LMFDB - Embedded newform 2156.2.c.b.1077.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(1077,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2156.1077");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.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:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.395868833065051360220676096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12 x^{14} + 134 x^{12} - 24 x^{11} - 524 x^{10} - 648 x^{9} + 567 x^{8} - 384 x^{7} + \cdots + 97582 $ x^16 - 12x^14 + 134x^12 - 24x^11 - 524x^10 - 648x^9 + 567x^8 - 384x^7 + 1948x^6 - 14496x^5 + 71082x^4 + 7896x^3 - 6140x^2 + 127512*x + 97582
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1077.7
Root			$0.792287 - 1.21752i$ of defining polynomial
Character	$\chi$	$=$	2156.1077
Dual form			2156.2.c.b.1077.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	− 1.22568i	− 0.707647i	−0.935312	−	0.353823i	$-0.884881\pi$
$3$	− 1.22568i	− 0.707647i	0.935312	−	0.353823i	$-0.115119\pi$
$4$	0	0
$5$	− 2.54143i	− 1.13656i	−0.822835	−	0.568281i	$-0.807609\pi$
$5$	− 2.54143i	− 1.13656i	0.822835	−	0.568281i	$-0.192391\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.49771	0.499236
$10$	0	0
$11$	3.31662	1.00000
$11$	3.31662	1.00000
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−3.11498	−0.804284
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.83493	1.63370	0.816848	−	0.576854i	$-0.195720\pi$
$23$	7.83493	1.63370	0.816848	+	0.576854i	$0.195720\pi$
$24$	0	0
$25$	−1.45886	−0.291771
$26$	0	0
$27$	− 5.51275i	− 1.06093i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 4.84996i	− 0.871078i	−0.900170	−	0.435539i	$-0.856558\pi$
$31$	− 4.84996i	− 0.871078i	0.900170	−	0.435539i	$-0.143442\pi$
$32$	0	0
$33$	− 4.06512i	− 0.707647i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0371	1.81449	0.907245	−	0.420602i	$-0.138181\pi$
$37$	11.0371	1.81449	0.907245	+	0.420602i	$0.138181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 3.80632i	− 0.567413i
$46$	0	0
$47$	8.54812i	1.24687i	0.781875	+	0.623436i	$0.214264\pi$
$47$	8.54812i	1.24687i	−0.781875	+	0.623436i	$0.785736\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.13819	−0.705785	−0.352892	−	0.935664i	$-0.614802\pi$
$53$	−5.13819	−0.705785	−0.352892	+	0.935664i	$0.614802\pi$
$54$	0	0
$55$	− 8.42896i	− 1.13656i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.24614i	0.682990i	0.939884	+	0.341495i	$0.110933\pi$
$59$	5.24614i	0.682990i	−0.939884	+	0.341495i	$0.889067\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.24619	−0.763094	−0.381547	−	0.924349i	$-0.624609\pi$
$67$	−6.24619	−0.763094	−0.381547	+	0.924349i	$0.624609\pi$
$68$	0	0
$69$	− 9.60311i	− 1.15608i
$70$	0	0
$71$	−15.2416	−1.80885	−0.904423	−	0.426636i	$-0.859698\pi$
$71$	−15.2416	−1.80885	−0.904423	+	0.426636i	$0.859698\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	1.78809i	0.206471i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−2.26374	−0.251527
$82$	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
$89$	− 15.2665i	− 1.61825i	−0.587638	−	0.809124i	$-0.699942\pi$
$89$	− 15.2665i	− 1.61825i	0.587638	−	0.809124i	$-0.300058\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.94450	−0.616416
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.0837i	− 1.22691i	−0.789728	−	0.613457i	$-0.789778\pi$
$97$	− 12.0837i	− 1.22691i	0.789728	−	0.613457i	$-0.210222\pi$
$98$	0	0
$99$	4.96734	0.499236
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	16.8542i	1.66070i	0.557244	+	0.830349i	$0.311859\pi$
$103$	16.8542i	1.66070i	−0.557244	+	0.830349i	$0.688141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	− 13.5280i	− 1.28402i
$112$	0	0
$113$	−19.8448	−1.86685	−0.933423	−	0.358778i	$-0.883194\pi$
