LMFDB - Embedded newform 160.4.c.c.129.4

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			129.4
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.4.c.c.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.23607i q^{3} -11.1803 q^{5} -33.5967i q^{7} -58.3050 q^{9} +O(q^{10})$	$q+9.23607i q^{3} -11.1803 q^{5} -33.5967i q^{7} -58.3050 q^{9} -103.262i q^{15} +310.302 q^{21} -73.8723i q^{23} +125.000 q^{25} -289.135i q^{27} -306.000 q^{29} +375.623i q^{35} -460.630 q^{41} -563.092i q^{43} +651.869 q^{45} +41.1184i q^{47} -785.741 q^{49} -40.2492 q^{61} +1958.86i q^{63} -1.16335i q^{67} +682.289 q^{69} +1154.51i q^{75} +1096.23 q^{81} +989.604i q^{83} -2826.24i q^{87} +1386.00 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$	9.23607i	1.77748i	0.458410	+	0.888741i	$0.348419\pi$
$3$	9.23607i	1.77748i	−0.458410	+	0.888741i	$0.651581\pi$
$4$	0	0
$5$	−11.1803	−1.00000
$5$	−11.1803	−1.00000
$6$	0	0
$7$	− 33.5967i	− 1.81405i	−0.421073	−	0.907027i	$-0.638346\pi$
$7$	− 33.5967i	− 1.81405i	0.421073	−	0.907027i	$-0.361654\pi$
$8$	0	0
$9$	−58.3050	−2.15944
$10$	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
$15$	− 103.262i	− 1.77748i
$16$	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
$21$	310.302	3.22445
$22$	0	0
$23$	− 73.8723i	− 0.669715i	−0.942269	−	0.334857i	$-0.891312\pi$
$23$	− 73.8723i	− 0.669715i	0.942269	−	0.334857i	$-0.108688\pi$
$24$	0	0
$25$	125.000	1.00000
$26$	0	0
$27$	− 289.135i	− 2.06089i
$28$	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
$35$	375.623i	1.81405i
$36$	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	$-0.840655\pi$
$41$	−460.630	−1.75459	−0.877297	+	0.479949i	$0.840655\pi$
$42$	0	0
$43$	− 563.092i	− 1.99699i	−0.0548071	−	0.998497i	$-0.517454\pi$
$43$	− 563.092i	− 1.99699i	0.0548071	−	0.998497i	$-0.482546\pi$
$44$	0	0
$45$	651.869	2.15944
$46$	0	0
$47$	41.1184i	0.127611i	0.997962	+	0.0638057i	$0.0203238\pi$
$47$	41.1184i	0.127611i	−0.997962	+	0.0638057i	$0.979676\pi$
$48$	0	0
$49$	−785.741	−2.29079
$50$	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	$-0.513450\pi$
$61$	−40.2492	−0.0844817	−0.0422409	+	0.999107i	$0.513450\pi$
$62$	0	0
$63$	1958.86i	3.91735i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.16335i	− 0.00212127i	−0.999999	−	0.00106064i	$-0.999662\pi$
$67$	− 1.16335i	− 0.00212127i	0.999999	−	0.00106064i	$-0.000337611\pi$
$68$	0	0
$69$	682.289	1.19041
$70$	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
$75$	1154.51i	1.77748i
$76$	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
$81$	1096.23	1.50375
$82$	0	0
$83$	989.604i	1.30871i	0.756186	+	0.654357i	$0.227060\pi$
$83$	989.604i	1.30871i	−0.756186	+	0.654357i	$0.772940\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2826.24i	− 3.48281i
$88$	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
$103$	663.360i	0.634590i	0.948327	+	0.317295i	$0.102775\pi$
$103$	663.360i	0.634590i	−0.948327	+	0.317295i	$0.897225\pi$
$104$	0	0
$105$	−3469.28	−3.22445
$106$	0	0
$107$	1328.60i	1.20038i	0.799859	+	0.600188i	$0.204907\pi$
$107$	1328.60i	1.20038i	−0.799859	+	0.600188i	$0.795093\pi$
$108$	0	0
$109$	−1972.21	−1.73306	−0.866530	−	0.499124i	$-0.833655\pi$
$109$	−1972.21	−1.73306	−0.866530	+	0.499124i	$0.833655\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	825.917i	0.669715i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	− 4254.41i	− 3.11876i
$124$	0	0
