LMFDB - Embedded newform 1638.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -2.00000 q^{20} +4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +10.0000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +2.00000 q^{35} +6.00000 q^{37} +2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} +1.00000 q^{56} -10.0000 q^{58} -8.00000 q^{59} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{67} +6.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{77} +8.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -12.0000 q^{85} -4.00000 q^{86} +4.00000 q^{88} -18.0000 q^{89} +1.00000 q^{91} -8.00000 q^{92} -8.00000 q^{94} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	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	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	2.00000	0.338062
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	2.00000	0.316228
$41$	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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	1.00000	0.133631
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	−2.00000	−0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−4.00000	−0.431331
$87$	0	0
$88$	4.00000	0.426401
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	16.0000	1.49201
$116$	10.0000	0.928477
$117$	0	0
$118$	8.00000	0.736460
$119$	−6.00000	−0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−8.00000	−0.671345
$143$	4.00000	0.334497
$144$	0	0
$145$	−20.0000	−1.66091
$146$	−2.00000	−0.165521
$147$	0	0
$148$	6.00000	0.493197
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	−4.00000	−0.322329
$155$	16.0000	1.28515
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	2.00000	0.158114
$161$	8.00000	0.630488
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	12.0000	0.920358
$171$	0	0
$172$	4.00000	0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−4.00000	−0.301511
$177$	0	0
$178$	18.0000	1.34916
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	8.00000	0.589768
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−24.0000	−1.75505
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−14.0000	−0.985037
$203$	−10.0000	−0.701862
$204$	0	0
$205$	−12.0000	−0.838116
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−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	$0.258301\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−8.00000	−0.545595
$216$	0	0
$217$	8.00000	0.543075
$218$	2.00000	0.135457
$219$	0	0
$220$	8.00000	0.539360
$221$	−6.00000	−0.403604
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	−10.0000	−0.656532
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	−8.00000	−0.520756
$237$	0	0
$238$	6.00000	0.388922
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	10.0000	0.640184
$245$	−2.00000	−0.127775
$246$	0	0
$247$	0	0
$248$	8.00000	0.508001
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	2.00000	0.124035
$261$	0	0
$262$	8.00000	0.494242
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	−2.00000	−0.119523
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−4.00000	−0.236525
$287$	−6.00000	−0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	20.0000	1.17444
$291$	0	0
$292$	2.00000	0.117041
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−18.0000	−1.04271
$299$	8.00000	0.462652
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	−16.0000	−0.908739
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−40.0000	−2.23957
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−8.00000	−0.445823
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	4.00000	0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	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
$340$	−12.0000	−0.650791
$341$	32.0000	1.73290
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	4.00000	0.213201
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	−18.0000	−0.953998
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.00000	0.315353
$363$	0	0
$364$	1.00000	0.0524142
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	12.0000	0.623850
$371$	6.00000	0.311504
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−10.0000	−0.515026
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	−2.00000	−0.101797
$387$	0	0
$388$	2.00000	0.101535
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−16.0000	−0.805047
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	14.0000	0.696526
$405$	0	0
$406$	10.0000	0.496292
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	16.0000	0.788263
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	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
$427$	−10.0000	−0.483934
$428$	4.00000	0.193347
$429$	0	0
$430$	8.00000	0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	6.00000	0.285391
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	0	0
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	16.0000	0.746004
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	10.0000	0.463241
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	16.0000	0.738025
$471$	0	0
$472$	8.00000	0.368230
$473$	−16.0000	−0.735681
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−24.0000	−1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−18.0000	−0.819878
$483$	0	0
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	2.00000	0.0903508
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	60.0000	2.70226
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−8.00000	−0.358849
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	24.0000	1.07117
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	−28.0000	−1.24598
$506$	−32.0000	−1.42257
$507$	0	0
$508$	16.0000	0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−32.0000	−1.41009
$516$	0	0
$517$	−32.0000	−1.40736
$518$	6.00000	0.263625
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	0	0
$527$	−48.0000	−2.09091
$528$	0	0
$529$	41.0000	1.78261
$530$	−12.0000	−0.521247
$531$	0	0
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−8.00000	−0.345870
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−30.0000	−1.29339
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	32.0000	1.37452
$543$	0	0
$544$	−6.00000	−0.257248
$545$	4.00000	0.171341
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	−4.00000	−0.170561
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−6.00000	−0.254916
$555$	0	0
$556$	−8.00000	−0.339276
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	2.00000	0.0845154
$561$	0	0
$562$	10.0000	0.421825
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	−16.0000	−0.672530
$567$	0	0
$568$	−8.00000	−0.335673
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	6.00000	0.250435
$575$	8.00000	0.333623
$576$	0	0
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	−20.0000	−0.830455
$581$	0	0
$582$	0	0
$583$	24.0000	0.993978
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−14.0000	−0.578335
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	0	0
$590$	−16.0000	−0.658710
