LMFDB - Embedded newform 6975.2.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,2,0,0,4,0,0,0,1,0,6,8,0,-4,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2325)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +4.00000 q^{7} +1.00000 q^{11} +6.00000 q^{13} +8.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} -5.00000 q^{19} +2.00000 q^{22} +5.00000 q^{23} +12.0000 q^{26} +8.00000 q^{28} +9.00000 q^{29} +1.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} -10.0000 q^{37} -10.0000 q^{38} +6.00000 q^{41} -6.00000 q^{43} +2.00000 q^{44} +10.0000 q^{46} -10.0000 q^{47} +9.00000 q^{49} +12.0000 q^{52} +9.00000 q^{53} +18.0000 q^{58} +14.0000 q^{59} +10.0000 q^{61} +2.00000 q^{62} -8.00000 q^{64} -13.0000 q^{67} +2.00000 q^{68} -6.00000 q^{71} +14.0000 q^{73} -20.0000 q^{74} -10.0000 q^{76} +4.00000 q^{77} -12.0000 q^{79} +12.0000 q^{82} +5.00000 q^{83} -12.0000 q^{86} +11.0000 q^{89} +24.0000 q^{91} +10.0000 q^{92} -20.0000 q^{94} +3.00000 q^{97} +18.0000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	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
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	8.00000	2.13809
$15$	0	0
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	0	0
$26$	12.0000	2.35339
$27$	0	0
$28$	8.00000	1.51186
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−8.00000	−1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−10.0000	−1.62221
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	10.0000	1.47442
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	12.0000	1.66410
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	18.0000	2.36352
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	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	$0.194369\pi$
$74$	−20.0000	−2.32495
$75$	0	0
$76$	−10.0000	−1.14708
$77$	4.00000	0.455842
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	12.0000	1.32518
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	10.0000	1.04257
$93$	0	0
$94$	−20.0000	−2.06284
$95$	0	0
$96$	0	0
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	18.0000	1.81827
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	0	0
$106$	18.0000	1.74831
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	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
$111$	0	0
$112$	−16.0000	−1.51186
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	18.0000	1.67126
$117$	0	0
$118$	28.0000	2.57761
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	20.0000	1.81071
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−20.0000	−1.73422
$134$	−26.0000	−2.24606
$135$	0	0
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	28.0000	2.31730
$147$	0	0
$148$	−20.0000	−1.64399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	8.00000	0.644658
$155$	0	0
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−24.0000	−1.90934
$159$	0	0
$160$	0	0
$161$	20.0000	1.57622
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	10.0000	0.776151
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	−12.0000	−0.914991
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	22.0000	1.64897
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	48.0000	3.55800
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.00000	0.0731272
$188$	−20.0000	−1.45865
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	21.0000	1.51161	0.755807	−	0.654795i	$-0.227245\pi$
$193$	21.0000	1.51161	0.755807	+	0.654795i	$0.227245\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	18.0000	1.28571
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	−28.0000	−1.97007
$203$	36.0000	2.52670
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−24.0000	−1.66410
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	18.0000	1.23625
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	−4.00000	−0.270914
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−32.0000	−2.13809
$225$	0	0
$226$	−12.0000	−0.798228
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	28.0000	1.82264
$237$	0	0
$238$	8.00000	0.518563
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	−20.0000	−1.28565
$243$	0	0
$244$	20.0000	1.28037
$245$	0	0
$246$	0	0
$247$	−30.0000	−1.90885
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	−8.00000	−0.501965
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−40.0000	−2.48548
$260$	0	0
$261$	0	0
$262$	−8.00000	−0.494242
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	−40.0000	−2.45256
