LMFDB - Embedded newform 2415.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,2,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2415.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:	$19.2838720881$
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	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{15} +4.00000 q^{16} -2.00000 q^{17} +7.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} +3.00000 q^{33} -1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +3.00000 q^{41} +10.0000 q^{43} +6.00000 q^{44} -1.00000 q^{45} -1.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +2.00000 q^{51} -3.00000 q^{53} +3.00000 q^{55} -7.00000 q^{57} -5.00000 q^{59} -2.00000 q^{60} +3.00000 q^{61} +1.00000 q^{63} -8.00000 q^{64} -14.0000 q^{67} +4.00000 q^{68} -1.00000 q^{69} +2.00000 q^{73} -1.00000 q^{75} -14.0000 q^{76} -3.00000 q^{77} -6.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -8.00000 q^{83} +2.00000 q^{84} +2.00000 q^{85} +2.00000 q^{87} -12.0000 q^{89} -2.00000 q^{92} +4.00000 q^{93} -7.00000 q^{95} -6.00000 q^{97} -3.00000 q^{99} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−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
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	2.00000	0.447214
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	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
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	−2.00000	−0.258199
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	−14.0000	−1.60591
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−4.00000	−0.447214
$81$	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	$0.644687\pi$
$84$	2.00000	0.218218
$85$	2.00000	0.216930
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	4.00000	0.414781
$94$	0	0
$95$	−7.00000	−0.718185
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	−2.00000	−0.200000
$101$	11.0000	1.09454	0.547270	−	0.836956i	$-0.315667\pi$
$101$	11.0000	1.09454	0.547270	+	0.836956i	$0.315667\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	2.00000	0.192450
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	4.00000	0.371391
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−3.00000	−0.270501
$124$	8.00000	0.718421
$125$	−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
$129$	−10.0000	−0.880451
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	−6.00000	−0.522233
$133$	7.00000	0.606977
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.00000	−0.598050	−0.299025	−	0.954245i	$-0.596661\pi$
$137$	−7.00000	−0.598050	−0.299025	+	0.954245i	$0.596661\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	2.00000	0.169031
$141$	1.00000	0.0842152
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	2.00000	0.166091
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	−4.00000	−0.328798
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	7.00000	0.548282	0.274141	−	0.961689i	$-0.411606\pi$
$163$	7.00000	0.548282	0.274141	+	0.961689i	$0.411606\pi$
$164$	−6.00000	−0.468521
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−13.0000	−1.00597	−0.502985	−	0.864295i	$-0.667765\pi$
$167$	−13.0000	−1.00597	−0.502985	+	0.864295i	$0.667765\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	7.00000	0.535303
$172$	−20.0000	−1.52499
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−12.0000	−0.904534
$177$	5.00000	0.375823
$178$	0	0
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−3.00000	−0.221766
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	6.00000	0.438763
$188$	2.00000	0.145865
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	8.00000	0.577350
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	−20.0000	−1.42494	−0.712470	−	0.701702i	$-0.752424\pi$
$197$	−20.0000	−1.42494	−0.712470	+	0.701702i	$0.752424\pi$
$198$	0	0
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	0	0
$203$	−2.00000	−0.140372
$204$	−4.00000	−0.280056
$205$	−3.00000	−0.209529
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−2.00000	−0.135147
$220$	−6.00000	−0.404520
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	14.0000	0.927173
$229$	11.0000	0.726900	0.363450	−	0.931614i	$-0.381599\pi$
$229$	11.0000	0.726900	0.363450	+	0.931614i	$0.381599\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	10.0000	0.650945
$237$	6.00000	0.389742
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	4.00000	0.258199
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	−2.00000	−0.125988
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−2.00000	−0.125245
$256$	16.0000	1.00000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	12.0000	0.734388
$268$	28.0000	1.71037
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	2.00000	0.120386
$277$	−15.0000	−0.901263	−0.450631	−	0.892710i	$-0.648801\pi$