$113$	−19.8448	−1.86685	−0.933423	+	0.358778i	$0.883194\pi$
$114$	0	0
$115$	− 19.9119i	− 1.85679i
$116$	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
$125$	− 8.99956i	− 0.804945i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	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
$135$	−14.0103	−1.20581
$136$	0	0
$137$	−18.6060	−1.58962	−0.794808	−	0.606861i	$-0.792428\pi$
$137$	−18.6060	−1.58962	−0.794808	+	0.606861i	$0.792428\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	10.4773	0.882344
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−12.3258	−0.990034
$156$	0	0
$157$	− 21.8953i	− 1.74743i	−0.486435	−	0.873717i	$-0.661703\pi$
$157$	− 21.8953i	− 1.74743i	0.486435	−	0.873717i	$-0.338297\pi$
$158$	0	0
$159$	6.29778i	0.499446i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.75754	0.215987	0.107994	−	0.994152i	$-0.465557\pi$
$163$	2.75754	0.215987	0.107994	+	0.994152i	$0.465557\pi$
$164$	0	0
$165$	−10.3312	−0.804284
$166$	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		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.43009	0.483315
$178$	0	0
$179$	26.4781	1.97907	0.989533	−	0.144308i	$-0.0460955\pi$
$179$	26.4781	1.97907	0.989533	+	0.144308i	$0.0460955\pi$
$180$	0	0
$181$	25.8909i	1.92446i	0.272242	+	0.962229i	$0.412235\pi$
$181$	25.8909i	1.92446i	−0.272242	+	0.962229i	$0.587765\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 28.0500i	− 2.06228i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.6060	0.912136	0.456068	−	0.889945i	$-0.349257\pi$
$191$	12.6060	0.912136	0.456068	+	0.889945i	$0.349257\pi$
$192$	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
$199$	26.0929i	1.84968i	0.380361	+	0.924838i	$0.375800\pi$
$199$	26.0929i	1.84968i	−0.380361	+	0.924838i	$0.624200\pi$
$200$	0	0
$201$	7.65583i	0.540001i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	11.7344	0.815600
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	18.6813i	1.28002i
$214$	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
$223$	4.88760i	0.327298i	0.986519	+	0.163649i	$0.0523265\pi$
$223$	4.88760i	0.327298i	−0.986519	+	0.163649i	$0.947674\pi$
$224$	0	0
$225$	−2.18494	−0.145663
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	28.9416i	1.91252i	0.292524	+	0.956258i	$0.405505\pi$
$229$	28.9416i	1.91252i	−0.292524	+	0.956258i	$0.594495\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	21.7244	1.41715
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 13.7636i	− 0.882937i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.28028i	0.207050i	0.994627	+	0.103525i	$0.0330121\pi$
$251$	3.28028i	0.207050i	−0.994627	+	0.103525i	$0.966988\pi$
$252$	0	0
$253$	25.9855	1.63370
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 31.4016i	− 1.95878i	−0.201984	−	0.979389i	$-0.564739\pi$
$257$	− 31.4016i	− 1.95878i	0.201984	−	0.979389i	$-0.435261\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	13.0583i	0.802168i
$266$	0	0
$267$	−18.7119	−1.14515
$268$	0	0
$269$	− 23.7372i	− 1.44728i	−0.690178	−	0.723640i	$-0.742468\pi$
$269$	− 23.7372i	− 1.44728i	0.690178	−	0.723640i	$-0.257532\pi$
$270$	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
$275$	−4.83848	−0.291771
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	− 7.26383i	− 0.434874i
$280$	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
$291$	−14.8107	−0.868221
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	13.3327	0.776260
$296$	0	0
$297$	− 18.2837i	− 1.06093i
$298$	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
$309$	20.6579	1.17519
$310$	0	0
$311$	35.2338i	1.99793i	0.0455232	+	0.998963i	$0.485505\pi$
$311$	35.2338i	1.99793i	−0.0455232	+	0.998963i	$0.514495\pi$
$312$	0	0
$313$	− 33.2450i	− 1.87912i	−0.342385	−	0.939560i	$-0.611235\pi$
