LMFDB - Embedded newform 455.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [455,2,Mod(1,455)] 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("455.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(455, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 455 = 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	455.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.63319329197$
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	$\chi$	$=$	455.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} +3.00000 q^{18} -1.00000 q^{20} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} +3.00000 q^{36} -10.0000 q^{37} +3.00000 q^{40} +2.00000 q^{41} -8.00000 q^{43} -3.00000 q^{45} +4.00000 q^{46} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -2.00000 q^{53} -3.00000 q^{56} +2.00000 q^{58} -2.00000 q^{61} +4.00000 q^{62} +3.00000 q^{63} +7.00000 q^{64} +1.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{70} +12.0000 q^{71} -9.00000 q^{72} -6.00000 q^{73} +10.0000 q^{74} +8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} +8.00000 q^{86} +2.00000 q^{89} +3.00000 q^{90} -1.00000 q^{91} +4.00000 q^{92} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.00000	1.06066
$9$	−3.00000	−1.00000
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	3.00000	0.707107
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−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	$0.616954\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	−1.00000	−0.169031
$36$	3.00000	0.500000
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	3.00000	0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	2.00000	0.262613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	3.00000	0.377964
$64$	7.00000	0.875000
$65$	1.00000	0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	1.00000	0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−9.00000	−1.06066
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	−2.00000	−0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	3.00000	0.316228
$91$	−1.00000	−0.104828
$92$	4.00000	0.417029
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	2.00000	0.194257
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	2.00000	0.185695
$117$	−3.00000	−0.277350
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000	0.181071
$123$	0	0
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	−3.00000	−0.267261
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	−18.0000	−1.54349
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	3.00000	0.250000
$145$	−2.00000	−0.166091
$146$	6.00000	0.496564
$147$	0	0
$148$	10.0000	0.821995
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	18.0000	1.45521
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−5.00000	−0.395285
$161$	4.00000	0.315244
$162$	−9.00000	−0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.00000	0.460179
$171$	0	0
$172$	8.00000	0.609994
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	3.00000	0.223607
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	−12.0000	−0.884652
$185$	−10.0000	−0.735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−6.00000	−0.422159
$203$	2.00000	0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	−4.00000	−0.278693
$207$	12.0000	0.834058
$208$	−1.00000	−0.0693375
$209$	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	$0.258301\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−8.00000	−0.546869
$215$	−8.00000	−0.545595
$216$	0	0
$217$	4.00000	0.271538
$218$	10.0000	0.677285
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	5.00000	0.334077
$225$	−3.00000	−0.200000
$226$	−18.0000	−1.19734
$227$	−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	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−6.00000	−0.388922
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	2.00000	0.128037
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	−12.0000	−0.762001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.00000	−0.188982
$253$	0	0
$254$	−12.0000	−0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	−1.00000	−0.0620174
$261$	6.00000	0.371391
$262$	−4.00000	−0.247121
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	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
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	−20.0000	−1.19952
$279$	12.0000	0.718421
$280$	−3.00000	−0.179284
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	15.0000	0.883883
$289$	19.0000	1.11765
$290$	2.00000	0.117444
$291$	0	0
$292$	6.00000	0.351123
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	0	0
$296$	−30.0000	−1.74371
$297$	0	0
$298$	2.00000	0.115857
$299$	−4.00000	−0.231326
$300$	0	0
$301$	8.00000	0.461112
$302$	−4.00000	−0.230174
$303$	0	0
$304$	0	0
$305$	−2.00000	−0.114520
$306$	−18.0000	−1.02899
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	18.0000	1.01580
$315$	3.00000	0.169031
$316$	−8.00000	−0.450035
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	0	0
$320$	7.00000	0.391312
$321$	0	0
$322$	−4.00000	−0.222911
$323$	0	0
$324$	−9.00000	−0.500000
$325$	1.00000	0.0554700
$326$	4.00000	0.221540
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	−4.00000	−0.219529
$333$	30.0000	1.64399
$334$	8.00000	0.437741
$335$	−4.00000	−0.218543
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−24.0000	−1.29399
$345$	0	0
$346$	18.0000	0.967686
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−2.00000	−0.106000
$357$	0	0
$358$	12.0000	0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	−9.00000	−0.474342
