LMFDB - Embedded newform 2760.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} -4.00000 q^{13} +1.00000 q^{15} +2.00000 q^{17} -4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.00000 q^{29} +8.00000 q^{31} +6.00000 q^{33} -4.00000 q^{35} -6.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} +6.00000 q^{43} -1.00000 q^{45} -12.0000 q^{47} +9.00000 q^{49} -2.00000 q^{51} +6.00000 q^{53} +6.00000 q^{55} +4.00000 q^{59} +2.00000 q^{61} +4.00000 q^{63} +4.00000 q^{65} +6.00000 q^{67} -1.00000 q^{69} -6.00000 q^{71} +14.0000 q^{73} -1.00000 q^{75} -24.0000 q^{77} +1.00000 q^{81} -2.00000 q^{85} -8.00000 q^{87} +12.0000 q^{89} -16.0000 q^{91} -8.00000 q^{93} +12.0000 q^{97} -6.00000 q^{99} +O(q^{100})$

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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−4.00000	−0.676123
$36$	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
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−24.0000	−2.73505
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	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
$105$	4.00000	0.390360
$106$	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
$109$	−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
$111$	6.00000	0.569495
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	24.0000	2.00698
$144$	0	0
$145$	−8.00000	−0.664364
$146$	0	0
$147$	−9.00000	−0.742307
$148$	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
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	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
$175$	4.00000	0.302372
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	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
$195$	−4.00000	−0.286446
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	32.0000	2.24596
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	32.0000	2.17230
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.00000	−0.574989
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	16.0000	0.968364
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	6.00000	0.348155
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−48.0000	−2.68748
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	−48.0000	−2.64633
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−48.0000	−2.59935
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−25.0000	−1.31216
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	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
$377$	−32.0000	−1.64808
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	36.0000	1.78445
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	0	0
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	16.0000	0.750092
$456$	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
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−36.0000	−1.65528
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	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
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	72.0000	3.16656
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−44.0000	−1.92767	−0.963837	−	0.266491i	$-0.914136\pi$
$521$	−44.0000	−1.92767	−0.963837	+	0.266491i	$0.914136\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	−54.0000	−2.32594
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.00000	−0.254686
$556$	0	0
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	0	0
$597$	16.0000	0.654836
$598$	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
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	−25.0000	−1.01639
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	−32.0000	−1.29671
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	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
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	48.0000	1.92308
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−18.0000	−0.714308
$636$	0	0
$637$	−36.0000	−1.42637
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	10.0000	0.394362	0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000	0.394362	0.197181	+	0.980367i	$0.436821\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	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
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−32.0000	−1.25418
$652$	0	0
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	48.0000	1.84207
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−2.00000	−0.0746914
$718$	0	0
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	10.0000	0.364420
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	0	0
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−72.0000	−2.56003
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−84.0000	−2.96430
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−12.0000	−0.422420
$808$	0	0
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−16.0000	−0.559085
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	18.0000	0.627441	0.313720	−	0.949515i	$-0.398425\pi$
$823$	18.0000	0.627441	0.313720	+	0.949515i	$0.398425\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	−20.0000	−0.688837
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	100.000	3.43604
$848$	0	0
$849$	−18.0000	−0.617758
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−12.0000	−0.410872	−0.205436	−	0.978671i	$-0.565861\pi$
$853$	−12.0000	−0.410872	−0.205436	+	0.978671i	$0.565861\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	−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
$861$	8.00000	0.272639
$862$	0	0
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	−4.00000	−0.135225
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	72.0000	2.41480
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	0	0
$894$	0	0
$895$	20.0000	0.668526
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	64.0000	2.13452
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	6.00000	0.196431
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	−56.0000	−1.81784
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	48.0000	1.55162
$958$	0	0
$959$	88.0000	2.84167
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−62.0000	−1.98356	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000	−1.98356	−0.991778	+	0.127971i	$0.959153\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	48.0000	1.52786
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.a.c.1.1	✓	1
3.2	odd	2		8280.2.a.w.1.1		1
4.3	odd	2		5520.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.c.1.1	✓	1	1.1	even	1	trivial
5520.2.a.s.1.1		1	4.3	odd	2
8280.2.a.w.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:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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