LMFDB - Embedded newform 160.4.c.c.129.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(129,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			129.3
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.4.c.c.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76393i q^{3} +11.1803 q^{5} +15.5967i q^{7} +4.30495 q^{9} +53.2624i q^{15} -74.3018 q^{21} +207.872i q^{23} +125.000 q^{25} +149.135i q^{27} -306.000 q^{29} +174.377i q^{35} +460.630 q^{41} -30.9079i q^{43} +48.1308 q^{45} -643.118i q^{47} +99.7415 q^{49} +40.2492 q^{61} +67.1432i q^{63} -1096.84i q^{67} -990.289 q^{69} +595.492i q^{75} -594.234 q^{81} -1143.60i q^{83} -1457.76i q^{87} +1386.00 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9} + 472 q^{21} + 500 q^{25} - 1224 q^{29} + 1400 q^{45} - 1372 q^{49} - 616 q^{69} + 1004 q^{81} + 5544 q^{89}+O(q^{100}) $ 4 * q - 108 * q^9 + 472 * q^21 + 500 * q^25 - 1224 * q^29 + 1400 * q^45 - 1372 * q^49 - 616 * q^69 + 1004 * q^81 + 5544 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	4.76393i	0.916819i	0.888741	+	0.458410i	$0.151581\pi$
$3$	4.76393i	0.916819i	−0.888741	+	0.458410i	$0.848419\pi$
$4$	0	0
$5$	11.1803	1.00000
$5$	11.1803	1.00000
$6$	0	0
$7$	15.5967i	0.842145i	0.907027	+	0.421073i	$0.138346\pi$
$7$	15.5967i	0.842145i	−0.907027	+	0.421073i	$0.861654\pi$
$8$	0	0
$9$	4.30495	0.159443
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	53.2624i	0.916819i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−74.3018	−0.772095
$22$	0	0
$23$	207.872i	1.88454i	0.334857	+	0.942269i	$0.391312\pi$
$23$	207.872i	1.88454i	−0.334857	+	0.942269i	$0.608688\pi$
$24$	0	0
$25$	125.000	1.00000
$26$	0	0
$27$	149.135i	1.06300i
$28$	0	0
$29$	−306.000	−1.95941	−0.979703	−	0.200455i	$-0.935758\pi$
$29$	−306.000	−1.95941	−0.979703	+	0.200455i	$0.935758\pi$
$30$	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
$35$	174.377i	0.842145i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	460.630	1.75459	0.877297	−	0.479949i	$-0.159345\pi$
$41$	460.630	1.75459	0.877297	+	0.479949i	$0.159345\pi$
$42$	0	0
$43$	− 30.9079i	− 0.109614i	−0.998497	−	0.0548071i	$-0.982546\pi$
$43$	− 30.9079i	− 0.109614i	0.998497	−	0.0548071i	$-0.0174544\pi$
$44$	0	0
$45$	48.1308	0.159443
$46$	0	0
$47$	− 643.118i	− 1.99592i	−0.0638057	−	0.997962i	$-0.520324\pi$
$47$	− 643.118i	− 1.99592i	0.0638057	−	0.997962i	$-0.479676\pi$
$48$	0	0
$49$	99.7415	0.290791
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	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
$61$	40.2492	0.0844817	0.0422409	−	0.999107i	$-0.486550\pi$
$61$	40.2492	0.0844817	0.0422409	+	0.999107i	$0.486550\pi$
$62$	0	0
$63$	67.1432i	0.134274i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1096.84i	− 2.00000i	−0.00106064	−	0.999999i	$-0.500338\pi$
$67$	− 1096.84i	− 2.00000i	0.00106064	−	0.999999i	$-0.499662\pi$
$68$	0	0
$69$	−990.289	−1.72778
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	595.492i	0.916819i
$76$	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
$81$	−594.234	−0.815135
$82$	0	0
$83$	− 1143.60i	− 1.51237i	−0.654357	−	0.756186i	$-0.727060\pi$
$83$	− 1143.60i	− 1.51237i	0.654357	−	0.756186i	$-0.272940\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1457.76i	− 1.79642i
$88$	0	0
$89$	1386.00	1.65074	0.825369	−	0.564593i	$-0.190967\pi$
