LMFDB - Embedded newform 960.2.o.a.959.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(959,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.959");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.o (of order $2$, degree $1$, not 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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			959.1
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	960.959
Dual form			960.2.o.a.959.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} +3.87298i q^{15} -4.47214 q^{17} +7.74597i q^{19} -3.46410i q^{23} +5.00000 q^{25} +5.19615i q^{27} +7.74597i q^{31} +6.70820 q^{45} +10.3923i q^{47} -7.00000 q^{49} +7.74597i q^{51} -4.47214 q^{53} +13.4164 q^{57} +2.00000 q^{61} -6.00000 q^{69} -8.66025i q^{75} +7.74597i q^{79} +9.00000 q^{81} -3.46410i q^{83} +10.0000 q^{85} +13.4164 q^{93} -17.3205i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 20 q^{25} - 28 q^{49} + 8 q^{61} - 24 q^{69} + 36 q^{81} + 40 q^{85}+O(q^{100}) $ 4 * q - 12 * q^9 + 20 * q^25 - 28 * q^49 + 8 * q^61 - 24 * q^69 + 36 * q^81 + 40 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	− 1.73205i	− 1.00000i
$3$	− 1.73205i	− 1.00000i
$4$	0	0
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	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
$15$	3.87298i	1.00000i
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	7.74597i	1.77705i	0.458831	+	0.888523i	$0.348268\pi$
$19$	7.74597i	1.77705i	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.46410i	− 0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	− 3.46410i	− 0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	5.19615i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	7.74597i	1.39122i	0.718421	+	0.695608i	$0.244865\pi$
$31$	7.74597i	1.39122i	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	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
$45$	6.70820	1.00000
$46$	0	0
$47$	10.3923i	1.51587i	0.652328	+	0.757937i	$0.273792\pi$
$47$	10.3923i	1.51587i	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	7.74597i	1.08465i
$52$	0	0
$53$	−4.47214	−0.614295	−0.307148	−	0.951662i	$-0.599375\pi$
$53$	−4.47214	−0.614295	−0.307148	+	0.951662i	$0.599375\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.4164	1.77705
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	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
$75$	− 8.66025i	− 1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.74597i	0.871489i	0.900070	+	0.435745i	$0.143515\pi$
$79$	7.74597i	0.871489i	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 3.46410i	− 0.380235i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$	− 3.46410i	− 0.380235i	0.981761	−	0.190117i	$-0.0608868\pi$
$84$	0	0
$85$	10.0000	1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13.4164	1.39122
$94$	0	0
$95$	− 17.3205i	− 1.77705i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.3923i	1.00466i	0.864675	+	0.502331i	$0.167524\pi$
$107$	10.3923i	1.00466i	−0.864675	+	0.502331i	$0.832476\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.47214	−0.420703	−0.210352	−	0.977626i	$-0.567461\pi$
$113$	−4.47214	−0.420703	−0.210352	+	0.977626i	$0.567461\pi$
$114$	0	0
$115$	7.74597i	0.722315i
$116$	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
$125$	−11.1803	−1.00000
$126$	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
$135$	− 11.6190i	− 1.00000i
$136$	0	0
$137$	−22.3607	−1.91040	−0.955201	−	0.295958i	$-0.904361\pi$
$137$	−22.3607	−1.91040	−0.955201	+	0.295958i	$0.904361\pi$
$138$	0	0
$139$	− 23.2379i	− 1.97101i	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	− 23.2379i	− 1.97101i	0.169638	−	0.985506i	$-0.445740\pi$
$140$	0	0
$141$	18.0000	1.51587
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.1244i	1.00000i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	− 23.2379i	− 1.89107i	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	− 23.2379i	− 1.89107i	0.325515	−	0.945537i	$-0.394462\pi$
$152$	0	0
$153$	13.4164	1.08465
$154$	0	0