$277$	−15.0000	−0.901263	−0.450631	+	0.892710i	$0.648801\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	0	0
$285$	7.00000	0.414644
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.00000	0.351726
$292$	−4.00000	−0.234082
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	5.00000	0.291111
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	0	0
$300$	2.00000	0.115470
$301$	10.0000	0.576390
$302$	0	0
$303$	−11.0000	−0.631933
$304$	28.0000	1.60591
$305$	−3.00000	−0.171780
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	6.00000	0.341882
$309$	5.00000	0.284440
$310$	0	0
$311$	23.0000	1.30421	0.652105	−	0.758129i	$-0.273886\pi$
$311$	23.0000	1.30421	0.652105	+	0.758129i	$0.273886\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	12.0000	0.675053
$317$	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
$319$	6.00000	0.335936
$320$	8.00000	0.447214
$321$	4.00000	0.223258
$322$	0	0
$323$	−14.0000	−0.778981
$324$	−2.00000	−0.111111
$325$	0	0
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	−1.00000	−0.0551318
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	16.0000	0.878114
$333$	2.00000	0.109599
$334$	0	0
$335$	14.0000	0.764902
$336$	−4.00000	−0.218218
$337$	−36.0000	−1.96104	−0.980522	−	0.196407i	$-0.937073\pi$
$337$	−36.0000	−1.96104	−0.980522	+	0.196407i	$0.937073\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	−4.00000	−0.216930
$341$	12.0000	0.649836
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−4.00000	−0.214423
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	0	0
$356$	24.0000	1.27200
$357$	2.00000	0.105851
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	4.00000	0.208514
$369$	3.00000	0.156174
$370$	0	0
$371$	−3.00000	−0.155752
$372$	−8.00000	−0.414781
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	14.0000	0.718185
$381$	1.00000	0.0512316
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	10.0000	0.508329
$388$	12.0000	0.609208
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	−9.00000	−0.453990
$394$	0	0
$395$	6.00000	0.301893
$396$	6.00000	0.301511
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	0	0
$399$	−7.00000	−0.350438
$400$	4.00000	0.200000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	0	0
$404$	−22.0000	−1.09454
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	7.00000	0.345285
$412$	10.0000	0.492665
$413$	−5.00000	−0.246034
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−2.00000	−0.0975900
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−1.00000	−0.0486217
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	3.00000	0.145180
$428$	8.00000	0.386695
$429$	0	0
$430$	0	0
$431$	−19.0000	−0.915198	−0.457599	−	0.889159i	$-0.651290\pi$
$431$	−19.0000	−0.915198	−0.457599	+	0.889159i	$0.651290\pi$
$432$	−4.00000	−0.192450
$433$	−13.0000	−0.624740	−0.312370	−	0.949960i	$-0.601123\pi$
$433$	−13.0000	−0.624740	−0.312370	+	0.949960i	$0.601123\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	4.00000	0.191565
$437$	7.00000	0.334855
$438$	0	0
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	22.0000	1.04525	0.522626	−	0.852562i	$-0.324953\pi$
$443$	22.0000	1.04525	0.522626	+	0.852562i	$0.324953\pi$
$444$	4.00000	0.189832
$445$	12.0000	0.568855
$446$	0	0
$447$	11.0000	0.520282
$448$	−8.00000	−0.377964
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	−12.0000	−0.564433
$453$	11.0000	0.516825
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	2.00000	0.0932505
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	−8.00000	−0.371391
$465$	−4.00000	−0.185496
$466$	0	0
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	0	0
$471$	11.0000	0.506853
$472$	0	0
$473$	−30.0000	−1.37940
$474$	0	0
$475$	7.00000	0.321182
$476$	4.00000	0.183340
$477$	−3.00000	−0.137361
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	4.00000	0.181818
$485$	6.00000	0.272446
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−7.00000	−0.316551
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000	0.270501
$493$	4.00000	0.180151
$494$	0	0
$495$	3.00000	0.134840
$496$	−16.0000	−0.718421
$497$	0	0
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	2.00000	0.0894427
$501$	13.0000	0.580797
$502$	0	0
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−11.0000	−0.489494
$506$	0	0
$507$	13.0000	0.577350
$508$	2.00000	0.0887357
$509$	19.0000	0.842160	0.421080	−	0.907023i	$-0.361651\pi$
$509$	19.0000	0.842160	0.421080	+	0.907023i	$0.361651\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	5.00000	0.220326
$516$	20.0000	0.880451
$517$	3.00000	0.131940
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	5.00000	0.218635	0.109317	−	0.994007i	$-0.465134\pi$
$523$	5.00000	0.218635	0.109317	+	0.994007i	$0.465134\pi$
$524$	−18.0000	−0.786334
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	8.00000	0.348485
