LMFDB - Embedded newform 4030.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
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	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} +3.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -3.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} -3.00000 q^{45} +2.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} +10.0000 q^{53} -2.00000 q^{55} +12.0000 q^{59} -4.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +4.00000 q^{67} +4.00000 q^{68} +3.00000 q^{72} -2.00000 q^{74} +4.00000 q^{76} -4.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{85} +8.00000 q^{86} +2.00000 q^{88} +6.00000 q^{89} +3.00000 q^{90} -2.00000 q^{92} +12.0000 q^{94} +4.00000 q^{95} +2.00000 q^{97} +7.00000 q^{98} +6.00000 q^{99} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	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
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	−3.00000	−1.00000
$10$	−1.00000	−0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	3.00000	0.707107
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	−3.00000	−0.500000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	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
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−2.00000	−0.301511
$45$	−3.00000	−0.447214
$46$	2.00000	0.294884
$47$	−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
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	9.00000	1.00000
$82$	−6.00000	−0.662589
$83$	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
$85$	4.00000	0.433861
$86$	8.00000	0.862662
$87$	0	0
$88$	2.00000	0.213201
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−2.00000	−0.208514
$93$	0	0
$94$	12.0000	1.23771
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	7.00000	0.707107
$99$	6.00000	0.603023
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−3.00000	−0.277350
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	4.00000	0.362143
$123$	0	0
$124$	1.00000	0.0898027
$125$	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	$0.205563\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−4.00000	−0.342997
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.00000	−0.167248
$144$	−3.00000	−0.250000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	2.00000	0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−4.00000	−0.324443
$153$	−12.0000	−0.970143
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−9.00000	−0.707107
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	−12.0000	−0.917663
$172$	−8.00000	−0.609994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	−3.00000	−0.223607
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	2.00000	0.147442
$185$	2.00000	0.147043
$186$	0	0
$187$	−8.00000	−0.585018
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	−6.00000	−0.426401
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−10.0000	−0.703598
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	4.00000	0.278693
$207$	6.00000	0.417029
$208$	1.00000	0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	6.00000	0.406371
$219$	0	0
$220$	−2.00000	−0.134840
$221$	4.00000	0.269069
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	−14.0000	−0.931266
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	3.00000	0.196116
$235$	−12.0000	−0.782794
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−4.00000	−0.256074
$245$	−7.00000	−0.447214
$246$	0	0
$247$	4.00000	0.254514
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	1.00000	0.0620174
$261$	0	0
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−2.00000	−0.120605
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−22.0000	−1.31947
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	3.00000	0.176777
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	−2.00000	−0.116248
$297$	0	0
$298$	18.0000	1.04271
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	12.0000	0.690522
$303$	0	0
$304$	4.00000	0.229416
$305$	−4.00000	−0.229039
$306$	12.0000	0.685994
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	34.0000	1.90963	0.954815	−	0.297200i	$-0.0960529\pi$
$317$	34.0000	1.90963	0.954815	+	0.297200i	$0.0960529\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	16.0000	0.890264
$324$	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
$331$	−30.0000	−1.64895	−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000	−1.64895	−0.824475	+	0.565899i	$0.808529\pi$
$332$	4.00000	0.219529
$333$	−6.00000	−0.328798
$334$	−6.00000	−0.328305
$335$	4.00000	0.218543
$336$	0	0
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	4.00000	0.216930
$341$	−2.00000	−0.108306
$342$	12.0000	0.648886
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	−2.00000	−0.107521
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	36.0000	1.91609	0.958043	−	0.286623i	$-0.0925328\pi$
$353$	36.0000	1.91609	0.958043	+	0.286623i	$0.0925328\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	10.0000	0.528516
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−2.00000	−0.104257
$369$	−18.0000	−0.937043
$370$	−2.00000	−0.103975
