LMFDB - Embedded newform 176.2.e.b.175.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("176.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	176.e (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:	$1.40536707557$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			175.1
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	176.175
Dual form			176.2.e.b.175.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52434i q^{3} -4.37228 q^{5} -3.37228 q^{9} -3.31662i q^{11} +11.0371i q^{15} -9.45254i q^{23} +14.1168 q^{25} +0.939764i q^{27} +0.644810i q^{31} -8.37228 q^{33} +5.11684 q^{37} +14.7446 q^{45} +6.63325i q^{47} -7.00000 q^{49} +6.00000 q^{53} +14.5012i q^{55} +11.3321i q^{59} -6.28339i q^{67} -23.8614 q^{69} -5.69349i q^{71} -35.6357i q^{75} -7.74456 q^{81} -9.86141 q^{89} +1.62772 q^{93} +17.1168 q^{97} +11.1846i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} - 14 q^{37} + 36 q^{45} - 28 q^{49} + 24 q^{53} - 38 q^{69} - 8 q^{81} + 18 q^{89} + 18 q^{93} + 34 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 - 2 * q^9 + 22 * q^25 - 22 * q^33 - 14 * q^37 + 36 * q^45 - 28 * q^49 + 24 * q^53 - 38 * q^69 - 8 * q^81 + 18 * q^89 + 18 * q^93 + 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	− 2.52434i	− 1.45743i	−0.684819	−	0.728714i	$-0.740119\pi$
$3$	− 2.52434i	− 1.45743i	0.684819	−	0.728714i	$-0.259881\pi$
$4$	0	0
$5$	−4.37228	−1.95534	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−4.37228	−1.95534	−0.977672	+	0.210138i	$0.932609\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.37228	−1.12409
$10$	0	0
$11$	− 3.31662i	− 1.00000i
$11$	− 3.31662i	− 1.00000i
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	11.0371i	2.84977i
$16$	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
$21$	0	0
$22$	0	0
$23$	− 9.45254i	− 1.97099i	−0.169701	−	0.985496i	$-0.554280\pi$
$23$	− 9.45254i	− 1.97099i	0.169701	−	0.985496i	$-0.445720\pi$
$24$	0	0
$25$	14.1168	2.82337
$26$	0	0
$27$	0.939764i	0.180858i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0.644810i	0.115811i	0.998322	+	0.0579057i	$0.0184423\pi$
$31$	0.644810i	0.115811i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$	0	0
$33$	−8.37228	−1.45743
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.11684	0.841204	0.420602	−	0.907245i	$-0.361819\pi$
$37$	5.11684	0.841204	0.420602	+	0.907245i	$0.361819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	14.7446	2.19799
$46$	0	0
$47$	6.63325i	0.967559i	0.875190	+	0.483779i	$0.160736\pi$
$47$	6.63325i	0.967559i	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	14.5012i	1.95534i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3321i	1.47531i	0.675178	+	0.737655i	$0.264067\pi$
$59$	11.3321i	1.47531i	−0.675178	+	0.737655i	$0.735933\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.28339i	− 0.767639i	−0.923408	−	0.383819i	$-0.874609\pi$
$67$	− 6.28339i	− 0.767639i	0.923408	−	0.383819i	$-0.125391\pi$
$68$	0	0
$69$	−23.8614	−2.87258
$70$	0	0
$71$	− 5.69349i	− 0.675692i	−0.941201	−	0.337846i	$-0.890302\pi$
$71$	− 5.69349i	− 0.675692i	0.941201	−	0.337846i	$-0.109698\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	− 35.6357i	− 4.11485i
$76$	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
$81$	−7.74456	−0.860507
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.86141	−1.04531	−0.522654	−	0.852545i	$-0.675058\pi$
$89$	−9.86141	−1.04531	−0.522654	+	0.852545i	$0.675058\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.62772	0.168787
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.1168	1.73795	0.868976	−	0.494854i	$-0.164778\pi$
$97$	17.1168	1.73795	0.868976	+	0.494854i	$0.164778\pi$
$98$	0	0
$99$	11.1846i	1.12409i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 19.8997i	− 1.96078i	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	− 19.8997i	− 1.96078i	0.197066	−	0.980390i	$-0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	− 12.9166i	− 1.22599i