$155$	− 17.3205i	− 1.39122i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	7.74597i	0.614295i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.2487i	1.87642i	0.346064	+	0.938211i	$0.387518\pi$
$167$	24.2487i	1.87642i	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 23.2379i	− 1.77705i
$172$	0	0
$173$	−22.3607	−1.70005	−0.850026	−	0.526742i	$-0.823414\pi$
$173$	−22.3607	−1.70005	−0.850026	+	0.526742i	$0.823414\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	− 3.46410i	− 0.256074i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	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
$197$	−4.47214	−0.318626	−0.159313	−	0.987228i	$-0.550928\pi$
$197$	−4.47214	−0.318626	−0.159313	+	0.987228i	$0.550928\pi$
$198$	0	0
$199$	− 23.2379i	− 1.64729i	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	− 23.2379i	− 1.64729i	0.567105	−	0.823646i	$-0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.3923i	0.722315i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.74597i	0.533254i	0.963800	+	0.266627i	$0.0859092\pi$
$211$	7.74597i	0.533254i	−0.963800	+	0.266627i	$0.914091\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	24.2487i	1.60944i	0.593652	+	0.804722i	$0.297686\pi$
$227$	24.2487i	1.60944i	−0.593652	+	0.804722i	$0.702314\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.3607	−1.46490	−0.732448	−	0.680823i	$-0.761622\pi$
$233$	−22.3607	−1.46490	−0.732448	+	0.680823i	$0.761622\pi$
$234$	0	0
$235$	− 23.2379i	− 1.51587i
$236$	0	0
$237$	13.4164	0.871489
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 15.5885i	− 1.00000i
$244$	0	0
$245$	15.6525	1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	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
$255$	− 17.3205i	− 1.08465i
$256$	0	0
$257$	31.3050	1.95275	0.976375	−	0.216085i	$-0.0693287\pi$
$257$	31.3050	1.95275	0.976375	+	0.216085i	$0.0693287\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 31.1769i	− 1.92245i	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	− 31.1769i	− 1.92245i	0.275764	−	0.961225i	$-0.411069\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	7.74597i	0.470534i	0.971931	+	0.235267i	$0.0755965\pi$
$271$	7.74597i	0.470534i	−0.971931	+	0.235267i	$0.924404\pi$
$272$	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
$279$	− 23.2379i	− 1.39122i
$280$	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
$285$	−30.0000	−1.77705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	31.3050	1.82885	0.914427	−	0.404750i	$-0.132641\pi$
$293$	31.3050	1.82885	0.914427	+	0.404750i	$0.132641\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.47214	−0.256074
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	−22.3607	−1.25590	−0.627950	−	0.778253i	$-0.716106\pi$
$317$	−22.3607	−1.25590	−0.627950	+	0.778253i	$0.716106\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	− 34.6410i	− 1.92748i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	24.2487i	1.34096i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 23.2379i	− 1.27727i	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	− 23.2379i	− 1.27727i	0.769510	−	0.638635i	$-0.220501\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	7.74597i	0.420703i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	13.4164	0.722315
$346$	0	0
$347$	10.3923i	0.557888i	0.960307	+	0.278944i	$0.0899844\pi$
$347$	10.3923i	0.557888i	−0.960307	+	0.278944i	$0.910016\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.3050	1.66619	0.833097	−	0.553127i	$-0.186565\pi$
$353$	31.3050	1.66619	0.833097	+	0.553127i	$0.186565\pi$
$354$	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
$361$	−41.0000	−2.15789
$362$	0	0
$363$	19.0526i	1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	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
$375$	19.3649i	1.00000i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	38.7298i	1.98942i	0.102733	+	0.994709i	$0.467241\pi$
$379$	38.7298i	1.98942i	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	38.1051i	1.94708i	0.228515	+	0.973540i	$0.426613\pi$
$383$	38.1051i	1.94708i	−0.228515	+	0.973540i	$0.573387\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	15.4919i	0.783461i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 17.3205i	− 0.871489i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−20.1246	−1.00000
