LMFDB - Embedded newform 800.6.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,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, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$128.307055850$
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\cdot 5 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+30.1803 q^{3} -194.180 q^{7} +667.853 q^{9} +O(q^{10})$	$q+30.1803 q^{3} -194.180 q^{7} +667.853 q^{9} -5860.43 q^{21} +5068.74 q^{23} +12822.2 q^{27} -1686.00 q^{29} +21041.4 q^{41} -10157.5 q^{43} -3217.05 q^{47} +20899.0 q^{49} -52256.9 q^{61} -129684. q^{63} +63577.5 q^{67} +152976. q^{69} +224690. q^{81} +116480. q^{83} -50884.1 q^{87} +149286. q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 38 q^{3} - 366 q^{7} + 486 q^{9}+O(q^{10}) $ 2 * q + 38 * q^3 - 366 * q^7 + 486 * q^9	$ 2 q + 38 q^{3} - 366 q^{7} + 486 q^{9} - 7204 q^{21} + 4838 q^{23} + 9500 q^{27} - 3372 q^{29} + 11862 q^{43} - 33334 q^{47} + 33614 q^{49} - 98438 q^{63} + 100434 q^{67} + 151172 q^{69} + 242902 q^{81} + 163262 q^{83} - 64068 q^{87} + 298572 q^{89}+O(q^{100}) $ 2 * q + 38 * q^3 - 366 * q^7 + 486 * q^9 - 7204 * q^21 + 4838 * q^23 + 9500 * q^27 - 3372 * q^29 + 11862 * q^43 - 33334 * q^47 + 33614 * q^49 - 98438 * q^63 + 100434 * q^67 + 151172 * q^69 + 242902 * q^81 + 163262 * q^83 - 64068 * q^87 + 298572 * q^89

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 $

$2$	0	0
$2$	0	0
$3$	30.1803	1.93607	0.968035	−	0.250816i	$-0.0806988\pi$
$3$	30.1803	1.93607	0.968035	+	0.250816i	$0.0806988\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−194.180	−1.49782	−0.748911	−	0.662671i	$-0.769423\pi$
$7$	−194.180	−1.49782	−0.748911	+	0.662671i	$0.769423\pi$
$8$	0	0
$9$	667.853	2.74837
$10$	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
$21$	−5860.43	−2.89989
$22$	0	0
$23$	5068.74	1.99793	0.998965	−	0.0454752i	$-0.0144802\pi$
$23$	5068.74	1.99793	0.998965	+	0.0454752i	$0.0144802\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	12822.2	3.38496
$28$	0	0
$29$	−1686.00	−0.372274	−0.186137	−	0.982524i	$-0.559597\pi$
$29$	−1686.00	−0.372274	−0.186137	+	0.982524i	$0.559597\pi$
$30$	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
$41$	21041.4	1.95486	0.977428	−	0.211267i	$-0.0677588\pi$
$41$	21041.4	1.95486	0.977428	+	0.211267i	$0.0677588\pi$
$42$	0	0
$43$	−10157.5	−0.837753	−0.418877	−	0.908043i	$-0.637576\pi$
$43$	−10157.5	−0.837753	−0.418877	+	0.908043i	$0.637576\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3217.05	−0.212429	−0.106214	−	0.994343i	$-0.533873\pi$
$47$	−3217.05	−0.212429	−0.106214	+	0.994343i	$0.533873\pi$
$48$	0	0
$49$	20899.0	1.24347
$50$	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
$61$	−52256.9	−1.79812	−0.899061	−	0.437824i	$-0.855749\pi$
$61$	−52256.9	−1.79812	−0.899061	+	0.437824i	$0.855749\pi$
$62$	0	0
$63$	−129684.	−4.11656
$64$	0	0
$65$	0	0
$66$	0	0
$67$	63577.5	1.73028	0.865140	−	0.501530i	$-0.167229\pi$
$67$	63577.5	1.73028	0.865140	+	0.501530i	$0.167229\pi$
$68$	0	0
$69$	152976.	3.86813
$70$	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
$81$	224690.	3.80515
$82$	0	0
$83$	116480.	1.85591	0.927954	−	0.372694i	$-0.121566\pi$
$83$	116480.	1.85591	0.927954	+	0.372694i	$0.121566\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−50884.1	−0.720748
$88$	0	0
$89$	149286.	1.99776	0.998882	−	0.0472789i	$-0.0150549\pi$
$89$	149286.	1.99776	0.998882	+	0.0472789i	$0.0150549\pi$
$90$	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
$101$	137502.	1.34124	0.670619	−	0.741802i	$-0.266029\pi$
