LMFDB - Embedded newform 4620.2.a.l.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:	$1$
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)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{13} +1.00000 q^{15} -3.00000 q^{17} +3.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.00000 q^{29} +4.00000 q^{31} +1.00000 q^{33} -1.00000 q^{35} -6.00000 q^{37} -6.00000 q^{39} -4.00000 q^{41} -9.00000 q^{43} +1.00000 q^{45} +1.00000 q^{49} -3.00000 q^{51} +3.00000 q^{53} +1.00000 q^{55} +3.00000 q^{57} +5.00000 q^{59} +5.00000 q^{61} -1.00000 q^{63} -6.00000 q^{65} -2.00000 q^{67} -1.00000 q^{69} -4.00000 q^{71} -16.0000 q^{73} +1.00000 q^{75} -1.00000 q^{77} +2.00000 q^{79} +1.00000 q^{81} -3.00000 q^{83} -3.00000 q^{85} -7.00000 q^{87} -9.00000 q^{89} +6.00000 q^{91} +4.00000 q^{93} +3.00000 q^{95} -7.00000 q^{97} +1.00000 q^{99} +O(q^{100})$

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
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	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
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−7.00000	−0.750479
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−13.0000	−1.22294	−0.611469	−	0.791269i	$-0.709421\pi$
$113$	−13.0000	−1.22294	−0.611469	+	0.791269i	$0.709421\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	0	0
$129$	−9.00000	−0.792406
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−7.00000	−0.581318
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	9.00000	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$157$	9.00000	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	1.00000	0.0788110
$162$	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
$165$	1.00000	0.0778499
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	5.00000	0.375823
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	7.00000	0.491304
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−18.0000	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$211$	−18.0000	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	−9.00000	−0.613795
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−16.0000	−1.08118
$220$	0	0
$221$	18.0000	1.21081
$222$	0	0
$223$	−15.0000	−1.00447	−0.502237	−	0.864730i	$-0.667490\pi$
$223$	−15.0000	−1.00447	−0.502237	+	0.864730i	$0.667490\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	13.0000	0.862840	0.431420	−	0.902151i	$-0.358013\pi$
$227$	13.0000	0.862840	0.431420	+	0.902151i	$0.358013\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.00000	0.129914
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−18.0000	−1.14531
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	−7.00000	−0.433289
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	−9.00000	−0.550791
$268$	0	0
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−7.00000	−0.410347
$292$	0	0
$293$	5.00000	0.292103	0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000	0.292103	0.146052	+	0.989277i	$0.453343\pi$
$294$	0	0
$295$	5.00000	0.291111
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	9.00000	0.518751
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−7.00000	−0.391925
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−6.00000	−0.332820
$326$	0	0
$327$	12.0000	0.663602
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−29.0000	−1.59398	−0.796992	−	0.603990i	$-0.793577\pi$
$331$	−29.0000	−1.59398	−0.796992	+	0.603990i	$0.793577\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−7.00000	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$338$	0	0
$339$	−13.0000	−0.706063
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	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
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	37.0000	1.95279	0.976393	−	0.216003i	$-0.0693022\pi$
$359$	37.0000	1.95279	0.976393	+	0.216003i	$0.0693022\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	−1.00000	−0.0521996	−0.0260998	−	0.999659i	$-0.508309\pi$
$367$	−1.00000	−0.0521996	−0.0260998	+	0.999659i	$0.508309\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	42.0000	2.16311
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−9.00000	−0.457496
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−5.00000	−0.246034
$414$	0	0
$415$	−3.00000	−0.147264
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	−5.00000	−0.241967
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−7.00000	−0.335624
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	31.0000	1.47955	0.739775	−	0.672855i	$-0.234932\pi$
$439$	31.0000	1.47955	0.739775	+	0.672855i	$0.234932\pi$
$440$	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
$445$	−9.00000	−0.426641
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	10.0000	0.469841
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	−9.00000	−0.413820
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	−22.0000	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$479$	−22.0000	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	21.0000	0.945792
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	−31.0000	−1.38775	−0.693875	−	0.720095i	$-0.744098\pi$
$499$	−31.0000	−1.38775	−0.693875	+	0.720095i	$0.744098\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	3.00000	0.133763	0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000	0.133763	0.0668817	+	0.997761i	$0.478695\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	23.0000	1.02147
$508$	0	0
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	7.00000	0.308457
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	17.0000	0.744784	0.372392	−	0.928076i	$-0.378538\pi$
$521$	17.0000	0.744784	0.372392	+	0.928076i	$0.378538\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	5.00000	0.216982
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	−23.0000	−0.983409	−0.491704	−	0.870762i	$-0.663626\pi$
$547$	−23.0000	−0.983409	−0.491704	+	0.870762i	$0.663626\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	−21.0000	−0.894630
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	0	0
$555$	−6.00000	−0.254686
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	54.0000	2.28396
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−13.0000	−0.546914