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	38.7298i	1.91040i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.74597i	0.380235i
$416$	0	0
$417$	−40.2492	−1.97101
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	0	0
$423$	− 31.1769i	− 1.51587i
$424$	0	0
$425$	−22.3607	−1.08465
$426$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.8328	1.28359
$438$	0	0
$439$	38.7298i	1.84847i	0.381819	+	0.924237i	$0.375298\pi$
$439$	38.7298i	1.84847i	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	0	0
$443$	38.1051i	1.81043i	0.424955	+	0.905214i	$0.360290\pi$
$443$	38.1051i	1.81043i	−0.424955	+	0.905214i	$0.639710\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−40.2492	−1.89107
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	− 23.2379i	− 1.08465i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−30.0000	−1.39122
$466$	0	0
$467$	24.2487i	1.12210i	0.827783	+	0.561048i	$0.189602\pi$
$467$	24.2487i	1.12210i	−0.827783	+	0.561048i	$0.810398\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	38.7298i	1.77705i
$476$	0	0
$477$	13.4164	0.614295
$478$	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
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	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
$499$	7.74597i	0.346757i	0.984855	+	0.173379i	$0.0554684\pi$
$499$	7.74597i	0.346757i	−0.984855	+	0.173379i	$0.944532\pi$
$500$	0	0
$501$	42.0000	1.87642
$502$	0	0
$503$	− 3.46410i	− 0.154457i	−0.997013	−	0.0772283i	$-0.975393\pi$
$503$	− 3.46410i	− 0.154457i	0.997013	−	0.0772283i	$-0.0246070\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 22.5167i	− 1.00000i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−40.2492	−1.77705
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	38.7298i	1.70005i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	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
$527$	− 34.6410i	− 1.50899i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 23.2379i	− 1.00466i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	38.1051i	1.63525i
$544$	0	0
$545$	31.3050	1.34096
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.3607	−0.947452	−0.473726	−	0.880672i	$-0.657091\pi$
$557$	−22.3607	−0.947452	−0.473726	+	0.880672i	$0.657091\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 31.1769i	− 1.31395i	−0.753912	−	0.656975i	$-0.771836\pi$
$563$	− 31.1769i	− 1.31395i	0.753912	−	0.656975i	$-0.228164\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	38.7298i	1.62079i	0.585882	+	0.810397i	$0.300748\pi$
$571$	38.7298i	1.62079i	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 17.3205i	− 0.722315i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 45.0333i	− 1.85872i	−0.369170	−	0.929362i	$-0.620358\pi$
$587$	− 45.0333i	− 1.85872i	0.369170	−	0.929362i	$-0.379642\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	0	0
$591$	7.74597i	0.318626i
$592$	0	0
$593$	−4.47214	−0.183649	−0.0918243	−	0.995775i	$-0.529270\pi$
$593$	−4.47214	−0.183649	−0.0918243	+	0.995775i	$0.529270\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−40.2492	−1.64729
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.5967	1.00000
$606$	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
$617$	49.1935	1.98046	0.990228	−	0.139459i	$-0.0445365\pi$
$617$	49.1935	1.98046	0.990228	+	0.139459i	$0.0445365\pi$
$618$	0	0
$619$	− 23.2379i	− 0.934010i	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	− 23.2379i	− 0.934010i	0.884255	−	0.467005i	$-0.154667\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	38.7298i	1.54181i	0.636950	+	0.770905i	$0.280196\pi$
$631$	38.7298i	1.54181i	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	13.4164	0.533254
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.2487i	0.953315i	0.879089	+	0.476658i	$0.158152\pi$
$647$	24.2487i	0.953315i	−0.879089	+	0.476658i	$0.841848\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	49.1935	1.92509	0.962545	−	0.271122i	$-0.0873945\pi$