$89$	1386.00	1.65074	0.825369	+	0.564593i	$0.190967\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−378.000	−0.372400	−0.186200	−	0.982512i	$-0.559617\pi$
$101$	−378.000	−0.372400	−0.186200	+	0.982512i	$0.559617\pi$
$102$	0	0
$103$	1982.64i	1.89665i	0.317295	+	0.948327i	$0.397225\pi$
$103$	1982.64i	1.89665i	−0.317295	+	0.948327i	$0.602775\pi$
$104$	0	0
$105$	−830.720	−0.772095
$106$	0	0
$107$	− 1770.60i	− 1.59972i	−0.600188	−	0.799859i	$-0.704907\pi$
$107$	− 1770.60i	− 1.59972i	0.600188	−	0.799859i	$-0.295093\pi$
$108$	0	0
$109$	1972.21	1.73306	0.866530	−	0.499124i	$-0.166345\pi$
$109$	1972.21	1.73306	0.866530	+	0.499124i	$0.166345\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	2324.08i	1.88454i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	2194.41i	1.60864i
$124$	0	0
$125$	1397.54	1.00000
$126$	0	0
$127$	617.120i	0.431185i	0.976483	+	0.215593i	$0.0691683\pi$
$127$	617.120i	0.431185i	−0.976483	+	0.215593i	$0.930832\pi$
$128$	0	0
$129$	147.243	0.100496
$130$	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
$135$	1667.38i	1.06300i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	3063.77	1.82990
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3421.18	−1.95941
$146$	0	0
$147$	475.162i	0.266603i
$148$	0	0
$149$	−1909.60	−1.04994	−0.524969	−	0.851121i	$-0.675923\pi$
$149$	−1909.60	−1.04994	−0.524969	+	0.851121i	$0.675923\pi$
$150$	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	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3242.13	−1.58705
$162$	0	0
$163$	2958.65i	1.42172i	0.703336	+	0.710858i	$0.251693\pi$
$163$	2958.65i	1.42172i	−0.703336	+	0.710858i	$0.748307\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4066.07i	− 1.88408i	−0.335494	−	0.942042i	$-0.608903\pi$
$167$	− 4066.07i	− 1.88408i	0.335494	−	0.942042i	$-0.391097\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1949.59i	0.842145i
$176$	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
$181$	1078.00	0.442691	0.221346	−	0.975195i	$-0.428955\pi$
$181$	1078.00	0.442691	0.221346	+	0.975195i	$0.428955\pi$
$182$	0	0
$183$	191.745i	0.0774545i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2326.02	−0.895200
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	5225.26	1.83364
$202$	0	0
$203$	− 4772.60i	− 1.65010i
$204$	0	0
$205$	5150.00	1.75459
$206$	0	0
$207$	894.880i	0.300476i
$208$	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
$215$	− 345.561i	− 0.109614i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 2999.43i	− 0.900703i	−0.892851	−	0.450352i	$-0.851299\pi$
$223$	− 2999.43i	− 0.900703i	0.892851	−	0.450352i	$-0.148701\pi$
$224$	0	0
$225$	538.119	0.159443
$226$	0	0
$227$	− 6677.07i	− 1.95230i	−0.217088	−	0.976152i	$-0.569656\pi$
$227$	− 6677.07i	− 1.95230i	0.217088	−	0.976152i	$-0.430344\pi$
$228$	0	0
$229$	−6874.00	−1.98361	−0.991805	−	0.127761i	$-0.959221\pi$
$229$	−6874.00	−1.98361	−0.991805	+	0.127761i	$0.959221\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	− 7190.28i	− 1.99592i
$236$	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
$241$	−7285.11	−1.94720	−0.973600	−	0.228261i	$-0.926696\pi$
$241$	−7285.11	−1.94720	−0.973600	+	0.228261i	$0.926696\pi$
$242$	0	0
$243$	1195.75i	0.315667i
$244$	0	0
$245$	1115.14	0.290791
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5448.05	1.38657
$250$	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	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1317.32	−0.312413