$125$	−1397.54	−1.00000
$126$	0	0
$127$	− 2795.12i	− 1.95297i	−0.215593	−	0.976483i	$-0.569168\pi$
$127$	− 2795.12i	− 1.95297i	0.215593	−	0.976483i	$-0.430832\pi$
$128$	0	0
$129$	5200.76	3.54962
$130$	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
$135$	3232.62i	2.06089i
$136$	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
$141$	−379.772	−0.226827
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3421.18	1.95941
$146$	0	0
$147$	− 7257.16i	− 4.07184i
$148$	0	0
$149$	1909.60	1.04994	0.524969	−	0.851121i	$-0.324077\pi$
$149$	1909.60	1.04994	0.524969	+	0.851121i	$0.324077\pi$
$150$	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
$161$	−2481.87	−1.21490
$162$	0	0
$163$	2927.35i	1.40667i	0.710858	+	0.703336i	$0.248307\pi$
$163$	2927.35i	1.40667i	−0.710858	+	0.703336i	$0.751693\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1448.07i	0.670989i	0.942042	+	0.335494i	$0.108903\pi$
$167$	1448.07i	0.670989i	−0.942042	+	0.335494i	$0.891097\pi$
$168$	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
$175$	− 4199.59i	− 1.81405i
$176$	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
$183$	− 371.745i	− 0.150165i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−9713.98	−3.73856
$190$	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
$201$	10.7447	0.00377052
$202$	0	0
$203$	10280.6i	3.55447i
$204$	0	0
$205$	5150.00	1.75459
$206$	0	0
$207$	4307.12i	1.44621i
$208$	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
$215$	6295.56i	1.99699i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 5946.57i	− 1.78570i	−0.450352	−	0.892851i	$-0.648701\pi$
$223$	− 5946.57i	− 1.78570i	0.450352	−	0.892851i	$-0.351299\pi$
$224$	0	0
$225$	−7288.12	−2.15944
$226$	0	0
$227$	− 1484.93i	− 0.434176i	−0.976152	−	0.217088i	$-0.930344\pi$
$227$	− 1484.93i	− 0.434176i	0.976152	−	0.217088i	$-0.0696558\pi$
$228$	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
$235$	− 459.718i	− 0.127611i
$236$	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	$-0.0733041\pi$
$241$	7285.11	1.94720	0.973600	+	0.228261i	$0.0733041\pi$
$242$	0	0
$243$	2318.25i	0.612000i
$244$	0	0
$245$	8784.86	2.29079
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−9140.05	−2.32621
$250$	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
$261$	17841.3	4.23122
$262$	0	0
$263$	− 5594.38i	− 1.31165i	−0.754912	−	0.655826i	$-0.772321\pi$
$263$	− 5594.38i	− 1.31165i	0.754912	−	0.655826i	$-0.227679\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12801.2i	2.93416i
$268$	0	0
$269$	6864.73	1.55595	0.777974	−	0.628297i	$-0.216248\pi$
$269$	6864.73	1.55595	0.777974	+	0.628297i	$0.216248\pi$
$270$	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	$-0.650448\pi$
$281$	−4288.78	−0.910488	−0.455244	+	0.890367i	$0.650448\pi$
$282$	0	0
$283$	− 7390.02i	− 1.55227i	−0.630569	−	0.776133i	$-0.717178\pi$
$283$	− 7390.02i	− 1.55227i	0.630569	−	0.776133i	$-0.282822\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15475.7i	3.18293i
$288$	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
$301$	−18918.1	−3.62265
$302$	0	0
$303$	− 3491.23i	− 0.661934i
$304$	0	0
$305$	450.000	0.0844817
$306$	0	0
$307$	− 9817.46i	− 1.82512i	−0.408942	−	0.912560i	$-0.634102\pi$
$307$	− 9817.46i	− 1.82512i	0.408942	−	0.912560i	$-0.365898\pi$
$308$	0	0
$309$	−6126.84	−1.12797
$310$	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