$653$	49.1935	1.92509	0.962545	+	0.271122i	$0.0873945\pi$
$654$	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
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	25.9808i	1.00000i
$676$	0	0
$677$	31.3050	1.20315	0.601574	−	0.798817i	$-0.294541\pi$
$677$	31.3050	1.20315	0.601574	+	0.798817i	$0.294541\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	42.0000	1.60944
$682$	0	0
$683$	38.1051i	1.45805i	0.684486	+	0.729026i	$0.260027\pi$
$683$	38.1051i	1.45805i	−0.684486	+	0.729026i	$0.739973\pi$
$684$	0	0
$685$	50.0000	1.91040
$686$	0	0
$687$	− 45.0333i	− 1.71813i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	7.74597i	0.294670i	0.989087	+	0.147335i	$0.0470696\pi$
$691$	7.74597i	0.294670i	−0.989087	+	0.147335i	$0.952930\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	51.9615i	1.97101i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	38.7298i	1.46490i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−40.2492	−1.51587
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	− 23.2379i	− 0.871489i
$712$	0	0
$713$	26.8328	1.00490
$714$	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
$721$	0	0
$722$	0	0
$723$	− 3.46410i	− 0.128831i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	− 27.1109i	− 1.00000i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 54.2218i	− 1.99458i	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	− 54.2218i	− 1.99458i	0.0735712	−	0.997290i	$-0.476560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 31.1769i	− 1.14377i	−0.820334	−	0.571885i	$-0.806212\pi$
$743$	− 31.1769i	− 1.14377i	0.820334	−	0.571885i	$-0.193788\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.3923i	0.380235i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 54.2218i	− 1.97858i	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	− 54.2218i	− 1.97858i	0.145962	−	0.989290i	$-0.453372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	51.9615i	1.89107i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−30.0000	−1.08465
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	− 54.2218i	− 1.95275i
$772$	0	0
$773$	−4.47214	−0.160852	−0.0804258	−	0.996761i	$-0.525628\pi$
$773$	−4.47214	−0.160852	−0.0804258	+	0.996761i	$0.525628\pi$
$774$	0	0
$775$	38.7298i	1.39122i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	−54.0000	−1.92245
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 17.3205i	− 0.614295i
$796$	0	0
$797$	49.1935	1.74252	0.871262	−	0.490819i	$-0.163302\pi$
$797$	49.1935	1.74252	0.871262	+	0.490819i	$0.163302\pi$
$798$	0	0
$799$	− 46.4758i	− 1.64420i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 23.2379i	− 0.815993i	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	− 23.2379i	− 0.815993i	0.912983	−	0.407997i	$-0.133772\pi$
$812$	0	0
$813$	13.4164	0.470534
$814$	0	0
$815$	0	0
$816$	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
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 45.0333i	− 1.56596i	−0.622046	−	0.782981i	$-0.713698\pi$
$827$	− 45.0333i	− 1.56596i	0.622046	−	0.782981i	$-0.286302\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.3050	1.08465
$834$	0	0
$835$	− 54.2218i	− 1.87642i
$836$	0	0
$837$	−40.2492	−1.39122
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−29.0689	−1.00000
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	51.9615i	1.77705i
$856$	0	0
$857$	−58.1378	−1.98595	−0.992974	−	0.118331i	$-0.962245\pi$
$857$	−58.1378	−1.98595	−0.992974	+	0.118331i	$0.962245\pi$
$858$	0	0
$859$	38.7298i	1.32144i	0.750630	+	0.660722i	$0.229750\pi$
$859$	38.7298i	1.32144i	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.1051i	1.29711i	0.761166	+	0.648557i	$0.224627\pi$
$863$	38.1051i	1.29711i	−0.761166	+	0.648557i	$0.775373\pi$
$864$	0	0
$865$	50.0000	1.70005
$866$	0	0
$867$	− 5.19615i	− 0.176471i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	− 54.2218i	− 1.82885i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 58.8897i	− 1.97732i	−0.150160	−	0.988662i	$-0.547979\pi$
$887$	− 58.8897i	− 1.97732i	0.150160	−	0.988662i	$-0.452021\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−80.4984	−2.69378
