LMFDB - Embedded newform 2736.2.f.b.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2736.1025
Dual form			2736.2.f.b.1025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{5} -2.00000 q^{7} +O(q^{10})$	$q+1.41421i q^{5} -2.00000 q^{7} -1.41421i q^{11} +1.41421i q^{17} +(-1.00000 - 4.24264i) q^{19} -1.41421i q^{23} +3.00000 q^{25} +6.00000 q^{29} -2.82843i q^{35} +8.48528i q^{37} +6.00000 q^{41} +4.00000 q^{43} +7.07107i q^{47} -3.00000 q^{49} +6.00000 q^{53} +2.00000 q^{55} -4.00000 q^{61} +8.48528i q^{67} +12.0000 q^{71} -10.0000 q^{73} +2.82843i q^{77} +8.48528i q^{79} +15.5563i q^{83} -2.00000 q^{85} +6.00000 q^{89} +(6.00000 - 1.41421i) q^{95} -8.48528i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^7	$ 2 q - 4 q^{7} - 2 q^{19} + 6 q^{25} + 12 q^{29} + 12 q^{41} + 8 q^{43} - 6 q^{49} + 12 q^{53} + 4 q^{55} - 8 q^{61} + 24 q^{71} - 20 q^{73} - 4 q^{85} + 12 q^{89} + 12 q^{95}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^19 + 6 * q^25 + 12 * q^29 + 12 * q^41 + 8 * q^43 - 6 * q^49 + 12 * q^53 + 4 * q^55 - 8 * q^61 + 24 * q^71 - 20 * q^73 - 4 * q^85 + 12 * q^89 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.82843i		−	0.478091i
$36$		0			0
$37$			8.48528i			1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$			8.48528i			1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.07107i			1.03142i	0.856763	+	0.515711i	$0.172472\pi$
$47$			7.07107i			1.03142i	−0.856763	+	0.515711i	$0.827528\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		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
$55$	2.00000			0.269680
$56$		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
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			8.48528i			1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$67$			8.48528i			1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.82843i			0.322329i
$78$		0			0
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			15.5563i			1.70753i	0.520658	+	0.853766i	$0.325687\pi$
$83$			15.5563i			1.70753i	−0.520658	+	0.853766i	$0.674313\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	−	1.41421i	0.615587	−	0.145095i
$96$		0			0
$97$		−	8.48528i		−	0.861550i	−0.902459	−	0.430775i	$-0.858240\pi$
$97$		−	8.48528i		−	0.861550i	0.902459	−	0.430775i	$-0.141760\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	15.5563i		−	1.54791i	−0.633238	−	0.773957i	$-0.718274\pi$
$101$		−	15.5563i		−	1.54791i	0.633238	−	0.773957i	$-0.281726\pi$
$102$		0			0
$103$			8.48528i			0.836080i	0.908429	+	0.418040i	$0.137283\pi$
$103$			8.48528i			0.836080i	−0.908429	+	0.418040i	$0.862717\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			16.9706i			1.62549i	0.582623	+	0.812743i	$0.302026\pi$
$109$			16.9706i			1.62549i	−0.582623	+	0.812743i	$0.697974\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	2.00000			0.186501
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.82843i		−	0.259281i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$			16.9706i			1.50589i	0.658081	+	0.752947i	$0.271368\pi$
$127$			16.9706i			1.50589i	−0.658081	+	0.752947i	$0.728632\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	9.89949i		−	0.864923i	−0.901652	−	0.432461i	$-0.857645\pi$
$131$		−	9.89949i		−	0.864923i	0.901652	−	0.432461i	$-0.142355\pi$
$132$		0			0
$133$	2.00000	+	8.48528i	0.173422	+	0.735767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	7.07107i		−	0.604122i	−0.953289	−	0.302061i	$-0.902325\pi$
$137$		−	7.07107i		−	0.604122i	0.953289	−	0.302061i	$-0.0976746\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$			8.48528i			0.704664i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			18.3848i			1.50614i	0.657941	+	0.753070i	$0.271428\pi$
$149$			18.3848i			1.50614i	−0.657941	+	0.753070i	$0.728572\pi$
$150$		0			0
$151$		−	8.48528i		−	0.690522i	−0.938507	−	0.345261i	$-0.887790\pi$
$151$		−	8.48528i		−	0.690522i	0.938507	−	0.345261i	$-0.112210\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.82843i			0.222911i
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$	−6.00000			−0.453557
$176$		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
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.0000			−0.882258