$262$	0	0
$263$	− 6439.62i	− 1.50982i	−0.655826	−	0.754912i	$-0.727679\pi$
$263$	− 6439.62i	− 1.50982i	0.655826	−	0.754912i	$-0.272321\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6602.81i	1.51343i
$268$	0	0
$269$	−6864.73	−1.55595	−0.777974	−	0.628297i	$-0.783752\pi$
$269$	−6864.73	−1.55595	−0.777974	+	0.628297i	$0.783752\pi$
$270$	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	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4288.78	0.910488	0.455244	−	0.890367i	$-0.349552\pi$
$281$	4288.78	0.910488	0.455244	+	0.890367i	$0.349552\pi$
$282$	0	0
$283$	6004.02i	1.26114i	0.776133	+	0.630569i	$0.217178\pi$
$283$	6004.02i	1.26114i	−0.776133	+	0.630569i	$0.782822\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7184.33i	1.47762i
$288$	0	0
$289$	4913.00	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	482.063	0.0923111
$302$	0	0
$303$	− 1800.77i	− 0.341424i
$304$	0	0
$305$	450.000	0.0844817
$306$	0	0
$307$	4399.46i	0.817884i	0.912560	+	0.408942i	$0.134102\pi$
$307$	4399.46i	0.817884i	−0.912560	+	0.408942i	$0.865898\pi$
$308$	0	0
$309$	−9445.16	−1.73889
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	750.684i	0.134274i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8434.99	1.46665
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9395.48i	1.58890i
$328$	0	0
$329$	10030.6	1.68086
$330$	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
$335$	− 12263.0i	− 2.00000i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6905.33i	1.08703i
$344$	0	0
$345$	−11071.8	−1.72778
$346$	0	0
$347$	12519.8i	1.93688i	0.249247	+	0.968440i	$0.419817\pi$
$347$	12519.8i	1.93688i	−0.249247	+	0.968440i	$0.580183\pi$
$348$	0	0
$349$	9646.00	1.47948	0.739740	−	0.672893i	$-0.234948\pi$
$349$	9646.00	1.47948	0.739740	+	0.672893i	$0.234948\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	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
$361$	−6859.00	−1.00000
$362$	0	0
$363$	− 6340.79i	− 0.916819i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12078.6i	1.71797i	0.512000	+	0.858985i	$0.328905\pi$
$367$	12078.6i	1.71797i	−0.512000	+	0.858985i	$0.671095\pi$
$368$	0	0
$369$	1982.99	0.279757
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	6657.80i	0.916819i
$376$	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
$381$	−2939.92	−0.395319
$382$	0	0
$383$	− 1290.76i	− 0.172205i	−0.996286	−	0.0861026i	$-0.972559\pi$
$383$	− 1290.76i	− 0.172205i	0.996286	−	0.0861026i	$-0.0274413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 133.057i	− 0.0174772i
$388$	0	0
$389$	5854.03	0.763010	0.381505	−	0.924367i	$-0.375406\pi$
$389$	5854.03	0.763010	0.381505	+	0.924367i	$0.375406\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15822.0	−1.97036	−0.985178	−	0.171534i	$-0.945128\pi$
$401$	−15822.0	−1.97036	−0.985178	+	0.171534i	$0.945128\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−6643.73	−0.815135
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−15817.9	−1.91234	−0.956170	−	0.292812i	$-0.905409\pi$
$409$	−15817.9	−1.91234	−0.956170	+	0.292812i	$0.905409\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 12785.9i	− 1.51237i
$416$	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
$421$	14369.0	1.66342	0.831711	−	0.555208i	$-0.187362\pi$
$421$	14369.0	1.66342	0.831711	+	0.555208i	$0.187362\pi$