$112$	0	0
$113$	7.62772	0.717555	0.358778	−	0.933423i	$-0.383194\pi$
$113$	7.62772	0.717555	0.358778	+	0.933423i	$0.383194\pi$
$114$	0	0
$115$	41.3292i	3.85396i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−39.8614	−3.56531
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	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
$135$	− 4.10891i	− 0.353639i
$136$	0	0
$137$	18.6060	1.58962	0.794808	−	0.606861i	$-0.207572\pi$
$137$	18.6060	1.58962	0.794808	+	0.606861i	$0.207572\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	16.7446	1.41015
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.6704i	1.45743i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	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
$155$	− 2.81929i	− 0.226451i
$156$	0	0
$157$	−2.88316	−0.230101	−0.115050	−	0.993360i	$-0.536703\pi$
$157$	−2.88316	−0.230101	−0.115050	+	0.993360i	$0.536703\pi$
$158$	0	0
$159$	− 15.1460i	− 1.20116i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.8997i	1.55867i	0.626608	+	0.779334i	$0.284443\pi$
$163$	19.8997i	1.55867i	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	36.6060	2.84977
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	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
$175$	0	0
$176$	0	0
$177$	28.6060	2.15016
$178$	0	0
$179$	− 26.4781i	− 1.97907i	−0.144308	−	0.989533i	$-0.546095\pi$
$179$	− 26.4781i	− 1.97907i	0.144308	−	0.989533i	$-0.453905\pi$
$180$	0	0
$181$	21.1168	1.56960	0.784801	−	0.619747i	$-0.212765\pi$
$181$	21.1168	1.56960	0.784801	+	0.619747i	$0.212765\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−22.3723	−1.64484
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.5986i	1.77989i	0.456068	+	0.889945i	$0.349257\pi$
$191$	24.5986i	1.77989i	−0.456068	+	0.889945i	$0.650743\pi$
$192$	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
$199$	− 19.8997i	− 1.41066i	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	− 19.8997i	− 1.41066i	0.708881	−	0.705328i	$-0.249200\pi$
$200$	0	0
$201$	−15.8614	−1.11878
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	31.8766i	2.21558i
$208$	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
$213$	−14.3723	−0.984772
$214$	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
$223$	− 15.7908i	− 1.05743i	−0.848799	−	0.528716i	$-0.822674\pi$
$223$	− 15.7908i	− 1.05743i	0.848799	−	0.528716i	$-0.177326\pi$
$224$	0	0
$225$	−47.6060	−3.17373
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−23.3505	−1.54305	−0.771523	−	0.636201i	$-0.780505\pi$
$229$	−23.3505	−1.54305	−0.771523	+	0.636201i	$0.780505\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	− 29.0024i	− 1.89191i
$236$	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
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	22.3692i	1.43498i
$244$	0	0
$245$	30.6060	1.95534
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.0911i	0.952543i	0.879298	+	0.476272i	$0.158012\pi$
$251$	15.0911i	0.952543i	−0.879298	+	0.476272i	$0.841988\pi$
$252$	0	0
$253$	−31.3505	−1.97099
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−26.2337	−1.61152
$266$	0	0
$267$	24.8935i	1.52346i
$268$	0	0
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 46.8203i	− 2.82337i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	− 2.17448i	− 0.130183i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	− 43.2087i	− 2.53294i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	− 49.5470i	− 2.88474i
$296$	0	0
$297$	3.11684	0.180858
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−50.2337	−2.85769
$310$	0	0
$311$	33.1662i	1.88069i	0.340229	+	0.940343i	$0.389495\pi$
$311$	33.1662i	1.88069i	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−35.3505	−1.99813	−0.999065	−	0.0432311i	$-0.986235\pi$
