LMFDB - Embedded newform 800.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, 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("800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{3} +5.23607 q^{7} -1.47214 q^{9} -6.47214 q^{21} +7.70820 q^{23} +5.52786 q^{27} -6.00000 q^{29} +4.47214 q^{41} +6.76393 q^{43} +0.291796 q^{47} +20.4164 q^{49} +13.4164 q^{61} -7.70820 q^{63} +14.1803 q^{67} -9.52786 q^{69} -2.41641 q^{81} -4.29180 q^{83} +7.41641 q^{87} +6.00000 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{7} + 6 q^{9} - 4 q^{21} + 2 q^{23} + 20 q^{27} - 12 q^{29} + 18 q^{43} + 14 q^{47} + 14 q^{49} - 2 q^{63} + 6 q^{67} - 28 q^{69} + 22 q^{81} - 22 q^{83} - 12 q^{87} + 12 q^{89}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^7 + 6 * q^9 - 4 * q^21 + 2 * q^23 + 20 * q^27 - 12 * q^29 + 18 * q^43 + 14 * q^47 + 14 * q^49 - 2 * q^63 + 6 * q^67 - 28 * q^69 + 22 * q^81 - 22 * q^83 - 12 * q^87 + 12 * q^89

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.23607	1.97905	0.989524	−	0.144370i	$-0.0461154\pi$
$7$	5.23607	1.97905	0.989524	+	0.144370i	$0.0461154\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−6.47214	−1.41234
$22$	0	0
$23$	7.70820	1.60727	0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820	1.60727	0.803636	+	0.595121i	$0.202896\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.52786	1.06384
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	0	0
$43$	6.76393	1.03149	0.515745	−	0.856742i	$-0.327515\pi$
$43$	6.76393	1.03149	0.515745	+	0.856742i	$0.327515\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.291796	0.0425628	0.0212814	−	0.999774i	$-0.493225\pi$
$47$	0.291796	0.0425628	0.0212814	+	0.999774i	$0.493225\pi$
$48$	0	0
$49$	20.4164	2.91663
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	0	0
$63$	−7.70820	−0.971142
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.1803	1.73240	0.866202	−	0.499694i	$-0.166554\pi$
$67$	14.1803	1.73240	0.866202	+	0.499694i	$0.166554\pi$
$68$	0	0
$69$	−9.52786	−1.14702
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−4.29180	−0.471086	−0.235543	−	0.971864i	$-0.575687\pi$
$83$	−4.29180	−0.471086	−0.235543	+	0.971864i	$0.575687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.41641	0.795122
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−2.18034	−0.214835	−0.107418	−	0.994214i	$-0.534258\pi$
$103$	−2.18034	−0.214835	−0.107418	+	0.994214i	$0.534258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.7082	−1.90526	−0.952632	−	0.304125i	$-0.901636\pi$
$107$	−19.7082	−1.90526	−0.952632	+	0.304125i	$0.901636\pi$
$108$	0	0
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−5.52786	−0.498431
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.6525	−1.12273	−0.561363	−	0.827570i	$-0.689723\pi$
$127$	−12.6525	−1.12273	−0.561363	+	0.827570i	$0.689723\pi$
$128$	0	0
$129$	−8.36068	−0.736117
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−0.360680	−0.0303747
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−25.2361	−2.08144
$148$	0	0
$149$	4.47214	0.366372	0.183186	−	0.983078i	$-0.441359\pi$
$149$	4.47214	0.366372	0.183186	+	0.983078i	$0.441359\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	40.3607	3.18087
$162$	0	0
$163$	24.6525	1.93093	0.965465	−	0.260531i	$-0.0838976\pi$
$163$	24.6525	1.93093	0.965465	+	0.260531i	$0.0838976\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.7082	−1.83460	−0.917298	−	0.398202i	$-0.869634\pi$
$167$	−23.7082	−1.83460	−0.917298	+	0.398202i	$0.869634\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−16.5836	−1.22589
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	28.9443	2.10539