$313$	− 33.2450i	− 1.87912i	0.342385	−	0.939560i	$-0.388765\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.60597	−0.371028	−0.185514	−	0.982642i	$-0.559395\pi$
$317$	−6.60597	−0.371028	−0.185514	+	0.982642i	$0.559395\pi$
$318$	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
$331$	35.2858	1.93948	0.969742	−	0.244131i	$-0.0785028\pi$
$331$	35.2858	1.93948	0.969742	+	0.244131i	$0.0785028\pi$
$332$	0	0
$333$	16.5304	0.905859
$334$	0	0
$335$	15.8743i	0.867303i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	24.3234i	1.32107i
$340$	0	0
$341$	− 16.0855i	− 0.871078i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−24.4056	−1.31395
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	23.4650i	1.24891i	0.781059	+	0.624457i	$0.214680\pi$
$353$	23.4650i	1.24891i	−0.781059	+	0.624457i	$0.785320\pi$
$354$	0	0
$355$	38.7355i	2.05586i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 13.4825i	− 0.707647i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	35.3847i	1.84707i	0.383516	+	0.923534i	$0.374713\pi$
$367$	35.3847i	1.84707i	−0.383516	+	0.923534i	$0.625287\pi$
$368$	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
$375$	−11.0306	−0.569617
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.4225	0.843567	0.421783	−	0.906697i	$-0.361404\pi$
$379$	16.4225	0.843567	0.421783	+	0.906697i	$0.361404\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 21.1104i	− 1.07869i	−0.842084	−	0.539347i	$-0.818671\pi$
$383$	− 21.1104i	− 1.07869i	0.842084	−	0.539347i	$-0.181329\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.04643	0.255864	0.127932	−	0.991783i	$-0.459166\pi$
$389$	5.04643	0.255864	0.127932	+	0.991783i	$0.459166\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.3828i	0.671666i	0.941922	+	0.335833i	$0.109018\pi$
$397$	13.3828i	0.671666i	−0.941922	+	0.335833i	$0.890982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−39.9749	−1.99625	−0.998125	−	0.0612120i	$-0.980503\pi$
$401$	−39.9749	−1.99625	−0.998125	+	0.0612120i	$0.980503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	5.75314i	0.285876i
$406$	0	0
$407$	36.6060	1.81449
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	22.8050i	1.12489i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.8653i	1.70328i	0.524127	+	0.851640i	$0.324392\pi$
$419$	34.8653i	1.70328i	−0.524127	+	0.851640i	$0.675608\pi$
$420$	0	0
$421$	35.2136	1.71620	0.858102	−	0.513479i	$-0.171644\pi$
$421$	35.2136	1.71620	0.858102	+	0.513479i	$0.171644\pi$
$422$	0	0
$423$	12.8026i	0.622483i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	32.6481i	1.56897i	0.620151	+	0.784483i	$0.287071\pi$
$433$	32.6481i	1.56897i	−0.620151	+	0.784483i	$0.712929\pi$
$434$	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
$441$	0	0
$442$	0	0
$443$	−0.0549029	−0.00260851	−0.00130426	−	0.999999i	$-0.500415\pi$
$443$	−0.0549029	−0.00260851	−0.00130426	+	0.999999i	$0.500415\pi$
$444$	0	0
$445$	−38.7988	−1.83924
$446$	0	0
$447$	0	0
$448$	0	0
$449$	41.9631	1.98036	0.990181	−	0.139788i	$-0.0446422\pi$
$449$	41.9631	1.98036	0.990181	+	0.139788i	$0.0446422\pi$
$450$	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		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−37.6694	−1.75064	−0.875322	−	0.483541i	$-0.839350\pi$
$463$	−37.6694	−1.75064	−0.875322	+	0.483541i	$0.839350\pi$
$464$	0	0
$465$	15.1075i	0.700594i
$466$	0	0
$467$	− 32.3800i	− 1.49837i	−0.662361	−	0.749185i	$-0.730446\pi$
$467$	− 32.3800i	− 1.49837i	0.662361	−	0.749185i	$-0.269554\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−26.8366	−1.23657
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.69551	−0.352353
$478$	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
$485$	−30.7098	−1.39446
$486$	0	0