$313$	−35.3505	−1.99813	−0.999065	+	0.0432311i	$0.986235\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.60597	0.371028	0.185514	−	0.982642i	$-0.440605\pi$
$317$	6.60597	0.371028	0.185514	+	0.982642i	$0.440605\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	35.2858i	1.93948i	0.244131	+	0.969742i	$0.421497\pi$
$331$	35.2858i	1.93948i	−0.244131	+	0.969742i	$0.578503\pi$
$332$	0	0
$333$	−17.2554	−0.945592
$334$	0	0
$335$	27.4728i	1.50100i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	− 19.2549i	− 1.04578i
$340$	0	0
$341$	2.13859	0.115811
$342$	0	0
$343$	0	0
$344$	0	0
$345$	104.329	5.61687
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	36.0951	1.92115	0.960574	−	0.278024i	$-0.0896796\pi$
$353$	36.0951	1.92115	0.960574	+	0.278024i	$0.0896796\pi$
$354$	0	0
$355$	24.8935i	1.32121i
$356$	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
$363$	27.7677i	1.45743i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 27.0680i	− 1.41294i	−0.707744	−	0.706469i	$-0.750287\pi$
$367$	− 27.0680i	− 1.41294i	0.707744	−	0.706469i	$-0.249713\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	100.624i	5.19618i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 36.5754i	− 1.87875i	−0.342885	−	0.939377i	$-0.611404\pi$
$379$	− 36.5754i	− 1.87875i	0.342885	−	0.939377i	$-0.388596\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.1167i	1.64109i	0.571585	+	0.820543i	$0.306329\pi$
$383$	32.1167i	1.64109i	−0.571585	+	0.820543i	$0.693671\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.0951	1.22167	0.610835	−	0.791758i	$-0.290834\pi$
$389$	24.0951	1.22167	0.610835	+	0.791758i	$0.290834\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	33.8614	1.68259
$406$	0	0
$407$	− 16.9707i	− 0.841204i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	− 46.9678i	− 2.31675i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 33.1662i	− 1.62028i	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	− 33.1662i	− 1.62028i	0.586238	−	0.810139i	$-0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	− 22.3692i	− 1.08763i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	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
$433$	−11.3505	−0.545472	−0.272736	−	0.962089i	$-0.587929\pi$
$433$	−11.3505	−0.545472	−0.272736	+	0.962089i	$0.587929\pi$
$434$	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
$441$	23.6060	1.12409
$442$	0	0
$443$	0.0549029i	0.00260851i	0.999999	+	0.00130426i	$0.000415158\pi$
$443$	0.0549029i	0.00260851i	−0.999999	+	0.00130426i	$0.999585\pi$
$444$	0	0
$445$	43.1168	2.04393
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.8614	−1.59802	−0.799009	−	0.601319i	$-0.794642\pi$
$449$	−33.8614	−1.59802	−0.799009	+	0.601319i	$0.794642\pi$
$450$	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
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	11.9220i	0.554061i	0.960861	+	0.277031i	$0.0893503\pi$
$463$	11.9220i	0.554061i	−0.960861	+	0.277031i	$0.910650\pi$
$464$	0	0
$465$	−7.11684	−0.330036
$466$	0	0
$467$	− 18.9600i	− 0.877363i	−0.898642	−	0.438682i	$-0.855446\pi$
$467$	− 18.9600i	− 0.877363i	0.898642	−	0.438682i	$-0.144554\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.27806i	0.335355i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−20.2337	−0.926437
$478$	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
$483$	0	0
$484$	0	0
$485$	−74.8397	−3.39829
$486$	0	0
$487$	42.2140i	1.91290i	0.291896	+	0.956450i	$0.405714\pi$
$487$	42.2140i	1.91290i	−0.291896	+	0.956450i	$0.594286\pi$
$488$	0	0
$489$	50.2337	2.27165
$490$	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
$495$	− 48.9022i	− 2.19799i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.8997i	0.890835i	0.895323	+	0.445418i	$0.146945\pi$
$499$	19.8997i	0.890835i	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 32.8164i	− 1.45743i
$508$	0	0
$509$	19.6277	0.869983	0.434992	−	0.900434i	$-0.356751\pi$