$315$	− 21900.7i	− 3.91735i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12271.0	−2.13365
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18215.5i	− 3.08048i
$328$	0	0
$329$	1381.44	0.231494
$330$	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
$335$	13.0066i	0.00212127i
$336$	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
$343$	14874.7i	2.34157i
$344$	0	0
$345$	−7628.23	−1.19041
$346$	0	0
$347$	3222.21i	0.498494i	0.968440	+	0.249247i	$0.0801832\pi$
$347$	3222.21i	0.498494i	−0.968440	+	0.249247i	$0.919817\pi$
$348$	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
$363$	− 12293.2i	− 1.77748i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7199.45i	1.02400i	0.858985	+	0.512000i	$0.171095\pi$
$367$	7199.45i	1.02400i	−0.858985	+	0.512000i	$0.828905\pi$
$368$	0	0
$369$	26857.0	3.78894
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	− 12907.8i	− 1.77748i
$376$	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
$381$	25815.9	3.47136
$382$	0	0
$383$	− 14935.2i	− 1.99257i	−0.0861026	−	0.996286i	$-0.527441\pi$
$383$	− 14935.2i	− 1.99257i	0.0861026	−	0.996286i	$-0.472559\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32831.1i	4.31239i
$388$	0	0
$389$	−5854.03	−0.763010	−0.381505	−	0.924367i	$-0.624594\pi$
$389$	−5854.03	−0.763010	−0.381505	+	0.924367i	$0.624594\pi$
$390$	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
$405$	−12256.3	−1.50375
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15817.9	1.91234	0.956170	−	0.292812i	$-0.0945911\pi$
$409$	15817.9	1.91234	0.956170	+	0.292812i	$0.0945911\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 11064.1i	− 1.30871i
$416$	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	$-0.812638\pi$
$421$	−14369.0	−1.66342	−0.831711	+	0.555208i	$0.812638\pi$
$422$	0	0
$423$	− 2397.41i	− 0.275569i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1352.24i	0.153254i
$428$	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
$435$	31598.3i	3.48281i
$436$	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
$441$	45812.6	4.94683
$442$	0	0
$443$	− 1925.54i	− 0.206512i	−0.994655	−	0.103256i	$-0.967074\pi$
$443$	− 1925.54i	− 0.206512i	0.994655	−	0.103256i	$-0.0329261\pi$
$444$	0	0
$445$	−15496.0	−1.65074
$446$	0	0
$447$	17637.2i	1.86625i
$448$	0	0
$449$	−18939.5	−1.99067	−0.995334	−	0.0964880i	$-0.969239\pi$
$449$	−18939.5	−1.99067	−0.995334	+	0.0964880i	$0.969239\pi$
$450$	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
$463$	19792.7i	1.98671i	0.115094	+	0.993355i	$0.463283\pi$
$463$	19792.7i	1.98671i	−0.115094	+	0.993355i	$0.536717\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19782.6i	− 1.96024i	−0.198408	−	0.980120i	$-0.563577\pi$
$467$	− 19782.6i	− 1.96024i	0.198408	−	0.980120i	$-0.436423\pi$
$468$	0	0
$469$	−39.0846	−0.00384810
$470$	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
$483$	− 22922.7i	− 2.15946i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15117.6i	1.40666i	0.710865	+	0.703328i	$0.248304\pi$
$487$	15117.6i	1.40666i	−0.710865	+	0.703328i	$0.751696\pi$
$488$	0	0
$489$	−27037.2	−2.50033
$490$	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
$501$	−13374.5	−1.19267
$502$	0	0
$503$	22321.9i	1.97870i	0.145556	+	0.989350i	$0.453503\pi$
$503$	22321.9i	1.97870i	−0.145556	+	0.989350i	$0.546497\pi$
$504$	0	0
$505$	4226.17	0.372400
$506$	0	0
$507$	20291.6i	1.77748i
