5.00000 q^{87} +6.00000 q^{91} -3.00000 q^{93} +2.00000 q^{95} -10.0000 q^{97} -2.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−3.00000	−0.311086
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	0	0
$113$	−7.00000	−0.658505	−0.329252	−	0.944242i	$-0.606797\pi$
$113$	−7.00000	−0.658505	−0.329252	+	0.944242i	$0.606797\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−15.0000	−1.37505
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−11.0000	−0.991837
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−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
$133$	−6.00000	−0.520266
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	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
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−5.00000	−0.415227
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	5.00000	0.396526
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	1.00000	0.0751646
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	−10.0000	−0.731272
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	15.0000	1.05279
$204$	0	0
$205$	−11.0000	−0.768273
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	−1.00000	−0.0685189
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−10.0000	−0.672673
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	5.00000	0.313112
$256$	0	0
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	21.0000	1.30488
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.00000	−0.426798	−0.213399	−	0.976965i	$-0.568453\pi$
$269$	−7.00000	−0.426798	−0.213399	+	0.976965i	$0.568453\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−9.00000	−0.534994	−0.267497	−	0.963559i	$-0.586197\pi$
$283$	−9.00000	−0.534994	−0.267497	+	0.963559i	$0.586197\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	33.0000	1.94793
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−33.0000	−1.92788	−0.963940	−	0.266119i	$-0.914259\pi$
$293$	−33.0000	−1.92788	−0.963940	+	0.266119i	$0.914259\pi$
$294$	0	0
$295$	1.00000	0.0582223
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	15.0000	0.861727
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−25.0000	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$313$	−25.0000	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	16.0000	0.898650	0.449325	−	0.893368i	$-0.351665\pi$
$317$	16.0000	0.898650	0.449325	+	0.893368i	$0.351665\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	10.0000	0.556415
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	0	0
$333$	−7.00000	−0.383598
$334$	0	0
$335$	9.00000	0.491723
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−7.00000	−0.380188
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	0	0
$355$	−1.00000	−0.0530745
$356$	0	0
$357$	−15.0000	−0.793884
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−33.0000	−1.72259	−0.861293	−	0.508109i	$-0.830345\pi$
$367$	−33.0000	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$368$	0	0
$369$	−11.0000	−0.572637
$370$	0	0
$371$	−15.0000	−0.778761
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	10.0000	0.515026
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	14.0000	0.693954
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	−15.0000	−0.736321
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	24.0000	1.16144
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	−5.00000	−0.239732
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−20.0000	−0.945968
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	22.0000	1.03594
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	41.0000	1.89725	0.948627	−	0.316397i	$-0.102473\pi$
$467$	41.0000	1.89725	0.948627	+	0.316397i	$0.102473\pi$
$468$	0	0
$469$	−27.0000	−1.24674
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	5.00000	0.228934
$478$	0	0
$479$	−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
$481$	14.0000	0.638345
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	35.0000	1.57953	0.789764	−	0.613411i	$-0.210203\pi$
$491$	35.0000	1.57953	0.789764	+	0.613411i	$0.210203\pi$
$492$	0	0
$493$	−25.0000	−1.12594
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	3.00000	0.134568
$498$	0	0
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−1.00000	−0.0445878	−0.0222939	−	0.999751i	$-0.507097\pi$
$503$	−1.00000	−0.0445878	−0.0222939	+	0.999751i	$0.507097\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−30.0000	−1.32712
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−15.0000	−0.653410
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.00000	0.0433963
$532$	0	0
$533$	22.0000	0.952926
$534$	0	0
$535$	−15.0000	−0.648507
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−10.0000	−0.426014
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.00000	−0.297133
$556$	0	0
$557$	−27.0000	−1.14403	−0.572013	−	0.820244i	$-0.693837\pi$
$557$	−27.0000	−1.14403	−0.572013	+	0.820244i	$0.693837\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	−7.00000	−0.294492
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	45.0000	1.86691
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	0	0
$597$	14.0000	0.572982
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	15.0000	0.607831
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	−11.0000	−0.443563
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	−35.0000	−1.39554
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	1.00000	0.0397464
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−1.00000	−0.0395594
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	10.0000	0.393141	0.196570	−	0.980490i	$-0.437020\pi$
$647$	10.0000	0.393141	0.196570	+	0.980490i	$0.437020\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	9.00000	0.352738
$652$	0	0
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	−10.0000	−0.388368
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	−5.00000	−0.193601
$668$	0	0
$669$	−26.0000	−1.00522
$670$	0	0
$671$	16.0000	0.617673
$672$	0	0