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49.1935	1.63525
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	7.74597i	0.256074i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 23.2379i	− 0.766548i	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	− 23.2379i	− 0.766548i	0.923635	−	0.383274i	$-0.125203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 54.2218i	− 1.77705i
$932$	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
$941$	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
$947$	− 58.8897i	− 1.91366i	−0.290650	−	0.956830i	$-0.593871\pi$
$947$	− 58.8897i	− 1.91366i	0.290650	−	0.956830i	$-0.406129\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	38.7298i	1.25590i
$952$	0	0
$953$	−58.1378	−1.88327	−0.941634	−	0.336640i	$-0.890710\pi$
$953$	−58.1378	−1.88327	−0.941634	+	0.336640i	$0.890710\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	− 31.1769i	− 1.00466i
$964$	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
$969$	−60.0000	−1.92748
$970$	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
$977$	−4.47214	−0.143076	−0.0715382	−	0.997438i	$-0.522791\pi$
$977$	−4.47214	−0.143076	−0.0715382	+	0.997438i	$0.522791\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	42.0000	1.34096
$982$	0	0
$983$	− 3.46410i	− 0.110488i	−0.998473	−	0.0552438i	$-0.982406\pi$
$983$	− 3.46410i	− 0.110488i	0.998473	−	0.0552438i	$-0.0175936\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 54.2218i	− 1.72241i	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	− 54.2218i	− 1.72241i	0.508257	−	0.861206i	$-0.330290\pi$
$992$	0	0
$993$	−40.2492	−1.27727
$994$	0	0
$995$	51.9615i	1.64729i
$996$	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	960.2.o.a.959.1		4
3.2	odd	2	inner	960.2.o.a.959.4		4
4.3	odd	2	inner	960.2.o.a.959.3		4
5.4	even	2	inner	960.2.o.a.959.4		4
8.3	odd	2		60.2.h.b.59.2	yes	4
8.5	even	2		60.2.h.b.59.1	✓	4
12.11	even	2	inner	960.2.o.a.959.2		4
15.14	odd	2	CM	960.2.o.a.959.1		4
20.19	odd	2	inner	960.2.o.a.959.2		4
24.5	odd	2		60.2.h.b.59.4	yes	4
24.11	even	2		60.2.h.b.59.3	yes	4
40.3	even	4		300.2.e.a.251.4		4
40.13	odd	4		300.2.e.a.251.2		4
40.19	odd	2		60.2.h.b.59.3	yes	4
40.27	even	4		300.2.e.a.251.1		4
40.29	even	2		60.2.h.b.59.4	yes	4
40.37	odd	4		300.2.e.a.251.3		4
60.59	even	2	inner	960.2.o.a.959.3		4
120.29	odd	2		60.2.h.b.59.1	✓	4
120.53	even	4		300.2.e.a.251.3		4
120.59	even	2		60.2.h.b.59.2	yes	4
120.77	even	4		300.2.e.a.251.2		4
120.83	odd	4		300.2.e.a.251.1		4
120.107	odd	4		300.2.e.a.251.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.h.b.59.1	✓	4	8.5	even	2
60.2.h.b.59.1	✓	4	120.29	odd	2
60.2.h.b.59.2	yes	4	8.3	odd	2
60.2.h.b.59.2	yes	4	120.59	even	2
60.2.h.b.59.3	yes	4	24.11	even	2
60.2.h.b.59.3	yes	4	40.19	odd	2
60.2.h.b.59.4	yes	4	24.5	odd	2
60.2.h.b.59.4	yes	4	40.29	even	2
300.2.e.a.251.1		4	40.27	even	4
300.2.e.a.251.1		4	120.83	odd	4
300.2.e.a.251.2		4	40.13	odd	4
300.2.e.a.251.2		4	120.77	even	4
300.2.e.a.251.3		4	40.37	odd	4
300.2.e.a.251.3		4	120.53	even	4
300.2.e.a.251.4		4	40.3	even	4
300.2.e.a.251.4		4	120.107	odd	4
960.2.o.a.959.1		4	1.1	even	1	trivial
960.2.o.a.959.1		4	15.14	odd	2	CM
960.2.o.a.959.2		4	12.11	even	2	inner
960.2.o.a.959.2		4	20.19	odd	2	inner
960.2.o.a.959.3		4	4.3	odd	2	inner
960.2.o.a.959.3		4	60.59	even	2	inner
960.2.o.a.959.4		4	3.2	odd	2	inner
960.2.o.a.959.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} +3.87298i q^{15} -4.47214 q^{17} +7.74597i q^{19} -3.46410i q^{23} +5.00000 q^{25} +5.19615i q^{27} +7.74597i q^{31} +6.70820 q^{45} +10.3923i q^{47} -7.00000 q^{49} +7.74597i q^{51} -4.47214 q^{53} +13.4164 q^{57} +2.00000 q^{61} -6.00000 q^{69} -8.66025i q^{75} +7.74597i q^{79} +9.00000 q^{81} -3.46410i q^{83} +10.0000 q^{85} +13.4164 q^{93} -17.3205i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 20 q^{25} - 28 q^{49} + 8 q^{61} - 24 q^{69} + 36 q^{81} + 40 q^{85}+O(q^{100}) \) 4 * q - 12 * q^9 + 20 * q^25 - 28 * q^49 + 8 * q^61 - 24 * q^69 + 36 * q^81 + 40 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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