$101$	137502.	1.34124	0.670619	+	0.741802i	$0.266029\pi$
$102$	0	0
$103$	110367.	1.02506	0.512528	−	0.858670i	$-0.328709\pi$
$103$	110367.	1.02506	0.512528	+	0.858670i	$0.328709\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6085.43	0.0513845	0.0256922	−	0.999670i	$-0.491821\pi$
$107$	6085.43	0.0513845	0.0256922	+	0.999670i	$0.491821\pi$
$108$	0	0
$109$	84456.3	0.680872	0.340436	−	0.940268i	$-0.389425\pi$
$109$	84456.3	0.680872	0.340436	+	0.940268i	$0.389425\pi$
$110$	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
$121$	−161051.	−1.00000
$122$	0	0
$123$	635037.	3.78474
$124$	0	0
$125$	0	0
$126$	0	0
$127$	58431.5	0.321468	0.160734	−	0.986998i	$-0.448614\pi$
$127$	58431.5	0.321468	0.160734	+	0.986998i	$0.448614\pi$
$128$	0	0
$129$	−306557.	−1.62195
$130$	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
$141$	−97091.7	−0.411277
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	630739.	2.40745
$148$	0	0
$149$	431583.	1.59257	0.796286	−	0.604920i	$-0.206795\pi$
$149$	431583.	1.59257	0.796286	+	0.604920i	$0.206795\pi$
$150$	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
$161$	−984250.	−2.99254
$162$	0	0
$163$	−169773.	−0.500494	−0.250247	−	0.968182i	$-0.580512\pi$
$163$	−169773.	−0.500494	−0.250247	+	0.968182i	$0.580512\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−330923.	−0.918198	−0.459099	−	0.888385i	$-0.651828\pi$
$167$	−330923.	−0.918198	−0.459099	+	0.888385i	$0.651828\pi$
$168$	0	0
$169$	−371293.	−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
$181$	−320402.	−0.726940	−0.363470	−	0.931606i	$-0.618408\pi$
$181$	−320402.	−0.726940	−0.363470	+	0.931606i	$0.618408\pi$
$182$	0	0
$183$	−1.57713e6	−3.48129
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.48982e6	−5.07006
$190$	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
$201$	1.91879e6	3.34994
$202$	0	0
$203$	327388.	0.557600
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.38517e6	5.49105
$208$	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
$223$	374640.	0.504490	0.252245	−	0.967663i	$-0.418831\pi$
$223$	374640.	0.504490	0.252245	+	0.967663i	$0.418831\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	979335.	1.26144	0.630720	−	0.776011i	$-0.282760\pi$
$227$	979335.	1.26144	0.630720	+	0.776011i	$0.282760\pi$
$228$	0	0
$229$	−1.06681e6	−1.34431	−0.672156	−	0.740410i	$-0.734632\pi$
$229$	−1.06681e6	−1.34431	−0.672156	+	0.740410i	$0.734632\pi$
$230$	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
$241$	−1.42086e6	−1.57583	−0.787916	−	0.615782i	$-0.788840\pi$
$241$	−1.42086e6	−1.57583	−0.787916	+	0.615782i	$0.788840\pi$
$242$	0	0
$243$	3.66543e6	3.98208
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.51541e6	3.59317
$250$	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
$261$	−1.12600e6	−1.02314
$262$	0	0
$263$	324007.	0.288845	0.144423	−	0.989516i	$-0.453868\pi$
$263$	324007.	0.288845	0.144423	+	0.989516i	$0.453868\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.50550e6	3.86781
$268$	0	0
$269$	−1.35718e6	−1.14356	−0.571778	−	0.820409i	$-0.693746\pi$
$269$	−1.35718e6	−1.14356	−0.571778	+	0.820409i	$0.693746\pi$
$270$	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
$281$	679161.	0.513106	0.256553	−	0.966530i	$-0.417413\pi$
$281$	679161.	0.513106	0.256553	+	0.966530i	$0.417413\pi$
$282$	0	0
$283$	−2.68372e6	−1.99192	−0.995958	−	0.0898251i	$-0.971369\pi$
$283$	−2.68372e6	−1.99192	−0.995958	+	0.0898251i	$0.971369\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.08583e6	−2.92803
$288$	0	0
$289$	−1.41986e6	−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
