LMFDB - Embedded newform 4620.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
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	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +4.00000 q^{13} -1.00000 q^{15} +2.00000 q^{17} +2.00000 q^{19} -1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.00000 q^{29} +8.00000 q^{31} +1.00000 q^{33} +1.00000 q^{35} -10.0000 q^{37} -4.00000 q^{39} -8.00000 q^{43} +1.00000 q^{45} +12.0000 q^{47} +1.00000 q^{49} -2.00000 q^{51} +10.0000 q^{53} -1.00000 q^{55} -2.00000 q^{57} +12.0000 q^{59} -2.00000 q^{61} +1.00000 q^{63} +4.00000 q^{65} -12.0000 q^{67} -4.00000 q^{69} -16.0000 q^{73} -1.00000 q^{75} -1.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +2.00000 q^{85} +4.00000 q^{87} -18.0000 q^{89} +4.00000 q^{91} -8.00000 q^{93} +2.00000 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	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
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	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	$0.525431\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	16.0000	1.08118
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	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
$245$	1.00000	0.0638877
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.00000	−0.105851
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.0000	0.519174
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	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
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	14.0000	0.657777
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	−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
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	10.0000	0.457869
$478$	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
$481$	−40.0000	−1.82384
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	2.00000	0.0908153
$486$	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
$489$	−12.0000	−0.542659
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−14.0000	−0.625474
$502$	0	0
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	10.0000	0.424476
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	−2.00000	−0.0842900	−0.0421450	−	0.999112i	$-0.513419\pi$
$563$	−2.00000	−0.0842900	−0.0421450	+	0.999112i	$0.513419\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	6.00000	0.251092	0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000	0.251092	0.125546	+	0.992088i	$0.459932\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	10.0000	0.397464
$634$	0	0
$635$	20.0000	0.793676
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	−16.0000	−0.624219
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−40.0000	−1.54189	−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000	−1.54189	−0.770943	+	0.636904i	$0.780215\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	40.0000	1.52388
$690$	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
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	22.0000	0.834508
$696$	0	0
$697$	0	0
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−48.0000	−1.77292	−0.886460	−	0.462805i	$-0.846843\pi$
$733$	−48.0000	−1.77292	−0.886460	+	0.462805i	$0.846843\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	46.0000	1.69214	0.846069	−	0.533074i	$-0.178963\pi$
$739$	46.0000	1.69214	0.846069	+	0.533074i	$0.178963\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	0	0
$789$	−30.0000	−1.06803
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	0	0
$813$	6.00000	0.210429
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−20.0000	−0.698005	−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000	−0.698005	−0.349002	+	0.937122i	$0.613479\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	14.0000	0.484490
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−4.00000	−0.137767
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−40.0000	−1.37118
$852$	0	0
$853$	−52.0000	−1.78045	−0.890223	−	0.455525i	$-0.849452\pi$
$853$	−52.0000	−1.78045	−0.890223	+	0.455525i	$0.849452\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	0	0
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	24.0000	0.803129
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	−32.0000	−1.06726
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−22.0000	−0.731305
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	2.00000	0.0661903
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−8.00000	−0.264183
$918$	0	0
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	32.0000	1.04763
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−32.0000	−1.04539	−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000	−1.04539	−0.522697	+	0.852518i	$0.675074\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	36.0000	1.15768	0.578841	−	0.815440i	$-0.303505\pi$
$967$	36.0000	1.15768	0.578841	+	0.815440i	$0.303505\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	22.0000	0.705288
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	52.0000	1.65854	0.829271	−	0.558846i	$-0.188756\pi$
$983$	52.0000	1.65854	0.829271	+	0.558846i	$0.188756\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	−20.0000	−0.634043
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.e.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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