$487$	−20.7401	−0.939822	−0.469911	−	0.882714i	$-0.655714\pi$
$487$	−20.7401	−0.939822	−0.469911	+	0.882714i	$0.655714\pi$
$488$	0	0
$489$	− 3.37986i	− 0.152843i
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 12.6241i	− 0.567413i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	42.3555	1.89609	0.948047	−	0.318131i	$-0.103055\pi$
$499$	42.3555	1.89609	0.948047	+	0.318131i	$0.103055\pi$
$500$	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
$507$	15.9338i	0.707647i
$508$	0	0
$509$	30.0255i	1.33086i	0.746461	+	0.665429i	$0.231751\pi$
$509$	30.0255i	1.33086i	−0.746461	+	0.665429i	$0.768249\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	42.8338	1.88748
$516$	0	0
$517$	28.3509i	1.24687i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.1477i	1.67128i	0.549276	+	0.835641i	$0.314904\pi$
$521$	38.1477i	1.67128i	−0.549276	+	0.835641i	$0.685096\pi$
$522$	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
$529$	38.3861	1.66896
$530$	0	0
$531$	7.85720i	0.340973i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 32.4537i	− 1.40048i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	31.7340	1.36184
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−34.3804	−1.45937
$556$	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
$565$	50.4343i	2.12178i
$566$	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	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	− 15.4509i	− 0.645470i
$574$	0	0
$575$	−11.4300	−0.476666
$576$	0	0
$577$	− 47.8966i	− 1.99396i	−0.0776414	−	0.996981i	$-0.524739\pi$
$577$	− 47.8966i	− 1.99396i	0.0776414	−	0.996981i	$-0.475261\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−17.0415	−0.705785
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.2405i	0.505219i	0.967568	+	0.252609i	$0.0812887\pi$
$587$	12.2405i	0.505219i	−0.967568	+	0.252609i	$0.918711\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.9815	1.30892
$598$	0	0
$599$	−2.00377	−0.0818716	−0.0409358	−	0.999162i	$-0.513034\pi$
$599$	−2.00377	−0.0818716	−0.0409358	+	0.999162i	$0.513034\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	−9.35498	−0.380964
$604$	0	0
$605$	− 27.9557i	− 1.13656i
$606$	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
$617$	26.5330	1.06818	0.534089	−	0.845428i	$-0.320655\pi$
$617$	26.5330	1.06818	0.534089	+	0.845428i	$0.320655\pi$
$618$	0	0
$619$	49.4543i	1.98774i	0.110575	+	0.993868i	$0.464731\pi$
$619$	49.4543i	1.98774i	−0.110575	+	0.993868i	$0.535269\pi$
$620$	0	0
$621$	− 43.1920i	− 1.73324i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.1660	−1.20664
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−30.9369	−1.23158	−0.615789	−	0.787911i	$-0.711162\pi$
$631$	−30.9369	−1.23158	−0.615789	+	0.787911i	$0.711162\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−22.8275	−0.903042
$640$	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
$643$	48.6405i	1.91819i	0.283076	+	0.959097i	$0.408645\pi$
$643$	48.6405i	1.91819i	−0.283076	+	0.959097i	$0.591355\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 46.2715i	− 1.81912i	−0.415574	−	0.909560i	$-0.636419\pi$
$647$	− 46.2715i	− 1.81912i	0.415574	−	0.909560i	$-0.363581\pi$
$648$	0	0
$649$	17.3995i	0.682990i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.99616	−0.391180	−0.195590	−	0.980686i	$-0.562662\pi$
$653$	−9.99616	−0.391180	−0.195590	+	0.980686i	$0.562662\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 40.5991i	− 1.57912i	−0.613672	−	0.789561i	$-0.710308\pi$
$661$	− 40.5991i	− 1.57912i	0.613672	−	0.789561i	$-0.289692\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	5.99064	0.231611
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	8.04232i	0.309549i
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	47.2857i	1.80670i
