LMFDB - Embedded newform 6240.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
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	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} -5.00000 q^{17} -2.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} +1.00000 q^{37} -1.00000 q^{39} +9.00000 q^{41} +2.00000 q^{43} +1.00000 q^{45} +8.00000 q^{47} -6.00000 q^{49} -5.00000 q^{51} -1.00000 q^{53} -3.00000 q^{55} -2.00000 q^{57} +4.00000 q^{59} -3.00000 q^{61} +1.00000 q^{63} -1.00000 q^{65} -16.0000 q^{67} -3.00000 q^{69} -15.0000 q^{71} -14.0000 q^{73} +1.00000 q^{75} -3.00000 q^{77} -15.0000 q^{79} +1.00000 q^{81} -8.00000 q^{83} -5.00000 q^{85} -9.00000 q^{89} -1.00000 q^{91} -4.00000 q^{93} -2.00000 q^{95} -5.00000 q^{97} -3.00000 q^{99} +O(q^{100})$

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	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	0	0
$87$	0	0
$88$	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
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	5.00000	0.483368	0.241684	−	0.970355i	$-0.422300\pi$
$107$	5.00000	0.483368	0.241684	+	0.970355i	$0.422300\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−5.00000	−0.458349
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	9.00000	0.811503
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	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
$139$	−1.00000	−0.0848189	−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000	−0.0848189	−0.0424094	+	0.999100i	$0.513503\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	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
$169$	1.00000	0.0769231
$170$	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
$177$	4.00000	0.300658
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	0	0
$183$	−3.00000	−0.221766
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	15.0000	1.09691
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.00000	0.628587
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	−15.0000	−1.02778
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	5.00000	0.336336
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−15.0000	−0.974355
$238$	0	0
$239$	−7.00000	−0.452792	−0.226396	−	0.974035i	$-0.572694\pi$
$239$	−7.00000	−0.452792	−0.226396	+	0.974035i	$0.572694\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	−5.00000	−0.313112
$256$	0	0
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	0	0
$267$	−9.00000	−0.550791
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−5.00000	−0.293105
$292$	0	0
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	10.0000	0.556415
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	−15.0000	−0.796117
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	9.00000	0.468521
$370$	0	0
$371$	−1.00000	−0.0519174
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	15.0000	0.758583
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	−15.0000	−0.754732
$396$	0	0
$397$	33.0000	1.65622	0.828111	−	0.560564i	$-0.189416\pi$
$397$	33.0000	1.65622	0.828111	+	0.560564i	$0.189416\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−1.00000	−0.0489702
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−3.00000	−0.145180
$428$	0	0
$429$	3.00000	0.144841
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	−17.0000	−0.811366	−0.405683	−	0.914014i	$-0.632966\pi$
$439$	−17.0000	−0.811366	−0.405683	+	0.914014i	$0.632966\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	9.00000	0.427603	0.213801	−	0.976877i	$-0.431415\pi$
$443$	9.00000	0.427603	0.213801	+	0.976877i	$0.431415\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	0	0
$447$	−5.00000	−0.236492
$448$	0	0
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	9.00000	0.418265	0.209133	−	0.977887i	$-0.432936\pi$
$463$	9.00000	0.418265	0.209133	+	0.977887i	$0.432936\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	9.00000	0.416470	0.208235	−	0.978079i	$-0.433228\pi$
$467$	9.00000	0.416470	0.208235	+	0.978079i	$0.433228\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−5.00000	−0.227038
$486$	0	0
$487$	3.00000	0.135943	0.0679715	−	0.997687i	$-0.478347\pi$
$487$	3.00000	0.135943	0.0679715	+	0.997687i	$0.478347\pi$
$488$	0	0
$489$	5.00000	0.226108
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3.00000	−0.134840
$496$	0	0
$497$	−15.0000	−0.672842
$498$	0	0
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	35.0000	1.55135	0.775674	−	0.631134i	$-0.217410\pi$
$509$	35.0000	1.55135	0.775674	+	0.631134i	$0.217410\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	20.0000	0.871214
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−9.00000	−0.389833
$534$	0	0
$535$	5.00000	0.216169
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	0	0
$543$	1.00000	0.0429141
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	−3.00000	−0.128037
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−15.0000	−0.637865
$554$	0	0
$555$	1.00000	0.0424476
$556$	0	0
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	−15.0000	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$571$	−15.0000	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−5.00000	−0.208153	−0.104076	−	0.994569i	$-0.533189\pi$
$577$	−5.00000	−0.208153	−0.104076	+	0.994569i	$0.533189\pi$
$578$	0	0
$579$	17.0000	0.706496
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	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
$589$	8.00000	0.329634