$422$	0	0
$423$	− 2768.59i	− 0.318236i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	627.757i	0.0711459i
$428$	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	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	− 16298.3i	− 1.79642i
$436$	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
$441$	429.382	0.0463646
$442$	0	0
$443$	− 18548.5i	− 1.98931i	−0.103256	−	0.994655i	$-0.532926\pi$
$443$	− 18548.5i	− 1.98931i	0.103256	−	0.994655i	$-0.467074\pi$
$444$	0	0
$445$	15496.0	1.65074
$446$	0	0
$447$	− 9097.21i	− 0.962603i
$448$	0	0
$449$	18939.5	1.99067	0.995334	−	0.0964880i	$-0.0307609\pi$
$449$	18939.5	1.99067	0.995334	+	0.0964880i	$0.0307609\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16002.0	−1.61668	−0.808338	−	0.588719i	$-0.799632\pi$
$461$	−16002.0	−1.61668	−0.808338	+	0.588719i	$0.799632\pi$
$462$	0	0
$463$	2293.27i	0.230188i	0.993355	+	0.115094i	$0.0367169\pi$
$463$	2293.27i	0.230188i	−0.993355	+	0.115094i	$0.963283\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4004.65i	0.396816i	0.980120	+	0.198408i	$0.0635770\pi$
$467$	4004.65i	0.396816i	−0.980120	+	0.198408i	$0.936423\pi$
$468$	0	0
$469$	17107.1	1.68429
$470$	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
$483$	− 15445.3i	− 1.45504i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 15279.6i	− 1.42173i	−0.703328	−	0.710865i	$-0.748304\pi$
$487$	− 15279.6i	− 1.42173i	0.703328	−	0.710865i	$-0.251696\pi$
$488$	0	0
$489$	−14094.8	−1.30346
$490$	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
$501$	19370.5	1.72736
$502$	0	0
$503$	3284.06i	0.291111i	0.989350	+	0.145556i	$0.0464970\pi$
$503$	3284.06i	0.291111i	−0.989350	+	0.145556i	$0.953503\pi$
$504$	0	0
$505$	−4226.17	−0.372400
$506$	0	0
$507$	10466.4i	0.916819i
$508$	0	0
$509$	−8946.00	−0.779026	−0.389513	−	0.921021i	$-0.627357\pi$
$509$	−8946.00	−0.779026	−0.389513	+	0.921021i	$0.627357\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	22166.6i	1.89665i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8442.00	0.709886	0.354943	−	0.934888i	$-0.384500\pi$
$521$	8442.00	0.709886	0.354943	+	0.934888i	$0.384500\pi$
$522$	0	0
$523$	5596.77i	0.467934i	0.972245	+	0.233967i	$0.0751708\pi$
$523$	5596.77i	0.467934i	−0.972245	+	0.233967i	$0.924829\pi$
$524$	0	0
$525$	−9287.73	−0.772095
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−31043.9	−2.55148
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 19795.9i	− 1.59972i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6802.00	−0.540556	−0.270278	−	0.962782i	$-0.587116\pi$
$541$	−6802.00	−0.540556	−0.270278	+	0.962782i	$0.587116\pi$
$542$	0	0
$543$	5135.52i	0.405868i
$544$	0	0
$545$	22050.0	1.73306
$546$	0	0
$547$	− 14074.0i	− 1.10011i	−0.835129	−	0.550055i	$-0.814607\pi$
$547$	− 14074.0i	− 1.10011i	0.835129	−	0.550055i	$-0.185393\pi$
$548$	0	0
$549$	173.271	0.0134700
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20503.5i	1.53485i	0.641139	+	0.767425i	$0.278462\pi$
$563$	20503.5i	1.53485i	−0.641139	+	0.767425i	$0.721538\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 9268.11i	− 0.686462i
$568$	0	0
$569$	−22758.7	−1.67679	−0.838396	−	0.545062i	$-0.816506\pi$
$569$	−22758.7	−1.67679	−0.838396	+	0.545062i	$0.816506\pi$
$570$	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