$509$	19.6277	0.869983	0.434992	+	0.900434i	$0.356751\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	87.0073i	3.83400i
$516$	0	0
$517$	22.0000	0.967559
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−44.8397	−1.96446	−0.982231	−	0.187678i	$-0.939904\pi$
$521$	−44.8397	−1.96446	−0.982231	+	0.187678i	$0.939904\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−66.3505	−2.88481
$530$	0	0
$531$	− 38.2149i	− 1.65839i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−66.8397	−2.88434
$538$	0	0
$539$	23.2164i	1.00000i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	− 53.3060i	− 2.28758i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	56.4752i	2.39724i
$556$	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
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−33.3505	−1.40307
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	62.0951	2.59406
$574$	0	0
$575$	− 133.440i	− 5.56483i
$576$	0	0
$577$	−14.8832	−0.619594	−0.309797	−	0.950803i	$-0.600261\pi$
$577$	−14.8832	−0.619594	−0.309797	+	0.950803i	$0.600261\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 19.8997i	− 0.824163i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.63325i	− 0.273784i	−0.990586	−	0.136892i	$-0.956289\pi$
$587$	− 6.63325i	− 0.273784i	0.990586	−	0.136892i	$-0.0437113\pi$
$588$	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
$597$	−50.2337	−2.05593
$598$	0	0
$599$	33.1662i	1.35514i	0.735460	+	0.677568i	$0.236966\pi$
$599$	33.1662i	1.35514i	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	21.1894i	0.862898i
$604$	0	0
$605$	48.0951	1.95534
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	24.0087i	0.964990i	0.875899	+	0.482495i	$0.160269\pi$
$619$	24.0087i	0.964990i	−0.875899	+	0.482495i	$0.839731\pi$
$620$	0	0
$621$	8.88316	0.356469
$622$	0	0
$623$	0	0
$624$	0	0
$625$	103.701	4.14804
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	30.9369i	1.23158i	0.787911	+	0.615789i	$0.211162\pi$
$631$	30.9369i	1.23158i	−0.787911	+	0.615789i	$0.788838\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	19.2000i	0.759542i
$640$	0	0
$641$	42.6060	1.68283	0.841417	−	0.540386i	$-0.181722\pi$
$641$	42.6060	1.68283	0.841417	+	0.540386i	$0.181722\pi$
$642$	0	0
$643$	− 50.4319i	− 1.98884i	−0.105502	−	0.994419i	$-0.533645\pi$
$643$	− 50.4319i	− 1.98884i	0.105502	−	0.994419i	$-0.466355\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.82462i	0.0717334i	0.999357	+	0.0358667i	$0.0114192\pi$
$647$	1.82462i	0.0717334i	−0.999357	+	0.0358667i	$0.988581\pi$
$648$	0	0
$649$	37.5842	1.47531
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.3723	−1.11029	−0.555147	−	0.831753i	$-0.687338\pi$
$653$	−28.3723	−1.11029	−0.555147	+	0.831753i	$0.687338\pi$
$654$	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
$661$	49.5842	1.92860	0.964301	−	0.264807i	$-0.0853084\pi$
$661$	49.5842	1.92860	0.964301	+	0.264807i	$0.0853084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−39.8614	−1.54113
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	13.2665i	0.510628i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	46.4327i	1.77670i	0.459167	+	0.888350i	$0.348148\pi$
$683$	46.4327i	1.77670i	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−81.3505	−3.10824
$686$	0	0
$687$	58.9446i	2.24888i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	10.1523i	0.386210i	0.981178	+	0.193105i	$0.0618558\pi$
$691$	10.1523i	0.386210i	−0.981178	+	0.193105i	$0.938144\pi$
$692$	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
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−73.2119	−2.75732
$706$	0	0
$707$	0	0
$708$	0	0
$709$	33.5842	1.26128	0.630641	−	0.776075i	$-0.282792\pi$
$709$	33.5842	1.26128	0.630641	+	0.776075i	$0.282792\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.09509	0.228263
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.8757i	1.33794i	0.743290	+	0.668970i	$0.233264\pi$