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−17.5279	−1.23632
$202$	0	0
$203$	−31.4164	−2.20500
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.3475	−0.788707
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	18.7639	1.25653	0.628263	−	0.778001i	$-0.283766\pi$
$223$	18.7639	1.25653	0.628263	+	0.778001i	$0.283766\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	27.1246	1.80032	0.900162	−	0.435556i	$-0.143448\pi$
$227$	27.1246	1.80032	0.900162	+	0.435556i	$0.143448\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−13.4164	−0.864227	−0.432113	−	0.901819i	$-0.642232\pi$
$241$	−13.4164	−0.864227	−0.432113	+	0.901819i	$0.642232\pi$
$242$	0	0
$243$	−13.5967	−0.872232
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5.30495	0.336188
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.83282	0.546738
$262$	0	0
$263$	−31.1246	−1.91923	−0.959613	−	0.281324i	$-0.909226\pi$
$263$	−31.1246	−1.91923	−0.959613	+	0.281324i	$0.909226\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.41641	−0.453877
$268$	0	0
$269$	22.3607	1.36335	0.681677	−	0.731653i	$-0.261251\pi$
$269$	22.3607	1.36335	0.681677	+	0.731653i	$0.261251\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.3050	−1.86750	−0.933748	−	0.357930i	$-0.883483\pi$
$281$	−31.3050	−1.86750	−0.933748	+	0.357930i	$0.883483\pi$
$282$	0	0
$283$	9.81966	0.583718	0.291859	−	0.956461i	$-0.405726\pi$
$283$	9.81966	0.583718	0.291859	+	0.956461i	$0.405726\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.4164	1.38223
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	35.4164	2.04137
$302$	0	0
$303$	22.2492	1.27818
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.5967	−1.23259	−0.616296	−	0.787515i	$-0.711367\pi$
$307$	−21.5967	−1.23259	−0.616296	+	0.787515i	$0.711367\pi$
$308$	0	0
$309$	2.69505	0.153316
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	24.3607	1.35968
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.5836	0.917075
$328$	0	0
$329$	1.52786	0.0842339
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	70.2492	3.79310
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.12461	0.167738	0.0838690	−	0.996477i	$-0.473272\pi$
$347$	3.12461	0.167738	0.0838690	+	0.996477i	$0.473272\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	13.5967	0.713644
$364$	0	0
$365$	0	0
$366$	0	0
$367$	29.2361	1.52611	0.763055	−	0.646333i	$-0.223698\pi$
$367$	29.2361	1.52611	0.763055	+	0.646333i	$0.223698\pi$
$368$	0	0
$369$	−6.58359	−0.342728
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	15.6393	0.801227
$382$	0	0
$383$	−39.1246	−1.99917	−0.999587	−	0.0287325i	$-0.990853\pi$
$383$	−39.1246	−1.99917	−0.999587	+	0.0287325i	$0.990853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.95743	−0.506164
$388$	0	0
$389$	−31.3050	−1.58722	−0.793612	−	0.608424i	$-0.791802\pi$
$389$	−31.3050	−1.58722	−0.793612	+	0.608424i	$0.791802\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−40.2492	−1.99020	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$	−40.2492	−1.99020	−0.995098	+	0.0988936i	$0.968470\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−40.2492	−1.96163	−0.980814	−	0.194948i	$-0.937546\pi$
$421$	−40.2492	−1.96163	−0.980814	+	0.194948i	$0.937546\pi$
$422$	0	0
$423$	−0.429563	−0.0208861
$424$	0	0
$425$	0	0
$426$	0	0
$427$	70.2492	3.39960
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−30.0557	−1.43123
$442$	0	0
$443$	35.7082	1.69655	0.848274	−	0.529558i	$-0.177642\pi$
$443$	35.7082	1.69655	0.848274	+	0.529558i	$0.177642\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.52786	−0.261459
$448$	0	0
$449$	−22.3607	−1.05527	−0.527633	−	0.849473i	$-0.676920\pi$