$361$	−19.0000	−1.00000
$362$	10.0000	0.525588
$363$	0	0
$364$	1.00000	0.0524142
$365$	−6.00000	−0.314054
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	4.00000	0.208514
$369$	−6.00000	−0.312348
$370$	10.0000	0.519875
$371$	2.00000	0.103835
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	0	0
$382$	−16.0000	−0.818631
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	24.0000	1.21999
$388$	14.0000	0.710742
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	3.00000	0.151523
$393$	0	0
$394$	10.0000	0.503793
$395$	8.00000	0.402524
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−6.00000	−0.298511
$405$	9.00000	0.447214
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−12.0000	−0.589768
$415$	4.00000	0.196352
$416$	−5.00000	−0.245145
$417$	0	0
$418$	0	0
$419$	−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
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−6.00000	−0.291043
$426$	0	0
$427$	2.00000	0.0967868
$428$	−8.00000	−0.386695
$429$	0	0
$430$	8.00000	0.385794
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	6.00000	0.285391
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	16.0000	0.757622
$447$	0	0
$448$	−7.00000	−0.330719
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	3.00000	0.141421
$451$	0	0
$452$	−18.0000	−0.846649
$453$	0	0
$454$	4.00000	0.187729
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	4.00000	0.186501
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	22.0000	1.01913
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	3.00000	0.138675
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	6.00000	0.274721
$478$	4.00000	0.182956
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	22.0000	1.00207
$483$	0	0
$484$	11.0000	0.500000
$485$	−14.0000	−0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	−12.0000	−0.538274
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	9.00000	0.400892
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	−12.0000	−0.532414
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	11.0000	0.486136
$513$	0	0
$514$	6.00000	0.264649
$515$	4.00000	0.176261
$516$	0	0
$517$	0	0
$518$	−10.0000	−0.439375
$519$	0	0
$520$	3.00000	0.131559
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	−6.00000	−0.262613
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	24.0000	1.04546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	2.00000	0.0868744
$531$	0	0
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	8.00000	0.345870
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−14.0000	−0.603583
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	28.0000	1.20270
$543$	0	0
$544$	30.0000	1.28624
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	−10.0000	−0.427179
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−30.0000	−1.27458
$555$	0	0
$556$	−20.0000	−0.848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−12.0000	−0.508001
$559$	−8.00000	−0.338364
$560$	1.00000	0.0422577
$561$	0	0
$562$	6.00000	0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	−16.0000	−0.672530
$567$	−9.00000	−0.377964
$568$	36.0000	1.51053
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	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
$574$	2.00000	0.0834784
$575$	−4.00000	−0.166812
$576$	−21.0000	−0.875000
$577$	−46.0000	−1.91501	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000	−1.91501	−0.957503	+	0.288425i	$0.906868\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	2.00000	0.0830455
$581$	−4.00000	−0.165948
$582$	0	0
$583$	0	0
$584$	−18.0000	−0.744845
$585$	−3.00000	−0.124035
$586$	26.0000	1.07405
$587$	−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
$591$	0	0
$592$	10.0000	0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	2.00000	0.0819232
$597$	0	0
$598$	4.00000	0.163572
$599$	−48.0000	−1.96123	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000	−1.96123	−0.980613	+	0.195952i	$0.937220\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	−8.00000	−0.326056
$603$	12.0000	0.488678
$604$	−4.00000	−0.162758
$605$	−11.0000	−0.447214
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	−18.0000	−0.727607
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	−32.0000	−1.28308
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	0	0
$628$	18.0000	0.718278
$629$	60.0000	2.39236
$630$	−3.00000	−0.119523
$631$	36.0000	1.43314	0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000	1.43314	0.716569	+	0.697517i	$0.245712\pi$
$632$	24.0000	0.954669
$633$	0	0
$634$	−30.0000	−1.19145
$635$	12.0000	0.476205
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−36.0000	−1.42414
$640$	3.00000	0.118585
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	27.0000	1.06066
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	4.00000	0.156652
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	−2.00000	−0.0780869
$657$	18.0000	0.702247
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−24.0000	−0.932786
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	−30.0000	−1.16248
$667$	8.00000	0.309761
$668$	8.00000	0.309529
$669$	0	0
$670$	4.00000	0.154533
$671$	0	0
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	−18.0000	−0.690268
$681$	0	0
$682$	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
$685$	10.0000	0.382080
$686$	1.00000	0.0381802
$687$	0	0
$688$	8.00000	0.304997
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−24.0000	−0.911028
$695$	20.0000	0.758643
$696$	0	0
$697$	−12.0000	−0.454532
$698$	2.00000	0.0757011
$699$	0	0