$301$	1.97239e6	1.25480
$302$	0	0
$303$	4.14986e6	2.59673
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−586606.	−0.355223	−0.177611	−	0.984101i	$-0.556837\pi$
$307$	−586606.	−0.355223	−0.177611	+	0.984101i	$0.556837\pi$
$308$	0	0
$309$	3.33093e6	1.98458
$310$	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
$321$	183660.	0.0994840
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.54892e6	1.31822
$328$	0	0
$329$	624688.	0.318180
$330$	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
$343$	−794587.	−0.364675
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.82853e6	−0.815225	−0.407613	−	0.913155i	$-0.633639\pi$
$347$	−1.82853e6	−0.815225	−0.407613	+	0.913155i	$0.633639\pi$
$348$	0	0
$349$	−2.95461e6	−1.29849	−0.649243	−	0.760581i	$-0.724914\pi$
$349$	−2.95461e6	−1.29849	−0.649243	+	0.760581i	$0.724914\pi$
$350$	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
$361$	−2.47610e6	−1.00000
$362$	0	0
$363$	−4.86057e6	−1.93607
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.80709e6	1.86302	0.931510	−	0.363716i	$-0.118492\pi$
$367$	4.80709e6	1.86302	0.931510	+	0.363716i	$0.118492\pi$
$368$	0	0
$369$	1.40526e7	5.37266
$370$	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
$381$	1.76348e6	0.622384
$382$	0	0
$383$	5.68236e6	1.97939	0.989695	−	0.143188i	$-0.0457355\pi$
$383$	5.68236e6	1.97939	0.989695	+	0.143188i	$0.0457355\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.78372e6	−2.30245
$388$	0	0
$389$	5.91961e6	1.98344	0.991720	−	0.128419i	$-0.0409902\pi$
$389$	5.91961e6	1.98344	0.991720	+	0.128419i	$0.0409902\pi$
$390$	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
$401$	4.66850e6	1.44983	0.724914	−	0.688840i	$-0.241880\pi$
$401$	4.66850e6	1.44983	0.724914	+	0.688840i	$0.241880\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−5.95306e6	−1.75967	−0.879837	−	0.475276i	$-0.842348\pi$
$409$	−5.95306e6	−1.75967	−0.879837	+	0.475276i	$0.842348\pi$
$410$	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
$421$	−4.04525e6	−1.11235	−0.556173	−	0.831067i	$-0.687731\pi$
$421$	−4.04525e6	−1.11235	−0.556173	+	0.831067i	$0.687731\pi$
$422$	0	0
$423$	−2.14852e6	−0.583832
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.01473e7	2.69327
$428$	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
$441$	1.39575e7	3.41751
$442$	0	0
$443$	7.75688e6	1.87792	0.938961	−	0.344023i	$-0.111790\pi$
$443$	7.75688e6	1.87792	0.938961	+	0.344023i	$0.111790\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.30253e7	3.08333
$448$	0	0
$449$	−3.02998e6	−0.709291	−0.354646	−	0.935001i	$-0.615398\pi$
$449$	−3.02998e6	−0.709291	−0.354646	+	0.935001i	$0.615398\pi$
$450$	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
$461$	−4.54780e6	−0.996665	−0.498333	−	0.866986i	$-0.666054\pi$
$461$	−4.54780e6	−0.996665	−0.498333	+	0.866986i	$0.666054\pi$
$462$	0	0
$463$	6.05420e6	1.31252	0.656258	−	0.754537i	$-0.272138\pi$
$463$	6.05420e6	1.31252	0.656258	+	0.754537i	$0.272138\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.90843e6	−1.89020	−0.945102	−	0.326774i	$-0.894038\pi$
$467$	−8.90843e6	−1.89020	−0.945102	+	0.326774i	$0.894038\pi$
$468$	0	0
$469$	−1.23455e7	−2.59165
$470$	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
$483$	−2.97050e7	−5.79377
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.00866e7	−1.92718	−0.963589	−	0.267388i	$-0.913839\pi$
$487$	−1.00866e7	−1.92718	−0.963589	+	0.267388i	$0.913839\pi$
$488$	0	0
$489$	−5.12380e6	−0.968991
$490$	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