$449$	−22.3607	−1.05527	−0.527633	+	0.849473i	$0.676920\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	−20.0689	−0.932680	−0.466340	−	0.884606i	$-0.654428\pi$
$463$	−20.0689	−0.932680	−0.466340	+	0.884606i	$0.654428\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	43.1246	1.99557	0.997785	−	0.0665285i	$-0.0211923\pi$
$467$	43.1246	1.99557	0.997785	+	0.0665285i	$0.0211923\pi$
$468$	0	0
$469$	74.2492	3.42851
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−49.8885	−2.27001
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.3475	0.514205	0.257103	−	0.966384i	$-0.417232\pi$
$487$	11.3475	0.514205	0.257103	+	0.966384i	$0.417232\pi$
$488$	0	0
$489$	−30.4721	−1.37800
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	29.3050	1.30925
$502$	0	0
$503$	−24.2918	−1.08312	−0.541559	−	0.840663i	$-0.682166\pi$
$503$	−24.2918	−1.08312	−0.541559	+	0.840663i	$0.682166\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.0689	0.713644
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	45.5967	1.99381	0.996903	−	0.0786374i	$-0.0250569\pi$
$523$	45.5967	1.99381	0.996903	+	0.0786374i	$0.0250569\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	2.47214	0.106090
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−30.7639	−1.31537	−0.657685	−	0.753293i	$-0.728464\pi$
$547$	−30.7639	−1.31537	−0.657685	+	0.753293i	$0.728464\pi$
$548$	0	0
$549$	−19.7508	−0.842943
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.5410	1.45573	0.727865	−	0.685720i	$-0.240513\pi$
$563$	34.5410	1.45573	0.727865	+	0.685720i	$0.240513\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−12.6525	−0.531354
$568$	0	0
$569$	31.3050	1.31237	0.656186	−	0.754599i	$-0.272169\pi$
$569$	31.3050	1.31237	0.656186	+	0.754599i	$0.272169\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22.4721	−0.932301
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.5410	−1.09547	−0.547733	−	0.836653i	$-0.684509\pi$
$587$	−26.5410	−1.09547	−0.547733	+	0.836653i	$0.684509\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	40.2492	1.64180	0.820900	−	0.571072i	$-0.193472\pi$
$601$	40.2492	1.64180	0.820900	+	0.571072i	$0.193472\pi$
$602$	0	0
$603$	−20.8754	−0.850112
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−21.8197	−0.885633	−0.442816	−	0.896612i	$-0.646021\pi$
$607$	−21.8197	−0.885633	−0.442816	+	0.896612i	$0.646021\pi$
$608$	0	0
$609$	38.8328	1.57359
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	42.6099	1.70988
$622$	0	0
$623$	31.4164	1.25867
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−49.1935	−1.94303	−0.971513	−	0.236986i	$-0.923841\pi$
$641$	−49.1935	−1.94303	−0.971513	+	0.236986i	$0.923841\pi$
$642$	0	0
$643$	−8.06888	−0.318206	−0.159103	−	0.987262i	$-0.550860\pi$
$643$	−8.06888	−0.318206	−0.159103	+	0.987262i	$0.550860\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.5410	−1.82972	−0.914858	−	0.403775i	$-0.867698\pi$
$647$	−46.5410	−1.82972	−0.914858	+	0.403775i	$0.867698\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−40.2492	−1.56551	−0.782757	−	0.622328i	$-0.786187\pi$
$661$	−40.2492	−1.56551	−0.782757	+	0.622328i	$0.786187\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−46.2492	−1.79078
$668$	0	0
$669$	−23.1935	−0.896712
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−33.5279	−1.28479
$682$	0	0
$683$	−51.1246	−1.95623	−0.978114	−	0.208068i	$-0.933283\pi$
$683$	−51.1246	−1.95623	−0.978114	+	0.208068i	$0.933283\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.3050	0.660225