$186$		0			0
$187$	2.00000			0.146254
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1.41421i		−	0.102329i	−0.998690	−	0.0511645i	$-0.983707\pi$
$191$		−	1.41421i		−	0.102329i	0.998690	−	0.0511645i	$-0.0162933\pi$
$192$		0			0
$193$		−	16.9706i		−	1.22157i	−0.791797	−	0.610784i	$-0.790854\pi$
$193$		−	16.9706i		−	1.22157i	0.791797	−	0.610784i	$-0.209146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	7.07107i		−	0.503793i	−0.967754	−	0.251896i	$-0.918946\pi$
$197$		−	7.07107i		−	0.503793i	0.967754	−	0.251896i	$-0.0810542\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0000			−0.842235
$204$		0			0
$205$			8.48528i			0.592638i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000	+	1.41421i	−0.415029	+	0.0978232i
$210$		0			0
$211$			25.4558i			1.75245i	0.481900	+	0.876226i	$0.339947\pi$
$211$			25.4558i			1.75245i	−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			5.65685i			0.385794i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	25.4558i		−	1.70465i	−0.523013	−	0.852325i	$-0.675192\pi$
$223$		−	25.4558i		−	1.70465i	0.523013	−	0.852325i	$-0.324808\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$	8.00000			0.528655			0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000			0.528655			0.264327	+	0.964433i	$0.414850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.89949i			0.648537i	0.945965	+	0.324269i	$0.105118\pi$
$233$			9.89949i			0.648537i	−0.945965	+	0.324269i	$0.894882\pi$
$234$		0			0
$235$	−10.0000			−0.652328
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	18.3848i		−	1.18921i	−0.804017	−	0.594606i	$-0.797308\pi$
$239$		−	18.3848i		−	1.18921i	0.804017	−	0.594606i	$-0.202692\pi$
$240$		0			0
$241$		−	25.4558i		−	1.63976i	−0.572539	−	0.819878i	$-0.694041\pi$
$241$		−	25.4558i		−	1.63976i	0.572539	−	0.819878i	$-0.305959\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	4.24264i		−	0.271052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			7.07107i			0.446322i	0.974782	+	0.223161i	$0.0716375\pi$
$251$			7.07107i			0.446322i	−0.974782	+	0.223161i	$0.928362\pi$
$252$		0			0
$253$	−2.00000			−0.125739
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$		−	16.9706i		−	1.05450i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.5563i			0.959246i	0.877475	+	0.479623i	$0.159226\pi$
$263$			15.5563i			0.959246i	−0.877475	+	0.479623i	$0.840774\pi$
$264$		0			0
$265$			8.48528i			0.521247i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4.24264i		−	0.255841i
$276$		0			0
$277$	−16.0000			−0.961347			−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000			−0.961347			−0.480673	+	0.876900i	$0.659608\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	28.0000			1.66443			0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000			1.66443			0.832214	+	0.554455i	$0.187073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.0000			1.05157			0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000			1.05157			0.525786	+	0.850617i	$0.323771\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.65685i		−	0.323911i
$306$		0			0
$307$			16.9706i			0.968561i	0.874913	+	0.484281i	$0.160919\pi$
$307$			16.9706i			0.968561i	−0.874913	+	0.484281i	$0.839081\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			24.0416i			1.36328i	0.731690	+	0.681638i	$0.238732\pi$
$311$			24.0416i			1.36328i	−0.731690	+	0.681638i	$0.761268\pi$
$312$		0			0
$313$	−28.0000			−1.58265			−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000			−1.58265			−0.791327	+	0.611393i	$0.790609\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	30.0000			1.68497			0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000			1.68497			0.842484	+	0.538721i	$0.181092\pi$
$318$		0			0
$319$		−	8.48528i		−	0.475085i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000	−	1.41421i	0.333849	−	0.0786889i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	14.1421i		−	0.779681i
$330$		0			0
$331$		−	16.9706i		−	0.932786i	−0.884577	−	0.466393i	$-0.845553\pi$
$331$		−	16.9706i		−	0.932786i	0.884577	−	0.466393i	$-0.154447\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$			8.48528i			0.462223i	0.972927	+	0.231111i	$0.0742362\pi$