$528$	12.0000	0.522233
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.00000	−0.216982
$532$	−14.0000	−0.606977
$533$	0	0
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	−3.00000	−0.129219
$540$	−2.00000	−0.0860663
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	14.0000	0.598050
$549$	3.00000	0.128037
$550$	0	0
$551$	−14.0000	−0.596420
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	2.00000	0.0848953
$556$	32.0000	1.35710
$557$	−34.0000	−1.44063	−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000	−1.44063	−0.720313	+	0.693649i	$0.756002\pi$
$558$	0	0
$559$	0	0
$560$	−4.00000	−0.169031
$561$	−6.00000	−0.253320
$562$	0	0
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	−2.00000	−0.0842152
$565$	−6.00000	−0.252422
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−7.00000	−0.293455	−0.146728	−	0.989177i	$-0.546874\pi$
$569$	−7.00000	−0.293455	−0.146728	+	0.989177i	$0.546874\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	1.00000	0.0417756
$574$	0	0
$575$	1.00000	0.0417029
$576$	−8.00000	−0.333333
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	0	0
$579$	−11.0000	−0.457144
$580$	−4.00000	−0.166091
$581$	−8.00000	−0.331896
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.0000	1.27951	0.639753	−	0.768580i	$-0.279036\pi$
$587$	31.0000	1.27951	0.639753	+	0.768580i	$0.279036\pi$
$588$	2.00000	0.0824786
$589$	−28.0000	−1.15372
$590$	0	0
$591$	20.0000	0.822690
$592$	8.00000	0.328798
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	22.0000	0.901155
$597$	−19.0000	−0.777618
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	22.0000	0.895167
$605$	2.00000	0.0813116
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	0	0
$612$	4.00000	0.161690
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	0	0
$615$	3.00000	0.120972
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	−8.00000	−0.321288
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.0000	0.838659
$628$	22.0000	0.877896
$629$	−4.00000	−0.159490
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	3.00000	0.119239
$634$	0	0
$635$	1.00000	0.0396838
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	0	0
$643$	11.0000	0.433798	0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$644$	−2.00000	−0.0788110
$645$	10.0000	0.393750
$646$	0	0
$647$	−44.0000	−1.72982	−0.864909	−	0.501928i	$-0.832624\pi$
$647$	−44.0000	−1.72982	−0.864909	+	0.501928i	$0.832624\pi$
$648$	0	0
$649$	15.0000	0.588802
$650$	0	0
$651$	4.00000	0.156772
$652$	−14.0000	−0.548282
$653$	44.0000	1.72185	0.860927	−	0.508729i	$-0.169885\pi$
$653$	44.0000	1.72185	0.860927	+	0.508729i	$0.169885\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	12.0000	0.468521
$657$	2.00000	0.0780274
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	6.00000	0.233550
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.00000	−0.271448
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	26.0000	1.00597
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−9.00000	−0.347441
$672$	0	0
$673$	27.0000	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$673$	27.0000	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	26.0000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−14.0000	−0.535303
$685$	7.00000	0.267456
$686$	0	0
$687$	−11.0000	−0.419676
$688$	40.0000	1.52499
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	28.0000	1.06440
$693$	−3.00000	−0.113961
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	26.0000	0.983410
$700$	−2.00000	−0.0755929
$701$	11.0000	0.415464	0.207732	−	0.978186i	$-0.433392\pi$
$701$	11.0000	0.415464	0.207732	+	0.978186i	$0.433392\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	24.0000	0.904534
$705$	−1.00000	−0.0376622
$706$	0	0
$707$	11.0000	0.413698
$708$	−10.0000	−0.375823
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	16.0000	0.597948
$717$	6.00000	0.224074
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	−4.00000	−0.149071
$721$	−5.00000	−0.186210
$722$	0	0
$723$	−25.0000	−0.929760
$724$	20.0000	0.743294
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.0000	−0.739727
$732$	6.00000	0.221766
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	42.0000	1.54709
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	0	0
$745$	11.0000	0.403009
$746$	0	0
$747$	−8.00000	−0.292705
$748$	−12.0000	−0.438763
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	−4.00000	−0.145865
$753$	−8.00000	−0.291536
$754$	0	0
$755$	11.0000	0.400331
$756$	2.00000	0.0727393
$757$	−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
$759$	3.00000	0.108893
$760$	0	0
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	2.00000	0.0723575
$765$	2.00000	0.0723102
$766$	0	0
$767$	0	0
$768$	−16.0000	−0.577350
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	−22.0000	−0.791797