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.3607	−0.844551	−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607	−0.844551	−0.422276	+	0.906467i	$0.638769\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−94.2492	−3.54461
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−11.4164	−0.425169
$722$	0	0
$723$	16.5836	0.616750
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.0132	1.52109	0.760547	−	0.649283i	$-0.224931\pi$
$727$	41.0132	1.52109	0.760547	+	0.649283i	$0.224931\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.5410	0.533458	0.266729	−	0.963772i	$-0.414057\pi$
$743$	14.5410	0.533458	0.266729	+	0.963772i	$0.414057\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.31811	0.231167
$748$	0	0
$749$	−103.193	−3.77061
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−70.2492	−2.54319
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−33.1672	−1.18530
$784$	0	0
$785$	0	0
$786$	0	0
$787$	56.0689	1.99864	0.999320	−	0.0368739i	$-0.0117400\pi$
$787$	56.0689	1.99864	0.999320	+	0.0368739i	$0.0117400\pi$
$788$	0	0
$789$	38.4721	1.36964
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−8.83282	−0.312092
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−27.6393	−0.972950
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.3050	1.09255	0.546275	−	0.837606i	$-0.316045\pi$
$821$	31.3050	1.09255	0.546275	+	0.837606i	$0.316045\pi$
$822$	0	0
$823$	−50.1803	−1.74918	−0.874588	−	0.484866i	$-0.838868\pi$
$823$	−50.1803	−1.74918	−0.874588	+	0.484866i	$0.838868\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.5410	−0.366547	−0.183274	−	0.983062i	$-0.558669\pi$
$827$	−10.5410	−0.366547	−0.183274	+	0.983062i	$0.558669\pi$
$828$	0	0
$829$	−13.4164	−0.465971	−0.232986	−	0.972480i	$-0.574849\pi$
$829$	−13.4164	−0.465971	−0.232986	+	0.972480i	$0.574849\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	38.6950	1.33273
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−57.5967	−1.97905
$848$	0	0
$849$	−12.1378	−0.416567
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	−28.9443	−0.986418
$862$	0	0
$863$	47.7082	1.62401	0.812003	−	0.583653i	$-0.198377\pi$
$863$	47.7082	1.62401	0.812003	+	0.583653i	$0.198377\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.0132	0.713644
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	58.1378	1.95871	0.979356	−	0.202145i	$-0.0647913\pi$
$881$	58.1378	1.95871	0.979356	+	0.202145i	$0.0647913\pi$
$882$	0	0
$883$	−23.3475	−0.785707	−0.392853	−	0.919601i	$-0.628512\pi$
$883$	−23.3475	−0.785707	−0.392853	+	0.919601i	$0.628512\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.8754	−0.566620	−0.283310	−	0.959028i	$-0.591433\pi$
$887$	−16.8754	−0.566620	−0.283310	+	0.959028i	$0.591433\pi$
$888$	0	0
$889$	−66.2492	−2.22193
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−43.7771	−1.45681
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−39.4853	−1.31109	−0.655544	−	0.755157i	$-0.727561\pi$
$907$	−39.4853	−1.31109	−0.655544	+	0.755157i	$0.727561\pi$
$908$	0	0
$909$	26.4984	0.878898
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	26.6950	0.879632
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.20976	0.105422
$928$	0	0
$929$	49.1935	1.61399	0.806993	−	0.590561i	$-0.201093\pi$
$929$	49.1935	1.61399	0.806993	+	0.590561i	$0.201093\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	34.4721	1.12257
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.2918	1.17932	0.589662	−	0.807650i	$-0.299261\pi$
$947$	36.2918	1.17932	0.589662	+	0.807650i	$0.299261\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	29.0132	0.934936
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.93112	−0.126416	−0.0632081	−	0.998000i	$-0.520133\pi$