$686$	0	0
$687$	35.4732	1.35339
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 10.3610i	− 0.394150i	−0.980388	−	0.197075i	$-0.936856\pi$
$691$	− 10.3610i	− 0.394150i	0.980388	−	0.197075i	$-0.0631442\pi$
$692$	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
$705$	− 26.6272i	− 1.00284i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	52.9718	1.98940	0.994698	−	0.102837i	$-0.0327921\pi$
$709$	52.9718	1.98940	0.994698	+	0.102837i	$0.0327921\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 37.9991i	− 1.42308i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.5562i	1.88543i	0.333603	+	0.942714i	$0.391736\pi$
$719$	50.5562i	1.88543i	−0.333603	+	0.942714i	$0.608264\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.1048i	1.78411i	0.451928	+	0.892055i	$0.350737\pi$
$727$	48.1048i	1.78411i	−0.451928	+	0.892055i	$0.649263\pi$
$728$	0	0
$729$	−23.6610	−0.876335
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.7163	−0.763094
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.34021	0.340829	0.170415	−	0.985372i	$-0.445489\pi$
$751$	9.34021	0.340829	0.170415	+	0.985372i	$0.445489\pi$
$752$	0	0
$753$	4.02058	0.146518
$754$	0	0
$755$	0	0
$756$	0	0
$757$	55.0126	1.99947	0.999733	−	0.0231238i	$-0.00736118\pi$
$757$	55.0126	1.99947	0.999733	+	0.0231238i	$0.00736118\pi$
$758$	0	0
$759$	− 31.8499i	− 1.15608i
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−38.4883	−1.38612
$772$	0	0
$773$	8.40826i	0.302424i	0.988501	+	0.151212i	$0.0483176\pi$
$773$	8.40826i	0.302424i	−0.988501	+	0.151212i	$0.951682\pi$
$774$	0	0
$775$	7.07540i	0.254156i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−50.5507	−1.80885
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−55.6453	−1.98607
$786$	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
$795$	16.0053	0.567651
$796$	0	0
$797$	− 56.2918i	− 1.99396i	−0.0776637	−	0.996980i	$-0.524746\pi$
$797$	− 56.2918i	− 1.99396i	0.0776637	−	0.996980i	$-0.475254\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	− 22.8648i	− 0.807888i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29.0941	−1.02416
$808$	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
$815$	− 7.00809i	− 0.245483i
$816$	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
$823$	−57.3601	−1.99945	−0.999723	−	0.0235383i	$-0.992507\pi$
$823$	−57.3601	−1.99945	−0.999723	+	0.0235383i	$0.992507\pi$
$824$	0	0
$825$	5.93043i	0.206471i
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	53.1090i	1.84455i	0.386531	+	0.922276i	$0.373673\pi$
$829$	53.1090i	1.84455i	−0.386531	+	0.922276i	$0.626327\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26.7366	−0.924153
$838$	0	0
$839$	− 57.9103i	− 1.99928i	−0.0267481	−	0.999642i	$-0.508515\pi$
$839$	− 57.9103i	− 1.99928i	0.0267481	−	0.999642i	$-0.491485\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	33.0386i	1.13656i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	86.4750	2.96432
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	58.3404i	1.99055i	0.0970965	+	0.995275i	$0.469044\pi$
$859$	58.3404i	1.99055i	−0.0970965	+	0.995275i	$0.530956\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20.8366i	0.707647i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 18.0979i	− 0.612520i
$874$	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
$881$	− 47.8557i	− 1.61230i	−0.591712	−	0.806149i	$-0.701548\pi$
$881$	− 47.8557i	− 1.61230i	0.591712	−	0.806149i	$-0.298452\pi$
$882$	0	0
$883$	−19.8997	−0.669680	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	−19.8997	−0.669680	−0.334840	+	0.942275i	$0.608682\pi$
$884$	0	0
$885$	− 16.3416i	− 0.549318i
$886$	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
$891$	−7.50798	−0.251527