$508$	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
$515$	− 7416.59i	− 0.634590i
$516$	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
$523$	23257.2i	1.94449i	0.233967	+	0.972245i	$0.424829\pi$
$523$	23257.2i	1.94449i	−0.233967	+	0.972245i	$0.575171\pi$
$524$	0	0
$525$	38787.7	3.22445
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6709.89	0.551482
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 14854.1i	− 1.20038i
$536$	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
$543$	9956.48i	0.786876i
$544$	0	0
$545$	22050.0	1.73306
$546$	0	0
$547$	− 21368.0i	− 1.67026i	−0.550055	−	0.835129i	$-0.685393\pi$
$547$	− 21368.0i	− 1.67026i	0.550055	−	0.835129i	$-0.314607\pi$
$548$	0	0
$549$	2346.73	0.182433
$550$	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
$563$	− 17129.5i	− 1.28228i	−0.767425	−	0.641139i	$-0.778462\pi$
$563$	− 17129.5i	− 1.28228i	0.767425	−	0.641139i	$-0.221538\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 36829.9i	− 2.72788i
$568$	0	0
$569$	22758.7	1.67679	0.838396	−	0.545062i	$-0.183494\pi$
$569$	22758.7	1.67679	0.838396	+	0.545062i	$0.183494\pi$
$570$	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
$575$	− 9234.04i	− 0.669715i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	33247.5	2.37408
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28042.6i	− 1.97179i	−0.167367	−	0.985895i	$-0.553527\pi$
$587$	− 28042.6i	− 1.97179i	0.167367	−	0.985895i	$-0.446473\pi$
$588$	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	$-0.580417\pi$
$601$	−7365.61	−0.499916	−0.249958	+	0.968257i	$0.580417\pi$
$602$	0	0
$603$	67.8288i	0.00458077i
$604$	0	0
$605$	14881.0	1.00000
$606$	0	0
$607$	− 29345.3i	− 1.96226i	−0.193359	−	0.981128i	$-0.561938\pi$
$607$	− 29345.3i	− 1.96226i	0.193359	−	0.981128i	$-0.438062\pi$
$608$	0	0
$609$	−94952.4	−6.31800
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	47565.8i	3.11876i
$616$	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
$621$	−21359.0	−1.38021
$622$	0	0
$623$	− 46565.1i	− 2.99453i
$624$	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
$635$	31250.4i	1.95297i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24449.2	−1.50653	−0.753264	−	0.657719i	$-0.771522\pi$
$641$	−24449.2	−1.50653	−0.753264	+	0.657719i	$0.771522\pi$
$642$	0	0
$643$	15039.5i	0.922397i	0.887297	+	0.461199i	$0.152580\pi$
$643$	15039.5i	0.922397i	−0.887297	+	0.461199i	$0.847420\pi$
$644$	0	0
$645$	−58146.2	−3.54962
$646$	0	0
$647$	10474.8i	0.636488i	0.948009	+	0.318244i	$0.103093\pi$
$647$	10474.8i	0.636488i	−0.948009	+	0.318244i	$0.896907\pi$
$648$	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	$-0.358561\pi$
$661$	14610.5	0.859730	0.429865	+	0.902893i	$0.358561\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22604.9i	1.31224i
$668$	0	0
$669$	54922.9	3.17405
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 36141.8i	− 2.06089i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	13714.9	0.771740
$682$	0	0
$683$	− 28871.4i	− 1.61747i	−0.588172	−	0.808736i	$-0.700152\pi$
$683$	− 28871.4i	− 1.61747i	0.588172	−	0.808736i	$-0.299848\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 63488.7i	− 3.52583i
$688$	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	$-0.0836943\pi$
$701$	35844.2	1.93126	0.965632	+	0.259914i	$0.0836943\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4245.98	0.226827
$706$	0	0