$591$	0	0
$592$	6.00000	0.246598
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	18.0000	0.737309
$597$	0	0
$598$	−8.00000	−0.327144
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−48.0000	−1.94826	−0.974130	−	0.225989i	$-0.927439\pi$
$607$	−48.0000	−1.94826	−0.974130	+	0.225989i	$0.927439\pi$
$608$	0	0
$609$	0	0
$610$	20.0000	0.809776
$611$	−8.00000	−0.323645
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	24.0000	0.962312
$623$	18.0000	0.721155
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	2.00000	0.0798087
$629$	36.0000	1.43541
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	6.00000	0.238290
$635$	−32.0000	−1.26988
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	40.0000	1.58362
$639$	0	0
$640$	2.00000	0.0790569
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−4.00000	−0.156652
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	6.00000	0.234261
$657$	0	0
$658$	8.00000	0.311872
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−80.0000	−3.09761
$668$	0	0
$669$	0	0
$670$	8.00000	0.309067
$671$	−40.0000	−1.54418
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−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
$679$	−2.00000	−0.0767530
$680$	12.0000	0.460179
$681$	0	0
$682$	−32.0000	−1.22534
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	1.00000	0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	6.00000	0.228582
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−12.0000	−0.455514
$695$	16.0000	0.606915
$696$	0	0
$697$	36.0000	1.36360
$698$	14.0000	0.529908
$699$	0	0
$700$	1.00000	0.0377964
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	−4.00000	−0.150756
$705$	0	0
$706$	10.0000	0.376355
$707$	−14.0000	−0.526524
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	18.0000	0.674579
$713$	64.0000	2.39682
$714$	0	0
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	19.0000	0.707107
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−10.0000	−0.371391
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	4.00000	0.148047
$731$	24.0000	0.887672
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	8.00000	0.294884
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	10.0000	0.366126
$747$	0	0
$748$	−24.0000	−0.877527
$749$	−4.00000	−0.146157
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	10.0000	0.364179
$755$	0	0
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	8.00000	0.289430
$765$	0	0
$766$	24.0000	0.867155
$767$	8.00000	0.288863
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	2.00000	0.0719816
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	30.0000	1.07555
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	48.0000	1.71648
$783$	0	0
$784$	1.00000	0.0357143
$785$	−4.00000	−0.142766
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	16.0000	0.569254
$791$	2.00000	0.0711118
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	1.00000	0.0353553
$801$	0	0
$802$	18.0000	0.635602
$803$	−8.00000	−0.282314
$804$	0	0
$805$	−16.0000	−0.563926
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−14.0000	−0.492518
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	−10.0000	−0.350931
$813$	0	0
$814$	24.0000	0.841200
$815$	8.00000	0.280228
$816$	0	0
$817$	0	0
$818$	38.0000	1.32864
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−14.0000	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$822$	0	0
$823$	48.0000	1.67317	0.836587	−	0.547833i	$-0.184547\pi$
$823$	48.0000	1.67317	0.836587	+	0.547833i	$0.184547\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	−8.00000	−0.278356
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	2.00000	0.0689246
$843$	0	0
$844$	20.0000	0.688428
$845$	−2.00000	−0.0688021
$846$	0	0
$847$	−5.00000	−0.171802
$848$	−6.00000	−0.206041
$849$	0	0
$850$	6.00000	0.205798
$851$	−48.0000	−1.64542
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	38.0000	1.29129
$867$	0	0
$868$	8.00000	0.271538
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−4.00000	−0.135535
$872$	2.00000	0.0677285
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	8.00000	0.269680
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−36.0000	−1.20944
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−36.0000	−1.20672
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−80.0000	−2.66815
$900$	0	0
$901$	−36.0000	−1.19933
$902$	24.0000	0.799113
$903$	0	0
$904$	2.00000	0.0665190
$905$	12.0000	0.398893
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	0	0
$910$	2.00000	0.0662994
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−6.00000	−0.198246
$917$	8.00000	0.264183
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	18.0000	0.592798
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−16.0000	−0.525793
$927$	0	0
$928$	−10.0000	−0.328266
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	−10.0000	−0.327561
$933$	0	0
$934$	8.00000	0.261768
$935$	48.0000	1.56977
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−16.0000	−0.521862
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	−8.00000	−0.260378
$945$	0	0
$946$	16.0000	0.520205
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	0	0
$952$	6.00000	0.194461
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	24.0000	0.776215
$957$	0	0
$958$	24.0000	0.775405
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	6.00000	0.193448
$963$	0	0
$964$	18.0000	0.579741
$965$	−4.00000	−0.128765
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	4.00000	0.128432
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	16.0000	0.512673
$975$	0	0
$976$	10.0000	0.320092
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	72.0000	2.30113
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	28.0000	0.893516
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	−60.0000	−1.91079
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	8.00000	0.253745
$995$	32.0000	1.01447
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.c.1.1		1
3.2	odd	2		182.2.a.c.1.1	✓	1
12.11	even	2		1456.2.a.i.1.1		1
15.14	odd	2		4550.2.a.g.1.1		1
21.2	odd	6		1274.2.f.f.1145.1		2
21.5	even	6		1274.2.f.g.1145.1		2
21.11	odd	6		1274.2.f.f.79.1		2
21.17	even	6		1274.2.f.g.79.1		2
21.20	even	2		1274.2.a.l.1.1		1
24.5	odd	2		5824.2.a.l.1.1		1
24.11	even	2		5824.2.a.m.1.1		1
39.5	even	4		2366.2.d.d.337.1		2
39.8	even	4		2366.2.d.d.337.2		2
39.38	odd	2		2366.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.c.1.1	✓	1	3.2	odd	2
1274.2.a.l.1.1		1	21.20	even	2
1274.2.f.f.79.1		2	21.11	odd	6
1274.2.f.f.1145.1		2	21.2	odd	6
1274.2.f.g.79.1		2	21.17	even	6
1274.2.f.g.1145.1		2	21.5	even	6
1456.2.a.i.1.1		1	12.11	even	2
1638.2.a.c.1.1		1	1.1	even	1	trivial
2366.2.a.d.1.1		1	39.38	odd	2
2366.2.d.d.337.1		2	39.5	even	4
2366.2.d.d.337.2		2	39.8	even	4
4550.2.a.g.1.1		1	15.14	odd	2
5824.2.a.l.1.1		1	24.5	odd	2
5824.2.a.m.1.1		1	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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