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	8.00000	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$593$	8.00000	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$594$	0	0
$595$	−5.00000	−0.204980
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−45.0000	−1.83559	−0.917794	−	0.397057i	$-0.870032\pi$
$601$	−45.0000	−1.83559	−0.917794	+	0.397057i	$0.870032\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	43.0000	1.73675	0.868377	−	0.495905i	$-0.165164\pi$
$613$	43.0000	1.73675	0.868377	+	0.495905i	$0.165164\pi$
$614$	0	0
$615$	9.00000	0.362915
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−9.00000	−0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	−5.00000	−0.199363
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	49.0000	1.93237	0.966186	−	0.257847i	$-0.0830131\pi$
$643$	49.0000	1.93237	0.966186	+	0.257847i	$0.0830131\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	5.00000	0.194184
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	9.00000	0.347441
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	0	0
$679$	−5.00000	−0.191882
$680$	0	0
$681$	2.00000	0.0766402
$682$	0	0
$683$	−10.0000	−0.382639	−0.191320	−	0.981528i	$-0.561277\pi$
$683$	−10.0000	−0.382639	−0.191320	+	0.981528i	$0.561277\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	1.00000	0.0380970
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	−1.00000	−0.0379322
$696$	0	0
$697$	−45.0000	−1.70450
$698$	0	0
$699$	21.0000	0.794293
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	36.0000	1.35201	0.676004	−	0.736898i	$-0.263710\pi$
$709$	36.0000	1.35201	0.676004	+	0.736898i	$0.263710\pi$
$710$	0	0
$711$	−15.0000	−0.562544
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	−7.00000	−0.261420
$718$	0	0
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	20.0000	0.743808
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0000	−0.369863
$732$	0	0
$733$	−15.0000	−0.554038	−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000	−0.554038	−0.277019	+	0.960864i	$0.589346\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	0	0
$745$	−5.00000	−0.183186
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	5.00000	0.182696
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	−5.00000	−0.180775
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	45.0000	1.61023
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	3.00000	0.106533
$794$	0	0
$795$	−1.00000	−0.0354663
$796$	0	0
$797$	21.0000	0.743858	0.371929	−	0.928261i	$-0.378696\pi$
$797$	21.0000	0.743858	0.371929	+	0.928261i	$0.378696\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	−9.00000	−0.317999
$802$	0	0
$803$	42.0000	1.48215
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	0	0
$813$	−14.0000	−0.491001
$814$	0	0
$815$	5.00000	0.175142
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−19.0000	−0.663105	−0.331552	−	0.943437i	$-0.607572\pi$
$821$	−19.0000	−0.663105	−0.331552	+	0.943437i	$0.607572\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	37.0000	1.27738	0.638691	−	0.769463i	$-0.279476\pi$
$839$	37.0000	1.27738	0.638691	+	0.769463i	$0.279476\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	39.0000	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$853$	39.0000	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	41.0000	1.40053	0.700267	−	0.713881i	$-0.253064\pi$
$857$	41.0000	1.40053	0.700267	+	0.713881i	$0.253064\pi$
$858$	0	0
$859$	51.0000	1.74010	0.870049	−	0.492966i	$-0.164087\pi$
$859$	51.0000	1.74010	0.870049	+	0.492966i	$0.164087\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	45.0000	1.52652
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	−5.00000	−0.169224
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	8.00000	0.269527	0.134763	−	0.990878i	$-0.456973\pi$
$881$	8.00000	0.269527	0.134763	+	0.990878i	$0.456973\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	39.0000	1.30949	0.654746	−	0.755849i	$-0.272776\pi$
$887$	39.0000	1.30949	0.654746	+	0.755849i	$0.272776\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	0	0
$900$	0	0
$901$	5.00000	0.166574
$902$	0	0
$903$	2.00000	0.0665558
$904$	0	0
$905$	1.00000	0.0332411
$906$	0	0
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	−3.00000	−0.0991769
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	21.0000	0.691974
$922$	0	0
$923$	15.0000	0.493731
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	35.0000	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$929$	35.0000	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	−14.0000	−0.458339
$934$	0	0
$935$	15.0000	0.490552
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	31.0000	1.01057	0.505286	−	0.862952i	$-0.331387\pi$
$941$	31.0000	1.01057	0.505286	+	0.862952i	$0.331387\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	5.00000	0.161123
$964$	0	0
$965$	17.0000	0.547249
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	10.0000	0.321246
$970$	0	0
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	0	0
$973$	−1.00000	−0.0320585
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	27.0000	0.862924
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−37.0000	−1.17534	−0.587672	−	0.809099i	$-0.699955\pi$
$991$	−37.0000	−1.17534	−0.587672	+	0.809099i	$0.699955\pi$
$992$	0	0
$993$	−18.0000	−0.571213
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bd.1.1	yes	1
4.3	odd	2		6240.2.a.j.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.j.1.1	✓	1	4.3	odd	2
6240.2.a.bd.1.1	yes	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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