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	3.00000	0.124461
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	0	0
$609$	7.00000	0.283654
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	9.00000	0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	18.0000	0.717707
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	−18.0000	−0.715436
$634$	0	0
$635$	−1.00000	−0.0396838
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	−41.0000	−1.61688	−0.808441	−	0.588577i	$-0.799688\pi$
$643$	−41.0000	−1.61688	−0.808441	+	0.588577i	$0.799688\pi$
$644$	0	0
$645$	−9.00000	−0.354375
$646$	0	0
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	25.0000	0.978326	0.489163	−	0.872192i	$-0.337302\pi$
$653$	25.0000	0.978326	0.489163	+	0.872192i	$0.337302\pi$
$654$	0	0
$655$	−14.0000	−0.547025
$656$	0	0
$657$	−16.0000	−0.624219
$658$	0	0
$659$	31.0000	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$660$	0	0
$661$	−44.0000	−1.71140	−0.855701	−	0.517471i	$-0.826874\pi$
$661$	−44.0000	−1.71140	−0.855701	+	0.517471i	$0.826874\pi$
$662$	0	0
$663$	18.0000	0.699062
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	7.00000	0.271041
$668$	0	0
$669$	−15.0000	−0.579934
$670$	0	0
$671$	5.00000	0.193023
$672$	0	0
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−51.0000	−1.96009	−0.980045	−	0.198778i	$-0.936303\pi$
$677$	−51.0000	−1.96009	−0.980045	+	0.198778i	$0.936303\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	0	0
$681$	13.0000	0.498161
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	−16.0000	−0.610438
$688$	0	0
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	9.00000	0.338002	0.169001	−	0.985616i	$-0.445946\pi$
$709$	9.00000	0.338002	0.169001	+	0.985616i	$0.445946\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	9.00000	0.336111
$718$	0	0
$719$	5.00000	0.186469	0.0932343	−	0.995644i	$-0.470279\pi$
$719$	5.00000	0.186469	0.0932343	+	0.995644i	$0.470279\pi$
$720$	0	0
$721$	−7.00000	−0.260694
$722$	0	0
$723$	−22.0000	−0.818189
$724$	0	0
$725$	−7.00000	−0.259973
$726$	0	0
$727$	17.0000	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	27.0000	0.998631
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	0	0
$741$	−18.0000	−0.661247
$742$	0	0
$743$	34.0000	1.24734	0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000	1.24734	0.623670	+	0.781688i	$0.285641\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	−3.00000	−0.109764
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	10.0000	0.363937
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	−30.0000	−1.08324
$768$	0	0
$769$	−51.0000	−1.83911	−0.919554	−	0.392965i	$-0.871449\pi$
$769$	−51.0000	−1.83911	−0.919554	+	0.392965i	$0.871449\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−7.00000	−0.250160
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	13.0000	0.462227
$792$	0	0
$793$	−30.0000	−1.06533
$794$	0	0
$795$	3.00000	0.106399
$796$	0	0
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−9.00000	−0.317999
$802$	0	0
$803$	−16.0000	−0.564628
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	−5.00000	−0.176008
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	−27.0000	−0.944610
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	47.0000	1.64031	0.820156	−	0.572140i	$-0.193887\pi$
$821$	47.0000	1.64031	0.820156	+	0.572140i	$0.193887\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−19.0000	−0.655953	−0.327976	−	0.944686i	$-0.606367\pi$
$839$	−19.0000	−0.655953	−0.327976	+	0.944686i	$0.606367\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	−14.0000	−0.482186
$844$	0	0
$845$	23.0000	0.791224
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−26.0000	−0.892318
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	3.00000	0.102598
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	−41.0000	−1.39566	−0.697828	−	0.716265i	$-0.745850\pi$
$863$	−41.0000	−1.39566	−0.697828	+	0.716265i	$0.745850\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	−7.00000	−0.236914
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	0	0
$879$	5.00000	0.168646
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	5.00000	0.168073
$886$	0	0
$887$	29.0000	0.973725	0.486862	−	0.873479i	$-0.338141\pi$
$887$	29.0000	0.973725	0.486862	+	0.873479i	$0.338141\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	−28.0000	−0.933852
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	9.00000	0.299501
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−6.00000	−0.199227	−0.0996134	−	0.995026i	$-0.531761\pi$
$907$	−6.00000	−0.199227	−0.0996134	+	0.995026i	$0.531761\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	52.0000	1.72284	0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000	1.72284	0.861418	+	0.507896i	$0.169577\pi$
$912$	0	0
$913$	−3.00000	−0.0992855
$914$	0	0
$915$	5.00000	0.165295
$916$	0	0
$917$	14.0000	0.462321
$918$	0	0
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	0	0
$921$	34.0000	1.12034
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	7.00000	0.229910
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	−24.0000	−0.784046	−0.392023	−	0.919955i	$-0.628225\pi$
$937$	−24.0000	−0.784046	−0.392023	+	0.919955i	$0.628225\pi$
$938$	0	0
$939$	11.0000	0.358971
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	4.00000	0.130258
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	96.0000	3.11629
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	−7.00000	−0.226278
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	−9.00000	−0.289420	−0.144710	−	0.989474i	$-0.546225\pi$
$967$	−9.00000	−0.289420	−0.144710	+	0.989474i	$0.546225\pi$
$968$	0	0
$969$	−9.00000	−0.289122
$970$	0	0
$971$	53.0000	1.70085	0.850425	−	0.526096i	$-0.176345\pi$
$971$	53.0000	1.70085	0.850425	+	0.526096i	$0.176345\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	51.0000	1.63163	0.815817	−	0.578310i	$-0.196287\pi$
$977$	51.0000	1.63163	0.815817	+	0.578310i	$0.196287\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.00000	0.286183
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	0	0
$993$	−29.0000	−0.920287
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.l.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

Related objects

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