$501$	−9.98738e6	−1.77770
$502$	0	0
$503$	3.13785e6	0.552983	0.276491	−	0.961016i	$-0.410828\pi$
$503$	3.13785e6	0.552983	0.276491	+	0.961016i	$0.410828\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.12057e7	−1.93607
$508$	0	0
$509$	−7.23049e6	−1.23701	−0.618505	−	0.785781i	$-0.712261\pi$
$509$	−7.23049e6	−1.23701	−0.618505	+	0.785781i	$0.712261\pi$
$510$	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
$521$	−5.30540e6	−0.856295	−0.428148	−	0.903709i	$-0.640834\pi$
$521$	−5.30540e6	−0.856295	−0.428148	+	0.903709i	$0.640834\pi$
$522$	0	0
$523$	−1.15542e7	−1.84707	−0.923537	−	0.383509i	$-0.874715\pi$
$523$	−1.15542e7	−1.84707	−0.923537	+	0.383509i	$0.874715\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.92558e7	2.99173
$530$	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
$541$	−1.35842e7	−1.99545	−0.997725	−	0.0674113i	$-0.978526\pi$
$541$	−1.35842e7	−1.99545	−0.997725	+	0.0674113i	$0.978526\pi$
$542$	0	0
$543$	−9.66984e6	−1.40741
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.05134e6	−0.864736	−0.432368	−	0.901697i	$-0.642322\pi$
$547$	−6.05134e6	−0.864736	−0.432368	+	0.901697i	$0.642322\pi$
$548$	0	0
$549$	−3.48999e7	−4.94190
$550$	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
$563$	1.20975e7	1.60852	0.804258	−	0.594281i	$-0.202563\pi$
$563$	1.20975e7	1.60852	0.804258	+	0.594281i	$0.202563\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.36304e7	−5.69944
$568$	0	0
$569$	−6.53945e6	−0.846760	−0.423380	−	0.905952i	$-0.639157\pi$
$569$	−6.53945e6	−0.846760	−0.423380	+	0.905952i	$0.639157\pi$
$570$	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
$581$	−2.26181e7	−2.77982
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.02285e6	0.481880	0.240940	−	0.970540i	$-0.422544\pi$
$587$	4.02285e6	0.481880	0.240940	+	0.970540i	$0.422544\pi$
$588$	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
$601$	−1.76169e7	−1.98950	−0.994748	−	0.102358i	$-0.967361\pi$
$601$	−1.76169e7	−1.98950	−0.994748	+	0.102358i	$0.967361\pi$
$602$	0	0
$603$	4.24604e7	4.75544
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.36145e7	1.49979	0.749897	−	0.661555i	$-0.230103\pi$
$607$	1.36145e7	1.49979	0.749897	+	0.661555i	$0.230103\pi$
$608$	0	0
$609$	9.88068e6	1.07955
$610$	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
$621$	6.49924e7	6.76291
$622$	0	0
$623$	−2.89884e7	−2.99229
$624$	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
$641$	−7.61048e6	−0.731589	−0.365794	−	0.930696i	$-0.619203\pi$
$641$	−7.61048e6	−0.731589	−0.365794	+	0.930696i	$0.619203\pi$
$642$	0	0
$643$	1.50256e7	1.43319	0.716596	−	0.697489i	$-0.245699\pi$
$643$	1.50256e7	1.43319	0.716596	+	0.697489i	$0.245699\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.03477e7	−0.971818	−0.485909	−	0.874009i	$-0.661511\pi$
$647$	−1.03477e7	−0.971818	−0.485909	+	0.874009i	$0.661511\pi$
$648$	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
$661$	2.19397e7	1.95311	0.976554	−	0.215275i	$-0.0690647\pi$
$661$	2.19397e7	1.95311	0.976554	+	0.215275i	$0.0690647\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.54590e6	−0.743778
$668$	0	0
$669$	1.13068e7	0.976728
$670$	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
$681$	2.95566e7	2.44223
$682$	0	0
$683$	1.21745e7	0.998619	0.499310	−	0.866424i	$-0.333587\pi$
$683$	1.21745e7	0.998619	0.499310	+	0.866424i	$0.333587\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.21968e7	−2.60268
$688$	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
$701$	−2.13434e7	−1.64047	−0.820235	−	0.572027i	$-0.806157\pi$