$719$	35.8757i	1.33794i	−0.743290	+	0.668970i	$0.766736\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 40.9244i	− 1.51780i	−0.651206	−	0.758901i	$-0.725737\pi$
$727$	− 40.9244i	− 1.51780i	0.651206	−	0.758901i	$-0.274263\pi$
$728$	0	0
$729$	33.2337	1.23088
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	− 77.2598i	− 2.84977i
$736$	0	0
$737$	−20.8397	−0.767639
$738$	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
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	44.7933i	1.63453i	0.576262	+	0.817265i	$0.304511\pi$
$751$	44.7933i	1.63453i	−0.576262	+	0.817265i	$0.695489\pi$
$752$	0	0
$753$	38.0951	1.38826
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	79.1393i	2.87258i
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	− 45.4381i	− 1.63641i
$772$	0	0
$773$	54.0000	1.94225	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000	1.94225	0.971123	+	0.238581i	$0.0766824\pi$
$774$	0	0
$775$	9.10268i	0.326978i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−18.8832	−0.675692
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.6060	0.449926
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	66.2227i	2.34868i
$796$	0	0
$797$	−50.3288	−1.78274	−0.891368	−	0.453279i	$-0.850254\pi$
$797$	−50.3288	−1.78274	−0.891368	+	0.453279i	$0.850254\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	33.2554	1.17502
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 75.7301i	− 2.66583i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	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
$815$	− 87.0073i	− 3.04773i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 57.3601i	− 1.99945i	−0.0235383	−	0.999723i	$-0.507493\pi$
$823$	− 57.3601i	− 1.99945i	0.0235383	−	0.999723i	$-0.492507\pi$
$824$	0	0
$825$	−118.190	−4.11485
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	57.5842	1.99998	0.999991	−	0.00416865i	$-0.00132693\pi$
$829$	57.5842	1.99998	0.999991	+	0.00416865i	$0.00132693\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.605969	−0.0209454
$838$	0	0
$839$	− 20.7297i	− 0.715669i	−0.933785	−	0.357834i	$-0.883515\pi$
$839$	− 20.7297i	− 0.715669i	0.933785	−	0.357834i	$-0.116485\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−56.8397	−1.95534
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 48.3672i	− 1.65801i
$852$	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
$859$	51.7215i	1.76471i	0.470581	+	0.882357i	$0.344044\pi$
$859$	51.7215i	1.76471i	−0.470581	+	0.882357i	$0.655956\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 46.4327i	− 1.58059i	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	− 46.4327i	− 1.58059i	0.612727	−	0.790295i	$-0.290072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 42.9137i	− 1.45743i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−57.7228	−1.95362
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.1386	0.476341	0.238171	−	0.971223i	$-0.423452\pi$
$881$	14.1386	0.476341	0.238171	+	0.971223i	$0.423452\pi$
$882$	0	0
$883$	19.8997i	0.669680i	0.942275	+	0.334840i	$0.108682\pi$
$883$	19.8997i	0.669680i	−0.942275	+	0.334840i	$0.891318\pi$
$884$	0	0
$885$	−125.073	−4.20429
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	25.6858i	0.860507i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	115.770i	3.86975i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−92.3288	−3.06911
$906$	0	0
$907$	− 59.6992i	− 1.98228i	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	− 59.6992i	− 1.98228i	0.132818	−	0.991140i	$-0.457597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.63325i	0.219769i	0.993944	+	0.109885i	$0.0350482\pi$
$911$	6.63325i	0.219769i	−0.993944	+	0.109885i	$0.964952\pi$
$912$	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
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	72.2337	2.37503
$926$	0	0
$927$	67.1076i	2.20410i