$575$	25984.0i	1.88454i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	17836.5	1.27364
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4760.56i	0.334735i	0.985895	+	0.167367i	$0.0535266\pi$
$587$	4760.56i	0.334735i	−0.985895	+	0.167367i	$0.946473\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	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
$601$	7365.61	0.499916	0.249958	−	0.968257i	$-0.419583\pi$
$601$	7365.61	0.499916	0.249958	+	0.968257i	$0.419583\pi$
$602$	0	0
$603$	− 4721.83i	− 0.318885i
$604$	0	0
$605$	−14881.0	−1.00000
$606$	0	0
$607$	5783.31i	0.386717i	0.981128	+	0.193359i	$0.0619381\pi$
$607$	5783.31i	0.386717i	−0.981128	+	0.193359i	$0.938062\pi$
$608$	0	0
$609$	22736.4	1.51285
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	24534.2i	1.60864i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−31001.0	−2.00326
$622$	0	0
$623$	21617.1i	1.39016i
$624$	0	0
$625$	15625.0	1.00000
$626$	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
$635$	6899.61i	0.431185i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24449.2	1.50653	0.753264	−	0.657719i	$-0.228478\pi$
$641$	24449.2	1.50653	0.753264	+	0.657719i	$0.228478\pi$
$642$	0	0
$643$	28934.5i	1.77459i	0.461199	+	0.887297i	$0.347420\pi$
$643$	28934.5i	1.77459i	−0.461199	+	0.887297i	$0.652580\pi$
$644$	0	0
$645$	1646.23	0.100496
$646$	0	0
$647$	31203.2i	1.89602i	0.318244	+	0.948009i	$0.396907\pi$
$647$	31203.2i	1.89602i	−0.318244	+	0.948009i	$0.603093\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	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
$661$	−14610.5	−0.859730	−0.429865	−	0.902893i	$-0.641439\pi$
$661$	−14610.5	−0.859730	−0.429865	+	0.902893i	$0.641439\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 63608.9i	− 3.69257i
$668$	0	0
$669$	14289.1	0.825782
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	18641.8i	1.06300i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	31809.1	1.78991
$682$	0	0
$683$	20997.4i	1.17634i	0.808736	+	0.588172i	$0.200152\pi$
$683$	20997.4i	1.17634i	−0.808736	+	0.588172i	$0.799848\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 32747.3i	− 1.81861i
$688$	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
$701$	−35844.2	−1.93126	−0.965632	−	0.259914i	$-0.916306\pi$
$701$	−35844.2	−1.93126	−0.965632	+	0.259914i	$0.916306\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	34254.0	1.82990
$706$	0	0
$707$	− 5895.57i	− 0.313615i
$708$	0	0
$709$	−506.000	−0.0268029	−0.0134014	−	0.999910i	$-0.504266\pi$
$709$	−506.000	−0.0268029	−0.0134014	+	0.999910i	$0.504266\pi$
$710$	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
$721$	−30922.7	−1.59726
$722$	0	0
$723$	− 34705.8i	− 1.78523i
$724$	0	0
$725$	−38250.0	−1.95941
$726$	0	0
$727$	− 33440.3i	− 1.70596i	−0.521943	−	0.852980i	$-0.674793\pi$
$727$	− 33440.3i	− 1.70596i	0.521943	−	0.852980i	$-0.325207\pi$
$728$	0	0
$729$	−21740.8	−1.10455
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	5312.47i	0.266603i
$736$	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
$743$	− 27928.6i	− 1.37901i	−0.724282	−	0.689504i	$-0.757829\pi$
$743$	− 27928.6i	− 1.37901i	0.724282	−	0.689504i	$-0.242171\pi$
$744$	0	0
$745$	−21350.0	−1.04994
$746$	0	0
$747$	− 4923.16i	− 0.241137i
$748$	0	0
$749$	27615.5	1.34720
$750$	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	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21798.0	−1.03834	−0.519170	−	0.854671i	$-0.673759\pi$