$967$	−3.93112	−0.126416	−0.0632081	+	0.998000i	$0.520133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	19.7508	0.630594
$982$	0	0
$983$	62.5410	1.99475	0.997374	−	0.0724180i	$-0.0230716\pi$
$983$	62.5410	1.99475	0.997374	+	0.0724180i	$0.0230716\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.88854	−0.0601130
$988$	0	0
$989$	52.1378	1.65788
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.a.n.1.1		2
3.2	odd	2		7200.2.a.cr.1.2		2
4.3	odd	2		800.2.a.j.1.2		2
5.2	odd	4		160.2.c.b.129.3	yes	4
5.3	odd	4		160.2.c.b.129.2	✓	4
5.4	even	2		800.2.a.j.1.2		2
8.3	odd	2		1600.2.a.bd.1.1		2
8.5	even	2		1600.2.a.z.1.2		2
12.11	even	2		7200.2.a.cb.1.1		2
15.2	even	4		1440.2.f.i.289.4		4
15.8	even	4		1440.2.f.i.289.3		4
15.14	odd	2		7200.2.a.cb.1.1		2
20.3	even	4		160.2.c.b.129.3	yes	4
20.7	even	4		160.2.c.b.129.2	✓	4
20.19	odd	2	CM	800.2.a.n.1.1		2
40.3	even	4		320.2.c.d.129.2		4
40.13	odd	4		320.2.c.d.129.3		4
40.19	odd	2		1600.2.a.z.1.2		2
40.27	even	4		320.2.c.d.129.3		4
40.29	even	2		1600.2.a.bd.1.1		2
40.37	odd	4		320.2.c.d.129.2		4
60.23	odd	4		1440.2.f.i.289.4		4
60.47	odd	4		1440.2.f.i.289.3		4
60.59	even	2		7200.2.a.cr.1.2		2
80.3	even	4		1280.2.f.h.129.2		4
80.13	odd	4		1280.2.f.g.129.4		4
80.27	even	4		1280.2.f.h.129.1		4
80.37	odd	4		1280.2.f.g.129.3		4
80.43	even	4		1280.2.f.g.129.3		4
80.53	odd	4		1280.2.f.h.129.1		4
80.67	even	4		1280.2.f.g.129.4		4
80.77	odd	4		1280.2.f.h.129.2		4
120.53	even	4		2880.2.f.w.1729.1		4
120.77	even	4		2880.2.f.w.1729.2		4
120.83	odd	4		2880.2.f.w.1729.2		4
120.107	odd	4		2880.2.f.w.1729.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.c.b.129.2	✓	4	5.3	odd	4
160.2.c.b.129.2	✓	4	20.7	even	4
160.2.c.b.129.3	yes	4	5.2	odd	4
160.2.c.b.129.3	yes	4	20.3	even	4
320.2.c.d.129.2		4	40.3	even	4
320.2.c.d.129.2		4	40.37	odd	4
320.2.c.d.129.3		4	40.13	odd	4
320.2.c.d.129.3		4	40.27	even	4
800.2.a.j.1.2		2	4.3	odd	2
800.2.a.j.1.2		2	5.4	even	2
800.2.a.n.1.1		2	1.1	even	1	trivial
800.2.a.n.1.1		2	20.19	odd	2	CM
1280.2.f.g.129.3		4	80.37	odd	4
1280.2.f.g.129.3		4	80.43	even	4
1280.2.f.g.129.4		4	80.13	odd	4
1280.2.f.g.129.4		4	80.67	even	4
1280.2.f.h.129.1		4	80.27	even	4
1280.2.f.h.129.1		4	80.53	odd	4
1280.2.f.h.129.2		4	80.3	even	4
1280.2.f.h.129.2		4	80.77	odd	4
1440.2.f.i.289.3		4	15.8	even	4
1440.2.f.i.289.3		4	60.47	odd	4
1440.2.f.i.289.4		4	15.2	even	4
1440.2.f.i.289.4		4	60.23	odd	4
1600.2.a.z.1.2		2	8.5	even	2
1600.2.a.z.1.2		2	40.19	odd	2
1600.2.a.bd.1.1		2	8.3	odd	2
1600.2.a.bd.1.1		2	40.29	even	2
2880.2.f.w.1729.1		4	120.53	even	4
2880.2.f.w.1729.1		4	120.107	odd	4
2880.2.f.w.1729.2		4	120.77	even	4
2880.2.f.w.1729.2		4	120.83	odd	4
7200.2.a.cb.1.1		2	12.11	even	2
7200.2.a.cb.1.1		2	15.14	odd	2
7200.2.a.cr.1.2		2	3.2	odd	2
7200.2.a.cr.1.2		2	60.59	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{3} +5.23607 q^{7} -1.47214 q^{9} -6.47214 q^{21} +7.70820 q^{23} +5.52786 q^{27} -6.00000 q^{29} +4.47214 q^{41} +6.76393 q^{43} +0.291796 q^{47} +20.4164 q^{49} +13.4164 q^{61} -7.70820 q^{63} +14.1803 q^{67} -9.52786 q^{69} -2.41641 q^{81} -4.29180 q^{83} +7.41641 q^{87} +6.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{7} + 6 q^{9} - 4 q^{21} + 2 q^{23} + 20 q^{27} - 12 q^{29} + 18 q^{43} + 14 q^{47} + 14 q^{49} - 2 q^{63} + 6 q^{67} - 28 q^{69} + 22 q^{81} - 22 q^{83} - 12 q^{87} + 12 q^{89}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^7 + 6 * q^9 - 4 * q^21 + 2 * q^23 + 20 * q^27 - 12 * q^29 + 18 * q^43 + 14 * q^47 + 14 * q^49 - 2 * q^63 + 6 * q^67 - 28 * q^69 + 22 * q^81 - 22 * q^83 - 12 * q^87 + 12 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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