$267$	0	0
$268$	−26.0000	−1.58820
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	12.0000	0.709575
$287$	24.0000	1.41668
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	28.0000	1.63858
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	12.0000	0.695141
$299$	30.0000	1.73494
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	0	0
$304$	20.0000	1.14708
$305$	0	0
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	8.00000	0.455842
$309$	0	0
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−24.0000	−1.35011
$317$	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
$319$	9.00000	0.503903
$320$	0	0
$321$	0	0
$322$	40.0000	2.22911
$323$	−5.00000	−0.278207
$324$	0	0
$325$	0	0
$326$	22.0000	1.21847
$327$	0	0
$328$	0	0
$329$	−40.0000	−2.20527
$330$	0	0
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	10.0000	0.548821
$333$	0	0
$334$	−32.0000	−1.75096
$335$	0	0
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	46.0000	2.50207
$339$	0	0
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	13.0000	0.695874	0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000	0.695874	0.347937	+	0.937518i	$0.386882\pi$
$350$	0	0
$351$	0	0
$352$	−8.00000	−0.426401
$353$	−19.0000	−1.01127	−0.505634	−	0.862748i	$-0.668741\pi$
$353$	−19.0000	−1.01127	−0.505634	+	0.862748i	$0.668741\pi$
$354$	0	0
$355$	0	0
$356$	22.0000	1.16600
$357$	0	0
$358$	8.00000	0.422813
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−12.0000	−0.630706
$363$	0	0
$364$	48.0000	2.51588
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−20.0000	−1.04257
$369$	0	0
$370$	0	0
$371$	36.0000	1.86903
$372$	0	0
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	0	0
$377$	54.0000	2.78114
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	40.0000	2.04658
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	0	0
$385$	0	0
$386$	42.0000	2.13774
$387$	0	0
$388$	6.00000	0.304604
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	44.0000	2.21669
$395$	0	0
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	0	0
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	−28.0000	−1.39305
$405$	0	0
$406$	72.0000	3.57330
$407$	−10.0000	−0.495682
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	0	0
$412$	−14.0000	−0.689730
$413$	56.0000	2.75558
$414$	0	0
$415$	0	0
$416$	−48.0000	−2.35339
$417$	0	0
$418$	−10.0000	−0.489116
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	−26.0000	−1.26566
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	40.0000	1.93574
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−25.0000	−1.19591
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−32.0000	−1.51186
$449$	−37.0000	−1.74614	−0.873069	−	0.487597i	$-0.837874\pi$
$449$	−37.0000	−1.74614	−0.873069	+	0.487597i	$0.837874\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−12.0000	−0.564433
$453$	0	0
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	−36.0000	−1.67126
$465$	0	0
$466$	48.0000	2.22356
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	−52.0000	−2.40114
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	8.00000	0.366679
$477$	0	0
$478$	10.0000	0.457389
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	−8.00000	−0.364390
$483$	0	0
$484$	−20.0000	−0.909091
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	−60.0000	−2.69953
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	−10.0000	−0.446322
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	0	0
$506$	10.0000	0.444554
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	32.0000	1.41421
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10.0000	−0.439799
$518$	−80.0000	−3.51500
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	−24.0000	−1.04645
$527$	1.00000	0.0435607
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	−40.0000	−1.73422
$533$	36.0000	1.55933
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−36.0000	−1.55207
$539$	9.00000	0.387657
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	−8.00000	−0.342997
$545$	0	0
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	−45.0000	−1.91706
$552$	0	0
$553$	−48.0000	−2.04117
$554$	−32.0000	−1.35955
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−37.0000	−1.56774	−0.783870	−	0.620925i	$-0.786757\pi$
$557$	−37.0000	−1.56774	−0.783870	+	0.620925i	$0.786757\pi$
$558$	0	0
$559$	−36.0000	−1.52264
$560$	0	0
$561$	0	0
$562$	−60.0000	−2.53095
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	−10.0000	−0.420331