$707$	12699.6i	0.675554i
$708$	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
$721$	22286.7	1.15118
$722$	0	0
$723$	67285.8i	3.46111i
$724$	0	0
$725$	−38250.0	−1.95941
$726$	0	0
$727$	20462.3i	1.04389i	0.852980	+	0.521943i	$0.174793\pi$
$727$	20462.3i	1.04389i	−0.852980	+	0.521943i	$0.825207\pi$
$728$	0	0
$729$	8186.77	0.415931
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	81137.5i	4.07184i
$736$	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
$743$	− 29337.4i	− 1.44856i	−0.689504	−	0.724282i	$-0.742171\pi$
$743$	− 29337.4i	− 1.44856i	0.689504	−	0.724282i	$-0.257829\pi$
$744$	0	0
$745$	−21350.0	−1.04994
$746$	0	0
$747$	− 57698.8i	− 2.82609i
$748$	0	0
$749$	44636.5	2.17755
$750$	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
$763$	66259.9i	3.14387i
$764$	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
$783$	88475.2i	4.03812i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 43886.2i	− 1.98777i	−0.110421	−	0.993885i	$-0.535220\pi$
$787$	− 43886.2i	− 1.98777i	0.110421	−	0.993885i	$-0.464780\pi$
$788$	0	0
$789$	51670.1	2.33144
$790$	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
$801$	−80810.7	−3.56467
$802$	0	0
$803$	0	0
$804$	0	0
$805$	27748.1	1.21490
$806$	0	0
$807$	63403.1i	2.76567i
$808$	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
$815$	− 32728.7i	− 1.40667i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46425.2	−1.97351	−0.986755	−	0.162216i	$-0.948136\pi$
$821$	−46425.2	−1.97351	−0.986755	+	0.162216i	$0.948136\pi$
$822$	0	0
$823$	− 2462.17i	− 0.104284i	−0.998640	−	0.0521422i	$-0.983395\pi$
$823$	− 2462.17i	− 0.104284i	0.998640	−	0.0521422i	$-0.0166049\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24981.0i	− 1.05039i	−0.850981	−	0.525197i	$-0.823992\pi$
$827$	− 24981.0i	− 1.05039i	0.850981	−	0.525197i	$-0.176008\pi$
$828$	0	0
$829$	−30951.7	−1.29674	−0.648369	−	0.761326i	$-0.724548\pi$
$829$	−30951.7	−1.29674	−0.648369	+	0.761326i	$0.724548\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 16189.9i	− 0.670989i
$836$	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
$843$	− 39611.4i	− 1.61838i
$844$	0	0
$845$	−24563.2	−1.00000
$846$	0	0
$847$	44717.3i	1.81405i
$848$	0	0
$849$	68254.8	2.75913
$850$	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
$861$	−142934.	−5.65759
$862$	0	0
$863$	− 14929.2i	− 0.588871i	−0.955671	−	0.294436i	$-0.904868\pi$
$863$	− 14929.2i	− 0.588871i	0.955671	−	0.294436i	$-0.0951316\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	45376.8i	1.77748i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	46952.9i	1.81405i
$876$	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	$-0.194374\pi$
$881$	42847.5	1.63856	0.819279	+	0.573395i	$0.194374\pi$
$882$	0	0
$883$	49120.7i	1.87208i	0.351899	+	0.936038i	$0.385536\pi$
$883$	49120.7i	1.87208i	−0.351899	+	0.936038i	$0.614464\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 40099.7i	− 1.51794i	−0.651124	−	0.758971i	$-0.725702\pi$
$887$	− 40099.7i	− 1.51794i	0.651124	−	0.758971i	$-0.274298\pi$
$888$	0	0
$889$	−93906.9	−3.54279
$890$	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
$903$	− 174729.i	− 6.43920i
$904$	0	0
$905$	−12052.4	−0.442691
$906$	0	0
$907$	− 45878.4i	− 1.67957i	−0.542921	−	0.839784i	$-0.682682\pi$
$907$	− 45878.4i	− 1.67957i	0.542921	−	0.839784i	$-0.317318\pi$
$908$	0	0
$909$	22039.3	0.804177