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	83.7228	2.74096
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	89.2367i	2.91213i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 37.7553i	− 1.22688i	−0.789741	−	0.613441i	$-0.789785\pi$
$947$	− 37.7553i	− 1.22688i	0.789741	−	0.613441i	$-0.210215\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 16.6757i	− 0.540747i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 107.552i	− 3.48030i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	30.5842	0.986588
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 60.5292i	− 1.94247i	−0.238114	−	0.971237i	$-0.576529\pi$
$971$	− 60.5292i	− 1.94247i	0.238114	−	0.971237i	$-0.423471\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−62.3288	−1.99407	−0.997037	−	0.0769208i	$-0.975491\pi$
$977$	−62.3288	−1.99407	−0.997037	+	0.0769208i	$0.975491\pi$
$978$	0	0
$979$	32.7066i	1.04531i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	62.4087i	1.99053i	0.0972017	+	0.995265i	$0.469011\pi$
$983$	62.4087i	1.99053i	−0.0972017	+	0.995265i	$0.530989\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	59.6992i	1.89641i	0.317660	+	0.948205i	$0.397103\pi$
$991$	59.6992i	1.89641i	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	89.0733	2.82666
$994$	0	0
$995$	87.0073i	2.75832i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	4.80862i	0.152138i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.2.e.b.175.1	✓	4
3.2	odd	2		1584.2.o.e.703.4		4
4.3	odd	2	inner	176.2.e.b.175.4	yes	4
8.3	odd	2		704.2.e.c.703.1		4
8.5	even	2		704.2.e.c.703.4		4
11.10	odd	2	CM	176.2.e.b.175.1	✓	4
12.11	even	2		1584.2.o.e.703.3		4
16.3	odd	4		2816.2.g.c.1407.2		8
16.5	even	4		2816.2.g.c.1407.1		8
16.11	odd	4		2816.2.g.c.1407.7		8
16.13	even	4		2816.2.g.c.1407.8		8
33.32	even	2		1584.2.o.e.703.4		4
44.43	even	2	inner	176.2.e.b.175.4	yes	4
88.21	odd	2		704.2.e.c.703.4		4
88.43	even	2		704.2.e.c.703.1		4
132.131	odd	2		1584.2.o.e.703.3		4
176.21	odd	4		2816.2.g.c.1407.1		8
176.43	even	4		2816.2.g.c.1407.7		8
176.109	odd	4		2816.2.g.c.1407.8		8
176.131	even	4		2816.2.g.c.1407.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.2.e.b.175.1	✓	4	1.1	even	1	trivial
176.2.e.b.175.1	✓	4	11.10	odd	2	CM
176.2.e.b.175.4	yes	4	4.3	odd	2	inner
176.2.e.b.175.4	yes	4	44.43	even	2	inner
704.2.e.c.703.1		4	8.3	odd	2
704.2.e.c.703.1		4	88.43	even	2
704.2.e.c.703.4		4	8.5	even	2
704.2.e.c.703.4		4	88.21	odd	2
1584.2.o.e.703.3		4	12.11	even	2
1584.2.o.e.703.3		4	132.131	odd	2
1584.2.o.e.703.4		4	3.2	odd	2
1584.2.o.e.703.4		4	33.32	even	2
2816.2.g.c.1407.1		8	16.5	even	4
2816.2.g.c.1407.1		8	176.21	odd	4
2816.2.g.c.1407.2		8	16.3	odd	4
2816.2.g.c.1407.2		8	176.131	even	4
2816.2.g.c.1407.7		8	16.11	odd	4
2816.2.g.c.1407.7		8	176.43	even	4
2816.2.g.c.1407.8		8	16.13	even	4
2816.2.g.c.1407.8		8	176.109	odd	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:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	176.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.52434i q^{3} -4.37228 q^{5} -3.37228 q^{9} -3.31662i q^{11} +11.0371i q^{15} -9.45254i q^{23} +14.1168 q^{25} +0.939764i q^{27} +0.644810i q^{31} -8.37228 q^{33} +5.11684 q^{37} +14.7446 q^{45} +6.63325i q^{47} -7.00000 q^{49} +6.00000 q^{53} +14.5012i q^{55} +11.3321i q^{59} -6.28339i q^{67} -23.8614 q^{69} -5.69349i q^{71} -35.6357i q^{75} -7.74456 q^{81} -9.86141 q^{89} +1.62772 q^{93} +17.1168 q^{97} +11.1846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} - 14 q^{37} + 36 q^{45} - 28 q^{49} + 24 q^{53} - 38 q^{69} - 8 q^{81} + 18 q^{89} + 18 q^{93} + 34 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 - 2 * q^9 + 22 * q^25 - 22 * q^33 - 14 * q^37 + 36 * q^45 - 28 * q^49 + 24 * q^53 - 38 * q^69 - 8 * q^81 + 18 * q^89 + 18 * q^93 + 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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