$761$	−21798.0	−1.03834	−0.519170	+	0.854671i	$0.673759\pi$
$762$	0	0
$763$	30760.1i	1.45949i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	29554.0	1.38588	0.692942	−	0.720994i	$-0.256314\pi$
$769$	29554.0	1.38588	0.692942	+	0.720994i	$0.256314\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	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
$783$	− 45635.2i	− 2.08285i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4875.78i	− 0.220842i	−0.993885	−	0.110421i	$-0.964780\pi$
$787$	− 4875.78i	− 0.220842i	0.993885	−	0.110421i	$-0.0352199\pi$
$788$	0	0
$789$	30677.9	1.38424
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5966.66	0.263198
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−36248.1	−1.58705
$806$	0	0
$807$	− 32703.1i	− 1.42652i
$808$	0	0
$809$	−26406.0	−1.14757	−0.573786	−	0.819005i	$-0.694526\pi$
$809$	−26406.0	−1.14757	−0.573786	+	0.819005i	$0.694526\pi$
$810$	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
$815$	33078.7i	1.42172i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46425.2	1.97351	0.986755	−	0.162216i	$-0.0518641\pi$
$821$	46425.2	1.97351	0.986755	+	0.162216i	$0.0518641\pi$
$822$	0	0
$823$	47156.2i	1.99728i	0.0521422	+	0.998640i	$0.483395\pi$
$823$	47156.2i	1.99728i	−0.0521422	+	0.998640i	$0.516605\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 40477.0i	− 1.70196i	−0.525197	−	0.850981i	$-0.676008\pi$
$827$	− 40477.0i	− 1.70196i	0.525197	−	0.850981i	$-0.323992\pi$
$828$	0	0
$829$	30951.7	1.29674	0.648369	−	0.761326i	$-0.275452\pi$
$829$	30951.7	1.29674	0.648369	+	0.761326i	$0.275452\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 45460.1i	− 1.88408i
$836$	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
$841$	69247.0	2.83927
$842$	0	0
$843$	20431.4i	0.834753i
$844$	0	0
$845$	24563.2	1.00000
$846$	0	0
$847$	− 20759.3i	− 0.842145i
$848$	0	0
$849$	−28602.8	−1.15624
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	−34225.7	−1.35471
$862$	0	0
$863$	− 48456.8i	− 1.91134i	−0.294436	−	0.955671i	$-0.595132\pi$
$863$	− 48456.8i	− 1.91134i	0.294436	−	0.955671i	$-0.404868\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23405.2i	0.916819i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21797.1i	0.842145i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42847.5	−1.63856	−0.819279	−	0.573395i	$-0.805626\pi$
$881$	−42847.5	−1.63856	−0.819279	+	0.573395i	$0.805626\pi$
$882$	0	0
$883$	− 18466.7i	− 0.703798i	−0.936038	−	0.351899i	$-0.885536\pi$
$883$	− 18466.7i	− 0.703798i	0.936038	−	0.351899i	$-0.114464\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34401.7i	1.30225i	0.758971	+	0.651124i	$0.225702\pi$
$887$	34401.7i	1.30225i	−0.758971	+	0.651124i	$0.774298\pi$
$888$	0	0
$889$	−9625.06	−0.363121
$890$	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
$903$	2296.51i	0.0846326i
$904$	0	0
$905$	12052.4	0.442691
$906$	0	0
$907$	29660.4i	1.08584i	0.839784	+	0.542921i	$0.182682\pi$
$907$	29660.4i	1.08584i	−0.839784	+	0.542921i	$0.817318\pi$
$908$	0	0
$909$	−1627.27	−0.0593765
$910$	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
$915$	2143.77i	0.0774545i
$916$	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
$921$	−20958.7	−0.749852
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8535.17i	0.302407i