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 67.2922i	− 2.24933i
$896$	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
$905$	65.7999	2.18726
$906$	0	0
$907$	−47.8706	−1.58952	−0.794759	−	0.606926i	$-0.792403\pi$
$907$	−47.8706	−1.58952	−0.794759	+	0.606926i	$0.792403\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.1168	−1.56105	−0.780525	−	0.625125i	$-0.785048\pi$
$911$	−47.1168	−1.56105	−0.780525	+	0.625125i	$0.785048\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−16.1016	−0.529417
$926$	0	0
$927$	25.2427i	0.829080i
$928$	0	0
$929$	− 48.0239i	− 1.57561i	−0.615924	−	0.787806i	$-0.711217\pi$
$929$	− 48.0239i	− 1.57561i	0.615924	−	0.787806i	$-0.288783\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	43.1854	1.41383
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	−40.7477	−1.32975
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.6060	−1.57948	−0.789741	−	0.613441i	$-0.789785\pi$
$947$	−48.6060	−1.57948	−0.789741	+	0.613441i	$0.789785\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	8.09680i	0.262557i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 32.0372i	− 1.03670i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.47790	0.241223
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 50.2428i	− 1.61237i	−0.591665	−	0.806184i	$-0.701529\pi$
$971$	− 50.2428i	− 1.61237i	0.591665	−	0.806184i	$-0.298471\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.4733	−1.51881	−0.759403	−	0.650621i	$-0.774509\pi$
$977$	−47.4733	−1.51881	−0.759403	+	0.650621i	$0.774509\pi$
$978$	0	0
$979$	− 50.6333i	− 1.61825i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.3257i	1.76461i	0.470674	+	0.882307i	$0.344011\pi$
$983$	55.3257i	1.76461i	−0.470674	+	0.882307i	$0.655989\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−28.0716	−0.891724	−0.445862	−	0.895102i	$-0.647103\pi$
$991$	−28.0716	−0.891724	−0.445862	+	0.895102i	$0.647103\pi$
$992$	0	0
$993$	− 43.2491i	− 1.37247i
$994$	0	0
$995$	66.3132	2.10227
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	− 60.8449i	− 1.92505i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.c.b.1077.7	✓	16
7.2	even	3		2156.2.q.e.2089.10		32
7.3	odd	6		2156.2.q.e.901.10		32
7.4	even	3		2156.2.q.e.901.7		32
7.5	odd	6		2156.2.q.e.2089.7		32
7.6	odd	2	inner	2156.2.c.b.1077.10	yes	16
11.10	odd	2	CM	2156.2.c.b.1077.7	✓	16
77.10	even	6		2156.2.q.e.901.10		32
77.32	odd	6		2156.2.q.e.901.7		32
77.54	even	6		2156.2.q.e.2089.7		32
77.65	odd	6		2156.2.q.e.2089.10		32
77.76	even	2	inner	2156.2.c.b.1077.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2156.2.c.b.1077.7	✓	16	1.1	even	1	trivial
2156.2.c.b.1077.7	✓	16	11.10	odd	2	CM
2156.2.c.b.1077.10	yes	16	7.6	odd	2	inner
2156.2.c.b.1077.10	yes	16	77.76	even	2	inner
2156.2.q.e.901.7		32	7.4	even	3
2156.2.q.e.901.7		32	77.32	odd	6
2156.2.q.e.901.10		32	7.3	odd	6
2156.2.q.e.901.10		32	77.10	even	6
2156.2.q.e.2089.7		32	7.5	odd	6
2156.2.q.e.2089.7		32	77.54	even	6
2156.2.q.e.2089.10		32	7.2	even	3
2156.2.q.e.2089.10		32	77.65	odd	6

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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.22568i q^{3} -2.54143i q^{5} +1.49771 q^{9} +3.31662 q^{11} -3.11498 q^{15} +7.83493 q^{23} -1.45886 q^{25} -5.51275i q^{27} -4.84996i q^{31} -4.06512i q^{33} +11.0371 q^{37} -3.80632i q^{45} +8.54812i q^{47} -5.13819 q^{53} -8.42896i q^{55} +5.24614i q^{59} -6.24619 q^{67} -9.60311i q^{69} -15.2416 q^{71} +1.78809i q^{75} -2.26374 q^{81} -15.2665i q^{89} -5.94450 q^{93} -12.0837i q^{97} +4.96734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 80 q^{25} + 144 q^{81}+O(q^{100}) \) 16 * q - 48 * q^9 - 80 * q^25 + 144 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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