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	51.0000	1.96009	0.980045	−	0.198778i	$-0.0636972\pi$
$677$	51.0000	1.96009	0.980045	+	0.198778i	$0.0636972\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	24.0000	0.915657
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	−5.00000	−0.189661
$696$	0	0
$697$	−55.0000	−2.08327
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	4.00000	0.151078	0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000	0.151078	0.0755390	+	0.997143i	$0.475932\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	−45.0000	−1.69240
$708$	0	0
$709$	−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
$711$	0	0
$712$	0	0
$713$	−3.00000	−0.112351
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	1.00000	0.0373457
$718$	0	0
$719$	9.00000	0.335643	0.167822	−	0.985817i	$-0.446327\pi$
$719$	9.00000	0.335643	0.167822	+	0.985817i	$0.446327\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	−5.00000	−0.185695
$726$	0	0
$727$	31.0000	1.14973	0.574863	−	0.818250i	$-0.305055\pi$
$727$	31.0000	1.14973	0.574863	+	0.818250i	$0.305055\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	−18.0000	−0.663039
$738$	0	0
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	0	0
$747$	−15.0000	−0.548821
$748$	0	0
$749$	45.0000	1.64426
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	20.0000	0.727875
$756$	0	0
$757$	7.00000	0.254419	0.127210	−	0.991876i	$-0.459398\pi$
$757$	7.00000	0.254419	0.127210	+	0.991876i	$0.459398\pi$
$758$	0	0
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	0	0
$765$	5.00000	0.180775
$766$	0	0
$767$	−2.00000	−0.0722158
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−3.00000	−0.107763
$776$	0	0
$777$	21.0000	0.753371
$778$	0	0
$779$	−22.0000	−0.788232
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	−13.0000	−0.463990
$786$	0	0
$787$	29.0000	1.03374	0.516869	−	0.856064i	$-0.327097\pi$
$787$	29.0000	1.03374	0.516869	+	0.856064i	$0.327097\pi$
$788$	0	0
$789$	11.0000	0.391610
$790$	0	0
$791$	21.0000	0.746674
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	5.00000	0.177332
$796$	0	0
$797$	−29.0000	−1.02723	−0.513616	−	0.858020i	$-0.671695\pi$
$797$	−29.0000	−1.02723	−0.513616	+	0.858020i	$0.671695\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−7.00000	−0.246412
$808$	0	0
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	33.0000	1.15879	0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000	1.15879	0.579393	+	0.815048i	$0.303290\pi$
$812$	0	0
$813$	−13.0000	−0.455930
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−29.0000	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$827$	−29.0000	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$828$	0	0
$829$	45.0000	1.56291	0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000	1.56291	0.781457	+	0.623959i	$0.214477\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	10.0000	0.346479
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−3.00000	−0.103695
$838$	0	0
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	21.0000	0.721569
$848$	0	0
$849$	−9.00000	−0.308879
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	0	0
$861$	33.0000	1.12464
$862$	0	0
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−18.0000	−0.609907
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	0	0
$879$	−33.0000	−1.11306
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	1.00000	0.0336146
$886$	0	0
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	16.0000	0.534821
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	15.0000	0.500278
$900$	0	0
$901$	25.0000	0.832871
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	−15.0000	−0.498067	−0.249033	−	0.968495i	$-0.580113\pi$
$907$	−15.0000	−0.498067	−0.249033	+	0.968495i	$0.580113\pi$
$908$	0	0
$909$	15.0000	0.497519
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	30.0000	0.992855
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	36.0000	1.18882
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	0	0
$923$	2.00000	0.0658308
$924$	0	0
$925$	−7.00000	−0.230159
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−41.0000	−1.34517	−0.672583	−	0.740022i	$-0.734815\pi$
$929$	−41.0000	−1.34517	−0.672583	+	0.740022i	$0.734815\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−10.0000	−0.327035
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	−25.0000	−0.815844
$940$	0	0
$941$	16.0000	0.521585	0.260793	−	0.965395i	$-0.416016\pi$
$941$	16.0000	0.521585	0.260793	+	0.965395i	$0.416016\pi$
$942$	0	0
$943$	−11.0000	−0.358209
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	16.0000	0.518836
$952$	0	0
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	−54.0000	−1.74375
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−15.0000	−0.483368
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	0	0
$969$	10.0000	0.321246
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−25.0000	−0.799821	−0.399910	−	0.916554i	$-0.630959\pi$
$977$	−25.0000	−0.799821	−0.399910	+	0.916554i	$0.630959\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	49.0000	1.55654	0.778268	−	0.627932i	$-0.216098\pi$
$991$	49.0000	1.55654	0.778268	+	0.627932i	$0.216098\pi$
$992$	0	0
$993$	−3.00000	−0.0952021
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	0	0
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bc.1.1		1
4.3	odd	2		345.2.a.f.1.1	✓	1
12.11	even	2		1035.2.a.a.1.1		1
20.3	even	4		1725.2.b.b.1174.1		2
20.7	even	4		1725.2.b.b.1174.2		2
20.19	odd	2		1725.2.a.a.1.1		1
60.59	even	2		5175.2.a.x.1.1		1
92.91	even	2		7935.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.f.1.1	✓	1	4.3	odd	2
1035.2.a.a.1.1		1	12.11	even	2
1725.2.a.a.1.1		1	20.19	odd	2
1725.2.b.b.1174.1		2	20.3	even	4
1725.2.b.b.1174.2		2	20.7	even	4
5175.2.a.x.1.1		1	60.59	even	2
5520.2.a.bc.1.1		1	1.1	even	1	trivial
7935.2.a.k.1.1		1	92.91	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

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