$701$	−2.13434e7	−1.64047	−0.820235	+	0.572027i	$0.806157\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.67002e7	−2.00893
$708$	0	0
$709$	2.34765e7	1.75395	0.876977	−	0.480533i	$-0.159557\pi$
$709$	2.34765e7	1.75395	0.876977	+	0.480533i	$0.159557\pi$
$710$	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
$721$	−2.14312e7	−1.53535
$722$	0	0
$723$	−4.28822e7	−3.05092
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.63437e7	1.84859	0.924294	−	0.381681i	$-0.124655\pi$
$727$	2.63437e7	1.84859	0.924294	+	0.381681i	$0.124655\pi$
$728$	0	0
$729$	5.60243e7	3.90443
$730$	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
$743$	−2.93648e7	−1.95144	−0.975721	−	0.219018i	$-0.929715\pi$
$743$	−2.93648e7	−1.95144	−0.975721	+	0.219018i	$0.929715\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.77916e7	5.10072
$748$	0	0
$749$	−1.18167e6	−0.0769648
$750$	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
$761$	−2.95982e7	−1.85269	−0.926347	−	0.376672i	$-0.877069\pi$
$761$	−2.95982e7	−1.85269	−0.926347	+	0.376672i	$0.877069\pi$
$762$	0	0
$763$	−1.63998e7	−1.01983
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	3.13824e7	1.91369	0.956843	−	0.290607i	$-0.0938571\pi$
$769$	3.13824e7	1.91369	0.956843	+	0.290607i	$0.0938571\pi$
$770$	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
$783$	−2.16182e7	−1.26013
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.41617e7	−1.96609	−0.983044	−	0.183368i	$-0.941300\pi$
$787$	−3.41617e7	−1.96609	−0.983044	+	0.183368i	$0.941300\pi$
$788$	0	0
$789$	9.77864e6	0.559224
$790$	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
$801$	9.97011e7	5.49058
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.09602e7	−2.21400
$808$	0	0
$809$	−1.06699e6	−0.0573175	−0.0286588	−	0.999589i	$-0.509124\pi$
$809$	−1.06699e6	−0.0573175	−0.0286588	+	0.999589i	$0.509124\pi$
$810$	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
$821$	9.63269e6	0.498758	0.249379	−	0.968406i	$-0.419774\pi$
$821$	9.63269e6	0.498758	0.249379	+	0.968406i	$0.419774\pi$
$822$	0	0
$823$	−3.18414e7	−1.63868	−0.819338	−	0.573311i	$-0.805659\pi$
$823$	−3.18414e7	−1.63868	−0.819338	+	0.573311i	$0.805659\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.13336e7	1.59311	0.796556	−	0.604565i	$-0.206653\pi$
$827$	3.13336e7	1.59311	0.796556	+	0.604565i	$0.206653\pi$
$828$	0	0
$829$	−3.65262e7	−1.84594	−0.922972	−	0.384867i	$-0.874247\pi$
$829$	−3.65262e7	−1.84594	−0.922972	+	0.384867i	$0.874247\pi$
$830$	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
$841$	−1.76686e7	−0.861412
$842$	0	0
$843$	2.04973e7	0.993409
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.12729e7	1.49782
$848$	0	0
$849$	−8.09955e7	−3.85649
$850$	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
$861$	−1.23312e8	−5.66887
$862$	0	0
$863$	4.37437e7	1.99935	0.999675	−	0.0254879i	$-0.00811393\pi$
$863$	4.37437e7	1.99935	0.999675	+	0.0254879i	$0.00811393\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.28518e7	−1.93607
$868$	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
$881$	2.42030e7	1.05058	0.525291	−	0.850922i	$-0.323956\pi$
$881$	2.42030e7	1.05058	0.525291	+	0.850922i	$0.323956\pi$
$882$	0	0
$883$	4.17673e7	1.80275	0.901373	−	0.433043i	$-0.142560\pi$
$883$	4.17673e7	1.80275	0.901373	+	0.433043i	$0.142560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.64402e7	1.98192	0.990958	−	0.134171i	$-0.0428373\pi$
$887$	4.64402e7	1.98192	0.990958	+	0.134171i	$0.0428373\pi$
$888$	0	0
$889$	−1.13462e7	−0.481502
$890$	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