$928$	0	0
$929$	−18054.0	−0.637603	−0.318801	−	0.947822i	$-0.603280\pi$
$929$	−18054.0	−0.637603	−0.318801	+	0.947822i	$0.603280\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	44478.0	1.54085	0.770426	−	0.637530i	$-0.220044\pi$
$941$	44478.0	1.54085	0.770426	+	0.637530i	$0.220044\pi$
$942$	0	0
$943$	95752.2i	3.30660i
$944$	0	0
$945$	−26005.6	−0.895200
$946$	0	0
$947$	18396.7i	0.631271i	0.948881	+	0.315635i	$0.102218\pi$
$947$	18396.7i	0.631271i	−0.948881	+	0.315635i	$0.897782\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	− 7622.33i	− 0.255063i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	59061.4i	1.96410i	0.188614	+	0.982051i	$0.439600\pi$
$967$	59061.4i	1.96410i	−0.188614	+	0.982051i	$0.560400\pi$
$968$	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	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8490.28	0.276324
$982$	0	0
$983$	13297.8i	0.431470i	0.976452	+	0.215735i	$0.0692147\pi$
$983$	13297.8i	0.431470i	−0.976452	+	0.215735i	$0.930785\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	47784.9i	1.54104i
$988$	0	0
$989$	6424.90	0.206572
$990$	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	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.c.c.129.3	yes	4
3.2	odd	2		1440.4.f.g.289.2		4
4.3	odd	2	inner	160.4.c.c.129.2	✓	4
5.2	odd	4		800.4.a.t.1.1		2
5.3	odd	4		800.4.a.l.1.2		2
5.4	even	2	inner	160.4.c.c.129.2	✓	4
8.3	odd	2		320.4.c.f.129.3		4
8.5	even	2		320.4.c.f.129.2		4
12.11	even	2		1440.4.f.g.289.1		4
15.14	odd	2		1440.4.f.g.289.1		4
20.3	even	4		800.4.a.t.1.1		2
20.7	even	4		800.4.a.l.1.2		2
20.19	odd	2	CM	160.4.c.c.129.3	yes	4
40.3	even	4		1600.4.a.cb.1.2		2
40.13	odd	4		1600.4.a.cp.1.1		2
40.19	odd	2		320.4.c.f.129.2		4
40.27	even	4		1600.4.a.cp.1.1		2
40.29	even	2		320.4.c.f.129.3		4
40.37	odd	4		1600.4.a.cb.1.2		2
60.59	even	2		1440.4.f.g.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.c.129.2	✓	4	4.3	odd	2	inner
160.4.c.c.129.2	✓	4	5.4	even	2	inner
160.4.c.c.129.3	yes	4	1.1	even	1	trivial
160.4.c.c.129.3	yes	4	20.19	odd	2	CM
320.4.c.f.129.2		4	8.5	even	2
320.4.c.f.129.2		4	40.19	odd	2
320.4.c.f.129.3		4	8.3	odd	2
320.4.c.f.129.3		4	40.29	even	2
800.4.a.l.1.2		2	5.3	odd	4
800.4.a.l.1.2		2	20.7	even	4
800.4.a.t.1.1		2	5.2	odd	4
800.4.a.t.1.1		2	20.3	even	4
1440.4.f.g.289.1		4	12.11	even	2
1440.4.f.g.289.1		4	15.14	odd	2
1440.4.f.g.289.2		4	3.2	odd	2
1440.4.f.g.289.2		4	60.59	even	2
1600.4.a.cb.1.2		2	40.3	even	4
1600.4.a.cb.1.2		2	40.37	odd	4
1600.4.a.cp.1.1		2	40.13	odd	4
1600.4.a.cp.1.1		2	40.27	even	4

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.76393i q^{3} +11.1803 q^{5} +15.5967i q^{7} +4.30495 q^{9} +53.2624i q^{15} -74.3018 q^{21} +207.872i q^{23} +125.000 q^{25} +149.135i q^{27} -306.000 q^{29} +174.377i q^{35} +460.630 q^{41} -30.9079i q^{43} +48.1308 q^{45} -643.118i q^{47} +99.7415 q^{49} +40.2492 q^{61} +67.1432i q^{63} -1096.84i q^{67} -990.289 q^{69} +595.492i q^{75} -594.234 q^{81} -1143.60i q^{83} -1457.76i q^{87} +1386.00 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9} + 472 q^{21} + 500 q^{25} - 1224 q^{29} + 1400 q^{45} - 1372 q^{49} - 616 q^{69} + 1004 q^{81} + 5544 q^{89}+O(q^{100}) \) 4 * q - 108 * q^9 + 472 * q^21 + 500 * q^25 - 1224 * q^29 + 1400 * q^45 - 1372 * q^49 - 616 * q^69 + 1004 * q^81 + 5544 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more