$700$	1.00000	0.0377964
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−34.0000	−1.27961
$707$	−6.00000	−0.225653
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	−12.0000	−0.450352
$711$	−24.0000	−0.900070
$712$	6.00000	0.224860
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−12.0000	−0.447836
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	3.00000	0.111803
$721$	−4.00000	−0.148968
$722$	19.0000	0.707107
$723$	0	0
$724$	10.0000	0.371647
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	−3.00000	−0.111187
$729$	−27.0000	−1.00000
$730$	6.00000	0.222070
$731$	48.0000	1.77534
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	20.0000	0.737210
$737$	0	0
$738$	6.00000	0.220863
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	2.00000	0.0732252
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	2.00000	0.0728357
$755$	4.00000	0.145575
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	32.0000	1.16229
$759$	0	0
$760$	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
$763$	10.0000	0.362024
$764$	−16.0000	−0.578860
$765$	18.0000	0.650791
$766$	0	0
$767$	0	0
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−24.0000	−0.862662
$775$	−4.00000	−0.143684
$776$	−42.0000	−1.50771
$777$	0	0
$778$	−6.00000	−0.215110
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−24.0000	−0.858238
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	−18.0000	−0.642448
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−14.0000	−0.496841
$795$	0	0
$796$	24.0000	0.850657
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−6.00000	−0.212000
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	0	0
$805$	4.00000	0.140981
$806$	4.00000	0.140894
$807$	0	0
$808$	18.0000	0.633238
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	−9.00000	−0.316228
$811$	−48.0000	−1.68551	−0.842754	−	0.538299i	$-0.819067\pi$
$811$	−48.0000	−1.68551	−0.842754	+	0.538299i	$0.819067\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	0	0
$818$	14.0000	0.489499
$819$	3.00000	0.104828
$820$	−2.00000	−0.0698430
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−12.0000	−0.417029
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	7.00000	0.242681
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	20.0000	0.690889
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	2.00000	0.0689246
$843$	0	0
$844$	−20.0000	−0.688428
$845$	1.00000	0.0344010
$846$	0	0
$847$	11.0000	0.377964
$848$	2.00000	0.0686803
$849$	0	0
$850$	6.00000	0.205798
$851$	40.0000	1.37118
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	24.0000	0.820303
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	12.0000	0.408722
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	38.0000	1.29129
$867$	0	0
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−30.0000	−1.01593
$873$	42.0000	1.42148
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	3.00000	0.101015
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	16.0000	0.537531
$887$	−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
$889$	−12.0000	−0.402467
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	16.0000	0.535720
$893$	0	0
$894$	0	0
$895$	−12.0000	−0.401116
$896$	−3.00000	−0.100223
$897$	0	0
$898$	30.0000	1.00111
$899$	8.00000	0.266815
$900$	3.00000	0.100000
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	54.0000	1.79601
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	4.00000	0.132745
$909$	−18.0000	−0.597022
$910$	1.00000	0.0331497
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	−6.00000	−0.198246
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	−30.0000	−0.987997
$923$	12.0000	0.394985
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−24.0000	−0.788689
$927$	−12.0000	−0.394132
$928$	10.0000	0.328266
$929$	58.0000	1.90292	0.951459	−	0.307775i	$-0.0995844\pi$
$929$	58.0000	1.90292	0.951459	+	0.307775i	$0.0995844\pi$
$930$	0	0
$931$	0	0
$932$	22.0000	0.720634
$933$	0	0
$934$	−32.0000	−1.04707
$935$	0	0
$936$	−9.00000	−0.294174
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	18.0000	0.583383
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−6.00000	−0.194257
$955$	16.0000	0.517748
$956$	4.00000	0.129369
$957$	0	0
$958$	4.00000	0.129234
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−15.0000	−0.483871
$962$	10.0000	0.322413
$963$	−24.0000	−0.773389
$964$	22.0000	0.708572
$965$	2.00000	0.0643823
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	14.0000	0.449513
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	−16.0000	−0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	30.0000	0.957826
$982$	−4.00000	−0.127645
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	−12.0000	−0.382158
$987$	0	0
$988$	0	0
$989$	32.0000	1.01754
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	20.0000	0.635001
$993$	0	0
$994$	12.0000	0.380617
$995$	−24.0000	−0.760851
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	455.2.a.a.1.1	✓	1
3.2	odd	2		4095.2.a.i.1.1		1
4.3	odd	2		7280.2.a.o.1.1		1
5.4	even	2		2275.2.a.e.1.1		1
7.6	odd	2		3185.2.a.c.1.1		1
13.12	even	2		5915.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.a.a.1.1	✓	1	1.1	even	1	trivial
2275.2.a.e.1.1		1	5.4	even	2
3185.2.a.c.1.1		1	7.6	odd	2
4095.2.a.i.1.1		1	3.2	odd	2
5915.2.a.g.1.1		1	13.12	even	2
7280.2.a.o.1.1		1	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 455 = 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	455.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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