$910$	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
$915$	4156.23i	0.150165i
$916$	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
$921$	90674.7	3.24412
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 38677.2i	− 1.37036i
$928$	0	0
$929$	18054.0	0.637603	0.318801	−	0.947822i	$-0.396720\pi$
$929$	18054.0	0.637603	0.318801	+	0.947822i	$0.396720\pi$
$930$	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
$943$	34027.8i	1.17508i
$944$	0	0
$945$	108606.	3.73856
$946$	0	0
$947$	55305.3i	1.89776i	0.315635	+	0.948881i	$0.397782\pi$
$947$	55305.3i	1.89776i	−0.315635	+	0.948881i	$0.602218\pi$
$948$	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
$963$	− 77463.7i	− 2.59214i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11343.4i	− 0.377228i	−0.982051	−	0.188614i	$-0.939600\pi$
$967$	− 11343.4i	− 0.377228i	0.982051	−	0.188614i	$-0.0603995\pi$
$968$	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
$981$	114990.	3.74245
$982$	0	0
$983$	60188.2i	1.95290i	0.215735	+	0.976452i	$0.430785\pi$
$983$	60188.2i	1.95290i	−0.215735	+	0.976452i	$0.569215\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12759.1i	0.411476i
$988$	0	0
$989$	−41596.9	−1.33742
$990$	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.129.4	yes	4
3.2	odd	2		1440.4.f.g.289.3		4
4.3	odd	2	inner	160.4.c.c.129.1	✓	4
5.2	odd	4		800.4.a.t.1.2		2
5.3	odd	4		800.4.a.l.1.1		2
5.4	even	2	inner	160.4.c.c.129.1	✓	4
8.3	odd	2		320.4.c.f.129.4		4
8.5	even	2		320.4.c.f.129.1		4
12.11	even	2		1440.4.f.g.289.4		4
15.14	odd	2		1440.4.f.g.289.4		4
20.3	even	4		800.4.a.t.1.2		2
20.7	even	4		800.4.a.l.1.1		2
20.19	odd	2	CM	160.4.c.c.129.4	yes	4
40.3	even	4		1600.4.a.cb.1.1		2
40.13	odd	4		1600.4.a.cp.1.2		2
40.19	odd	2		320.4.c.f.129.1		4
40.27	even	4		1600.4.a.cp.1.2		2
40.29	even	2		320.4.c.f.129.4		4
40.37	odd	4		1600.4.a.cb.1.1		2
60.59	even	2		1440.4.f.g.289.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.c.129.1	✓	4	4.3	odd	2	inner
160.4.c.c.129.1	✓	4	5.4	even	2	inner
160.4.c.c.129.4	yes	4	1.1	even	1	trivial
160.4.c.c.129.4	yes	4	20.19	odd	2	CM
320.4.c.f.129.1		4	8.5	even	2
320.4.c.f.129.1		4	40.19	odd	2
320.4.c.f.129.4		4	8.3	odd	2
320.4.c.f.129.4		4	40.29	even	2
800.4.a.l.1.1		2	5.3	odd	4
800.4.a.l.1.1		2	20.7	even	4
800.4.a.t.1.2		2	5.2	odd	4
800.4.a.t.1.2		2	20.3	even	4
1440.4.f.g.289.3		4	3.2	odd	2
1440.4.f.g.289.3		4	60.59	even	2
1440.4.f.g.289.4		4	12.11	even	2
1440.4.f.g.289.4		4	15.14	odd	2
1600.4.a.cb.1.1		2	40.3	even	4
1600.4.a.cb.1.1		2	40.37	odd	4
1600.4.a.cp.1.2		2	40.13	odd	4
1600.4.a.cp.1.2		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)

\(f(q)\)	\(=\)	\(q+9.23607i q^{3} -11.1803 q^{5} -33.5967i q^{7} -58.3050 q^{9} +O(q^{10})\)	\(q+9.23607i q^{3} -11.1803 q^{5} -33.5967i q^{7} -58.3050 q^{9} -103.262i q^{15} +310.302 q^{21} -73.8723i q^{23} +125.000 q^{25} -289.135i q^{27} -306.000 q^{29} +375.623i q^{35} -460.630 q^{41} -563.092i q^{43} +651.869 q^{45} +41.1184i q^{47} -785.741 q^{49} -40.2492 q^{61} +1958.86i q^{63} -1.16335i q^{67} +682.289 q^{69} +1154.51i q^{75} +1096.23 q^{81} +989.604i q^{83} -2826.24i q^{87} +1386.00 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} + 472 q^{21} + 500 q^{25} - 1224 q^{29} + 1400 q^{45} - 1372 q^{49} - 616 q^{69} + 1004 q^{81} + 5544 q^{89}+O(q^{100}) \) 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

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