$567$	0	0
$568$	0	0
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	48.0000	2.00348
$575$	0	0
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	0	0
$581$	20.0000	0.829740
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	0	0
$586$	−28.0000	−1.15667
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	−5.00000	−0.206021
$590$	0	0
$591$	0	0
$592$	40.0000	1.64399
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	0	0
$598$	60.0000	2.45358
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−48.0000	−1.95633
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	0	0
$611$	−60.0000	−2.42734
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	56.0000	2.25998
$615$	0	0
$616$	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
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	60.0000	2.40578
$623$	44.0000	1.76282
$624$	0	0
$625$	0	0
$626$	−16.0000	−0.639489
$627$	0	0
$628$	22.0000	0.877896
$629$	−10.0000	−0.398726
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	0	0
$634$	12.0000	0.476581
$635$	0	0
$636$	0	0
$637$	54.0000	2.13956
$638$	18.0000	0.712627
$639$	0	0
$640$	0	0
$641$	−31.0000	−1.22443	−0.612213	−	0.790693i	$-0.709721\pi$
$641$	−31.0000	−1.22443	−0.612213	+	0.790693i	$0.709721\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	40.0000	1.57622
$645$	0	0
$646$	−10.0000	−0.393445
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	0	0
$652$	22.0000	0.861586
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	0	0
$655$	0	0
$656$	−24.0000	−0.937043
$657$	0	0
$658$	−80.0000	−3.11872
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	−68.0000	−2.64290
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.0000	1.74241
$668$	−32.0000	−1.23812
$669$	0	0
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	68.0000	2.61926
$675$	0	0
$676$	46.0000	1.76923
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	0	0
$682$	2.00000	0.0765840
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	0	0
$685$	0	0
$686$	16.0000	0.610883
$687$	0	0
$688$	24.0000	0.914991
$689$	54.0000	2.05724
$690$	0	0
$691$	−45.0000	−1.71188	−0.855940	−	0.517075i	$-0.827021\pi$
$691$	−45.0000	−1.71188	−0.855940	+	0.517075i	$0.827021\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	6.00000	0.227266
$698$	26.0000	0.984115
$699$	0	0
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	50.0000	1.88579
$704$	−8.00000	−0.301511
$705$	0	0
$706$	−38.0000	−1.43015
$707$	−56.0000	−2.10610
$708$	0	0
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.00000	0.187251
$714$	0	0
$715$	0	0
$716$	8.00000	0.298974
$717$	0	0
$718$	−52.0000	−1.94062
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	12.0000	0.446594
$723$	0	0
$724$	−12.0000	−0.445976
$725$	0	0
$726$	0	0
$727$	−36.0000	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.00000	−0.221918
$732$	0	0
$733$	−17.0000	−0.627909	−0.313955	−	0.949438i	$-0.601654\pi$
$733$	−17.0000	−0.627909	−0.313955	+	0.949438i	$0.601654\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−40.0000	−1.47442
$737$	−13.0000	−0.478861
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	72.0000	2.64320
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	2.00000	0.0731272
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−35.0000	−1.27717	−0.638584	−	0.769552i	$-0.720480\pi$
$751$	−35.0000	−1.27717	−0.638584	+	0.769552i	$0.720480\pi$
$752$	40.0000	1.45865
$753$	0	0
$754$	108.000	3.93313
$755$	0	0
$756$	0	0
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	−40.0000	−1.45287
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	40.0000	1.44715
$765$	0	0
$766$	−18.0000	−0.650366
$767$	84.0000	3.03306
$768$	0	0
$769$	37.0000	1.33425	0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000	1.33425	0.667127	+	0.744944i	$0.267524\pi$
$770$	0	0
$771$	0	0
$772$	42.0000	1.51161
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	52.0000	1.86429
$779$	−30.0000	−1.07486
$780$	0	0
$781$	−6.00000	−0.214697
$782$	10.0000	0.357599
$783$	0	0
$784$	−36.0000	−1.28571
$785$	0	0
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	44.0000	1.56744
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	60.0000	2.13066
$794$	−14.0000	−0.496841
$795$	0	0
$796$	20.0000	0.708881
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	0	0
$801$	0	0
$802$	46.0000	1.62432
$803$	14.0000	0.494049
$804$	0	0
$805$	0	0
$806$	12.0000	0.422682
$807$	0	0
$808$	0	0
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	72.0000	2.52670