$903$	5.95274e7	2.42939
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.07876e7	−0.839046	−0.419523	−	0.907745i	$-0.637803\pi$
$907$	−2.07876e7	−0.839046	−0.419523	+	0.907745i	$0.637803\pi$
$908$	0	0
$909$	9.18311e7	3.68621
$910$	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
$921$	−1.77040e7	−0.687736
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.37092e7	2.81723
$928$	0	0
$929$	−5.26024e7	−1.99971	−0.999853	−	0.0171747i	$-0.994533\pi$
$929$	−5.26024e7	−1.99971	−0.999853	+	0.0171747i	$0.994533\pi$
$930$	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
$941$	3.19418e7	1.17594	0.587970	−	0.808883i	$-0.299927\pi$
$941$	3.19418e7	1.17594	0.587970	+	0.808883i	$0.299927\pi$
$942$	0	0
$943$	1.06653e8	3.90567
$944$	0	0
$945$	0	0
$946$	0	0
$947$	640893.	0.0232226	0.0116113	−	0.999933i	$-0.496304\pi$
$947$	640893.	0.0232226	0.0116113	+	0.999933i	$0.496304\pi$
$948$	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
$961$	−2.86292e7	−1.00000
$962$	0	0
$963$	4.06417e6	0.141223
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.80869e7	0.622012	0.311006	−	0.950408i	$-0.399334\pi$
$967$	1.80869e7	0.622012	0.311006	+	0.950408i	$0.399334\pi$
$968$	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
$981$	5.64044e7	1.87129
$982$	0	0
$983$	−5.66561e7	−1.87009	−0.935046	−	0.354526i	$-0.884642\pi$
$983$	−5.66561e7	−1.87009	−0.935046	+	0.354526i	$0.884642\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.88533e7	0.616020
$988$	0	0
$989$	−5.14858e7	−1.67377
$990$	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	800.6.a.m.1.2		2
4.3	odd	2		800.6.a.f.1.1		2
5.2	odd	4		160.6.c.b.129.1	✓	4
5.3	odd	4		160.6.c.b.129.4	yes	4
5.4	even	2		800.6.a.f.1.1		2
20.3	even	4		160.6.c.b.129.1	✓	4
20.7	even	4		160.6.c.b.129.4	yes	4
20.19	odd	2	CM	800.6.a.m.1.2		2
40.3	even	4		320.6.c.h.129.4		4
40.13	odd	4		320.6.c.h.129.1		4
40.27	even	4		320.6.c.h.129.1		4
40.37	odd	4		320.6.c.h.129.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.b.129.1	✓	4	5.2	odd	4
160.6.c.b.129.1	✓	4	20.3	even	4
160.6.c.b.129.4	yes	4	5.3	odd	4
160.6.c.b.129.4	yes	4	20.7	even	4
320.6.c.h.129.1		4	40.13	odd	4
320.6.c.h.129.1		4	40.27	even	4
320.6.c.h.129.4		4	40.3	even	4
320.6.c.h.129.4		4	40.37	odd	4
800.6.a.f.1.1		2	4.3	odd	2
800.6.a.f.1.1		2	5.4	even	2
800.6.a.m.1.2		2	1.1	even	1	trivial
800.6.a.m.1.2		2	20.19	odd	2	CM

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q+30.1803 q^{3} -194.180 q^{7} +667.853 q^{9} +O(q^{10})\)	\(q+30.1803 q^{3} -194.180 q^{7} +667.853 q^{9} -5860.43 q^{21} +5068.74 q^{23} +12822.2 q^{27} -1686.00 q^{29} +21041.4 q^{41} -10157.5 q^{43} -3217.05 q^{47} +20899.0 q^{49} -52256.9 q^{61} -129684. q^{63} +63577.5 q^{67} +152976. q^{69} +224690. q^{81} +116480. q^{83} -50884.1 q^{87} +149286. q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 38 q^{3} - 366 q^{7} + 486 q^{9}+O(q^{10}) \) 2 * q + 38 * q^3 - 366 * q^7 + 486 * q^9	\( 2 q + 38 q^{3} - 366 q^{7} + 486 q^{9} - 7204 q^{21} + 4838 q^{23} + 9500 q^{27} - 3372 q^{29} + 11862 q^{43} - 33334 q^{47} + 33614 q^{49} - 98438 q^{63} + 100434 q^{67} + 151172 q^{69} + 242902 q^{81} + 163262 q^{83} - 64068 q^{87} + 298572 q^{89}+O(q^{100}) \) 2 * q + 38 * q^3 - 366 * q^7 + 486 * q^9 - 7204 * q^21 + 4838 * q^23 + 9500 * q^27 - 3372 * q^29 + 11862 * q^43 - 33334 * q^47 + 33614 * q^49 - 98438 * q^63 + 100434 * q^67 + 151172 * q^69 + 242902 * q^81 + 163262 * q^83 - 64068 * q^87 + 298572 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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