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	12.0000	0.618853
$377$	0	0
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	8.00000	0.409316
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	24.0000	1.21999
$388$	2.00000	0.101535
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	7.00000	0.353553
$393$	0	0
$394$	−22.0000	−1.10834
$395$	−4.00000	−0.201262
$396$	6.00000	0.301511
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	10.0000	0.497519
$405$	9.00000	0.447214
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−6.00000	−0.294884
$415$	4.00000	0.196352
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	8.00000	0.391293
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	−12.0000	−0.584151
$423$	36.0000	1.75038
$424$	−10.0000	−0.485643
$425$	4.00000	0.194029
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	8.00000	0.385794
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	−8.00000	−0.382692
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	2.00000	0.0953463
$441$	21.0000	1.00000
$442$	−4.00000	−0.190261
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	3.00000	0.141421
$451$	−12.0000	−0.565058
$452$	14.0000	0.658505
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	12.0000	0.560723
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	−30.0000	−1.39422	−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000	−1.39422	−0.697109	+	0.716965i	$0.745531\pi$
$464$	0	0
$465$	0	0
$466$	2.00000	0.0926482
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−3.00000	−0.138675
$469$	0	0
$470$	12.0000	0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	16.0000	0.735681
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−30.0000	−1.37361
$478$	16.0000	0.731823
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	22.0000	1.00207
$483$	0	0
$484$	−7.00000	−0.318182
$485$	2.00000	0.0908153
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	7.00000	0.316228
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	0	0
$494$	−4.00000	−0.179969
$495$	6.00000	0.269680
$496$	1.00000	0.0449013
$497$	0	0
$498$	0	0
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	6.00000	0.267793
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	−4.00000	−0.177822
$507$	0	0
$508$	18.0000	0.798621
$509$	40.0000	1.77297	0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000	1.77297	0.886484	+	0.462758i	$0.153140\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	−4.00000	−0.176261
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	0	0
$525$	0	0
$526$	18.0000	0.784837
$527$	4.00000	0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−10.0000	−0.434372
$531$	−36.0000	−1.56227
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	12.0000	0.518805
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−24.0000	−1.03471
$539$	14.0000	0.603023
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−24.0000	−1.03089
$543$	0	0
$544$	−4.00000	−0.171499
$545$	−6.00000	−0.257012
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	12.0000	0.512615
$549$	12.0000	0.512148
$550$	2.00000	0.0852803
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	22.0000	0.933008
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	3.00000	0.127000
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	26.0000	1.09674
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	12.0000	0.504398
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	−3.00000	−0.125000
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.0000	−0.828315
$584$	0	0
$585$	−3.00000	−0.124035
$586$	−22.0000	−0.908812
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	−12.0000	−0.494032
$591$	0	0
$592$	2.00000	0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	2.00000	0.0817861
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−12.0000	−0.488273
$605$	−7.00000	−0.284590
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	4.00000	0.161955
$611$	−12.0000	−0.485468
$612$	−12.0000	−0.485071
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.0000	−1.11911
$627$	0	0
$628$	10.0000	0.399043
$629$	8.00000	0.318981
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−34.0000	−1.35031
$635$	18.0000	0.714308
$636$	0	0
$637$	−7.00000	−0.277350
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	−9.00000	−0.353553
$649$	−24.0000	−0.942082
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	4.00000	0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	30.0000	1.16598
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	6.00000	0.232147
$669$	0	0
$670$	−4.00000	−0.154533
$671$	8.00000	0.308837
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	−24.0000	−0.924445
$675$	0	0
$676$	1.00000	0.0384615
$677$	46.0000	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$678$	0	0
$679$	0	0
$680$	−4.00000	−0.153393
$681$	0	0
$682$	2.00000	0.0765840
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−12.0000	−0.458831
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	10.0000	0.380970
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−28.0000	−1.06287
$695$	22.0000	0.834508
$696$	0	0