$813$	0	0
$814$	−20.0000	−0.701000
$815$	0	0
$816$	0	0
$817$	30.0000	1.04957
$818$	24.0000	0.839140
$819$	0	0
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	112.000	3.89698
$827$	−33.0000	−1.14752	−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000	−1.14752	−0.573761	+	0.819023i	$0.694516\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	0	0
$832$	−48.0000	−1.66410
$833$	9.00000	0.311832
$834$	0	0
$835$	0	0
$836$	−10.0000	−0.345857
$837$	0	0
$838$	−40.0000	−1.38178
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	36.0000	1.24064
$843$	0	0
$844$	−26.0000	−0.894957
$845$	0	0
$846$	0	0
$847$	−40.0000	−1.37442
$848$	−36.0000	−1.23625
$849$	0	0
$850$	0	0
$851$	−50.0000	−1.71398
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	80.0000	2.73754
$855$	0	0
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	0	0
$862$	32.0000	1.08992
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	0	0
$865$	0	0
$866$	−36.0000	−1.22333
$867$	0	0
$868$	8.00000	0.271538
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−78.0000	−2.64293
$872$	0	0
$873$	0	0
$874$	−50.0000	−1.69128
$875$	0	0
$876$	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
$879$	0	0
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−8.00000	−0.268765
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	−4.00000	−0.133930
$893$	50.0000	1.67319
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−74.0000	−2.46941
$899$	9.00000	0.300167
$900$	0	0
$901$	9.00000	0.299833
$902$	12.0000	0.399556
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−57.0000	−1.89265	−0.946327	−	0.323211i	$-0.895238\pi$
$907$	−57.0000	−1.89265	−0.946327	+	0.323211i	$0.895238\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	0	0
$911$	41.0000	1.35839	0.679195	−	0.733958i	$-0.262329\pi$
$911$	41.0000	1.35839	0.679195	+	0.733958i	$0.262329\pi$
$912$	0	0
$913$	5.00000	0.165476
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−12.0000	−0.396491
$917$	−16.0000	−0.528367
$918$	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
$921$	0	0
$922$	−6.00000	−0.197599
$923$	−36.0000	−1.18495
$924$	0	0
$925$	0	0
$926$	−68.0000	−2.23462
$927$	0	0
$928$	−72.0000	−2.36352
$929$	5.00000	0.164045	0.0820223	−	0.996630i	$-0.473862\pi$
$929$	5.00000	0.164045	0.0820223	+	0.996630i	$0.473862\pi$
$930$	0	0
$931$	−45.0000	−1.47482
$932$	48.0000	1.57229
$933$	0	0
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	9.00000	0.294017	0.147009	−	0.989135i	$-0.453036\pi$
$937$	9.00000	0.294017	0.147009	+	0.989135i	$0.453036\pi$
$938$	−104.000	−3.39572
$939$	0	0
$940$	0	0
$941$	51.0000	1.66255	0.831276	−	0.555860i	$-0.187611\pi$
$941$	51.0000	1.66255	0.831276	+	0.555860i	$0.187611\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	−56.0000	−1.82264
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−9.00000	−0.292461	−0.146230	−	0.989251i	$-0.546714\pi$
$947$	−9.00000	−0.292461	−0.146230	+	0.989251i	$0.546714\pi$
$948$	0	0
$949$	84.0000	2.72676
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	0	0
$956$	10.0000	0.323423
$957$	0	0
$958$	20.0000	0.646171
$959$	36.0000	1.16250
$960$	0	0
$961$	1.00000	0.0322581
$962$	−120.000	−3.86896
$963$	0	0
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	4.00000	0.128168
$975$	0	0
$976$	−40.0000	−1.28037
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	0	0
$979$	11.0000	0.351562
$980$	0	0
$981$	0	0
$982$	6.00000	0.191468
$983$	−7.00000	−0.223265	−0.111633	−	0.993750i	$-0.535608\pi$
$983$	−7.00000	−0.223265	−0.111633	+	0.993750i	$0.535608\pi$
$984$	0	0
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	−60.0000	−1.90885
$989$	−30.0000	−0.953945
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	−48.0000	−1.52247
$995$	0	0
$996$	0	0
$997$	−57.0000	−1.80521	−0.902604	−	0.430472i	$-0.858347\pi$
$997$	−57.0000	−1.80521	−0.902604	+	0.430472i	$0.858347\pi$
$998$	−40.0000	−1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.s.1.1		1
3.2	odd	2		2325.2.a.a.1.1	✓	1
5.4	even	2		6975.2.a.a.1.1		1
15.2	even	4		2325.2.c.b.1024.1		2
15.8	even	4		2325.2.c.b.1024.2		2
15.14	odd	2		2325.2.a.l.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.a.1.1	✓	1	3.2	odd	2
2325.2.a.l.1.1	yes	1	15.14	odd	2
2325.2.c.b.1024.1		2	15.2	even	4
2325.2.c.b.1024.2		2	15.8	even	4
6975.2.a.a.1.1		1	5.4	even	2
6975.2.a.s.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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