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	21.0000	0.752403
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	4.00000	0.142857
$785$	11.0000	0.392607
$786$	0	0
$787$	35.0000	1.24762	0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000	1.24762	0.623808	+	0.781578i	$0.285585\pi$
$788$	40.0000	1.42494
$789$	9.00000	0.320408
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−3.00000	−0.106399
$796$	−38.0000	−1.34687
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	−6.00000	−0.211735
$804$	−28.0000	−0.987484
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	−14.0000	−0.492823
$808$	0	0
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	4.00000	0.140372
$813$	−14.0000	−0.491001
$814$	0	0
$815$	−7.00000	−0.245199
$816$	8.00000	0.280056
$817$	70.0000	2.44899
$818$	0	0
$819$	0	0
$820$	6.00000	0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	−1.00000	−0.0347734	−0.0173867	−	0.999849i	$-0.505535\pi$
$827$	−1.00000	−0.0347734	−0.0173867	+	0.999849i	$0.505535\pi$
$828$	−2.00000	−0.0695048
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	15.0000	0.520344
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	13.0000	0.449884
$836$	42.0000	1.45260
$837$	4.00000	0.138260
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	2.00000	0.0688837
$844$	6.00000	0.206529
$845$	13.0000	0.447214
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−12.0000	−0.412082
$849$	8.00000	0.274559
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	0	0
$855$	−7.00000	−0.239395
$856$	0	0
$857$	−15.0000	−0.512390	−0.256195	−	0.966625i	$-0.582469\pi$
$857$	−15.0000	−0.512390	−0.256195	+	0.966625i	$0.582469\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	20.0000	0.681994
$861$	−3.00000	−0.102240
$862$	0	0
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	13.0000	0.441503
$868$	8.00000	0.271538
$869$	18.0000	0.610608
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	4.00000	0.135147
$877$	25.0000	0.844190	0.422095	−	0.906552i	$-0.361295\pi$
$877$	25.0000	0.844190	0.422095	+	0.906552i	$0.361295\pi$
$878$	0	0
$879$	−4.00000	−0.134917
$880$	12.0000	0.404520
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	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
$885$	−5.00000	−0.168073
$886$	0	0
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−32.0000	−1.07144
$893$	−7.00000	−0.234246
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000	0.266815
$900$	−2.00000	−0.0666667
$901$	6.00000	0.199889
$902$	0	0
$903$	−10.0000	−0.332779
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	−24.0000	−0.796468
$909$	11.0000	0.364847
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	−28.0000	−0.927173
$913$	24.0000	0.794284
$914$	0	0
$915$	3.00000	0.0991769
$916$	−22.0000	−0.726900
$917$	9.00000	0.297206
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	0	0
$924$	−6.00000	−0.197386
$925$	2.00000	0.0657596
$926$	0	0
$927$	−5.00000	−0.164222
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	52.0000	1.70332
$933$	−23.0000	−0.752986
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	−2.00000	−0.0652328
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	−20.0000	−0.650945
$945$	1.00000	0.0325300
$946$	0	0
$947$	−10.0000	−0.324956	−0.162478	−	0.986712i	$-0.551949\pi$
$947$	−10.0000	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$948$	−12.0000	−0.389742
$949$	0	0
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	1.00000	0.0323592
$956$	12.0000	0.388108
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−7.00000	−0.226042
$960$	−8.00000	−0.258199
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−4.00000	−0.128898
$964$	−50.0000	−1.61039
$965$	−11.0000	−0.354103
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	14.0000	0.449745
$970$	0	0
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	2.00000	0.0641500
$973$	−16.0000	−0.512936
$974$	0	0
$975$	0	0
$976$	12.0000	0.384111
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	2.00000	0.0638877
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	54.0000	1.72233	0.861166	−	0.508323i	$-0.169735\pi$
$983$	54.0000	1.72233	0.861166	+	0.508323i	$0.169735\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	1.00000	0.0318304
$988$	0	0
$989$	10.0000	0.317982
$990$	0	0
$991$	1.00000	0.0317660	0.0158830	−	0.999874i	$-0.494944\pi$
$991$	1.00000	0.0317660	0.0158830	+	0.999874i	$0.494944\pi$
$992$	0	0
$993$	−17.0000	−0.539479
$994$	0	0
$995$	−19.0000	−0.602340
$996$	−16.0000	−0.506979
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.2.a.e.1.1	✓	1
3.2	odd	2		7245.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.e.1.1	✓	1	1.1	even	1	trivial
7245.2.a.k.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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