$697$	24.0000	0.909065
$698$	−14.0000	−0.529908
$699$	0	0
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−36.0000	−1.35488
$707$	0	0
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	−6.00000	−0.224860
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−10.0000	−0.373718
$717$	0	0
$718$	−32.0000	−1.19423
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	3.00000	0.111648
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	2.00000	0.0737210
$737$	−8.00000	−0.294684
$738$	18.0000	0.662589
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	0	0
$743$	−22.0000	−0.807102	−0.403551	−	0.914957i	$-0.632224\pi$
$743$	−22.0000	−0.807102	−0.403551	+	0.914957i	$0.632224\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	10.0000	0.366126
$747$	−12.0000	−0.439057
$748$	−8.00000	−0.292509
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	−4.00000	−0.145095
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	−12.0000	−0.433861
$766$	−6.00000	−0.216789
$767$	12.0000	0.433295
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	0	0
$772$	10.0000	0.359908
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	−24.0000	−0.862662
$775$	1.00000	0.0359211
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	20.0000	0.717035
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	−7.00000	−0.250000
$785$	10.0000	0.356915
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	4.00000	0.142314
$791$	0	0
$792$	−6.00000	−0.213201
$793$	−4.00000	−0.142044
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	20.0000	0.708881
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	2.00000	0.0706225
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	0	0
$808$	−10.0000	−0.351799
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	−9.00000	−0.316228
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	0	0
$814$	4.00000	0.140200
$815$	4.00000	0.140114
$816$	0	0
$817$	−32.0000	−1.11954
$818$	2.00000	0.0699284
$819$	0	0
$820$	6.00000	0.209529
$821$	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
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	4.00000	0.139347
$825$	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	$0.715277\pi$
$828$	6.00000	0.208514
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	1.00000	0.0346688
$833$	−28.0000	−0.970143
$834$	0	0
$835$	6.00000	0.207639
$836$	−8.00000	−0.276686
$837$	0	0
$838$	−32.0000	−1.10542
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−38.0000	−1.30957
$843$	0	0
$844$	12.0000	0.413057
$845$	1.00000	0.0344010
$846$	−36.0000	−1.23771
$847$	0	0
$848$	10.0000	0.343401
$849$	0	0
$850$	−4.00000	−0.137199
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	−12.0000	−0.410152
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	8.00000	0.272481
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	4.00000	0.135926
$867$	0	0
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	4.00000	0.135535
$872$	6.00000	0.203186
$873$	−6.00000	−0.203069
$874$	8.00000	0.270604
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−16.0000	−0.539974
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−21.0000	−0.707107
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	36.0000	1.20944
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	0	0
$890$	−6.00000	−0.201120
$891$	−18.0000	−0.603023
$892$	2.00000	0.0669650
$893$	−48.0000	−1.60626
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	0	0
$900$	−3.00000	−0.100000
$901$	40.0000	1.33259
$902$	12.0000	0.399556
$903$	0	0
$904$	−14.0000	−0.465633
$905$	0	0
$906$	0	0
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	4.00000	0.132745
$909$	−30.0000	−0.995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	4.00000	0.132308
$915$	0	0
$916$	−12.0000	−0.396491
$917$	0	0
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	2.00000	0.0659380
$921$	0	0
$922$	36.0000	1.18560
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	30.0000	0.985861
$927$	12.0000	0.394132
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	−2.00000	−0.0655122
$933$	0	0
$934$	20.0000	0.654420
$935$	−8.00000	−0.261628
$936$	3.00000	0.0980581
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	12.0000	0.390567
$945$	0	0
$946$	−16.0000	−0.520205
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	0	0
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	30.0000	0.971286
$955$	−8.00000	−0.258874
$956$	−16.0000	−0.517477
$957$	0	0
$958$	40.0000	1.29234
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	−2.00000	−0.0644826
$963$	−36.0000	−1.16008
$964$	−22.0000	−0.708572
$965$	10.0000	0.321911
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	−4.00000	−0.128037
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	−7.00000	−0.223607
$981$	18.0000	0.574696
$982$	18.0000	0.574403
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	16.0000	0.508770
$990$	−6.00000	−0.190693
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	0	0
$995$	20.0000	0.634043
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.a.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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