$337$			8.48528i			0.462223i	−0.972927	+	0.231111i	$0.925764\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1.41421i		−	0.0759190i	−0.999279	−	0.0379595i	$-0.987914\pi$
$347$		−	1.41421i		−	0.0759190i	0.999279	−	0.0379595i	$-0.0120858\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	15.5563i		−	0.827981i	−0.910281	−	0.413990i	$-0.864135\pi$
$353$		−	15.5563i		−	0.827981i	0.910281	−	0.413990i	$-0.135865\pi$
$354$		0			0
$355$			16.9706i			0.900704i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.3848i		−	0.970311i	−0.874428	−	0.485156i	$-0.838763\pi$
$359$		−	18.3848i		−	0.970311i	0.874428	−	0.485156i	$-0.161237\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	14.1421i		−	0.740233i
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000			−0.623009
$372$		0			0
$373$			25.4558i			1.31805i	0.752119	+	0.659027i	$0.229032\pi$
$373$			25.4558i			1.31805i	−0.752119	+	0.659027i	$0.770968\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	16.9706i		−	0.871719i	−0.900015	−	0.435860i	$-0.856444\pi$
$379$		−	16.9706i		−	0.871719i	0.900015	−	0.435860i	$-0.143556\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−36.0000			−1.83951			−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000			−1.83951			−0.919757	+	0.392488i	$0.871614\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$			35.3553i			1.79259i	0.443461	+	0.896293i	$0.353750\pi$
$389$			35.3553i			1.79259i	−0.443461	+	0.896293i	$0.646250\pi$
$390$		0			0
$391$	2.00000			0.101144
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.0000			−0.603786
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$		−	25.4558i		−	1.25871i	−0.777118	−	0.629355i	$-0.783319\pi$
$409$		−	25.4558i		−	1.25871i	0.777118	−	0.629355i	$-0.216681\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−22.0000			−1.07994
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	1.41421i		−	0.0690889i	−0.999403	−	0.0345444i	$-0.989002\pi$
$419$		−	1.41421i		−	0.0690889i	0.999403	−	0.0345444i	$-0.0109980\pi$
$420$		0			0
$421$		−	16.9706i		−	0.827095i	−0.910483	−	0.413547i	$-0.864290\pi$
$421$		−	16.9706i		−	0.827095i	0.910483	−	0.413547i	$-0.135710\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			4.24264i			0.205798i
$426$		0			0
$427$	8.00000			0.387147
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.00000	+	1.41421i	−0.287019	+	0.0676510i
$438$		0			0
$439$		−	8.48528i		−	0.404980i	−0.979284	−	0.202490i	$-0.935097\pi$
$439$		−	8.48528i		−	0.404980i	0.979284	−	0.202490i	$-0.0649034\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	9.89949i		−	0.470339i	−0.971954	−	0.235170i	$-0.924435\pi$
$443$		−	9.89949i		−	0.470339i	0.971954	−	0.235170i	$-0.0755646\pi$
$444$		0			0
$445$			8.48528i			0.402241i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		−	8.48528i		−	0.399556i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			18.3848i			0.856264i	0.903716	+	0.428132i	$0.140828\pi$
$461$			18.3848i			0.856264i	−0.903716	+	0.428132i	$0.859172\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.41421i		−	0.0654420i	−0.999465	−	0.0327210i	$-0.989583\pi$
$467$		−	1.41421i		−	0.0654420i	0.999465	−	0.0327210i	$-0.0104173\pi$
$468$		0			0
$469$		−	16.9706i		−	0.783628i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.65685i		−	0.260102i
$474$		0			0
$475$	−3.00000	−	12.7279i	−0.137649	−	0.583997i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	9.89949i		−	0.452319i	−0.974090	−	0.226160i	$-0.927383\pi$
$479$		−	9.89949i		−	0.452319i	0.974090	−	0.226160i	$-0.0726171\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000			0.544892
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.0416i			1.08498i	0.840061	+	0.542492i	$0.182519\pi$
$491$			24.0416i			1.08498i	−0.840061	+	0.542492i	$0.817481\pi$
$492$		0			0
$493$			8.48528i			0.382158i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.0000			−1.07655
$498$		0			0
$499$	−38.0000			−1.70111			−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000			−1.70111			−0.850557	+	0.525883i	$0.823735\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	18.3848i		−	0.819737i	−0.912145	−	0.409868i	$-0.865575\pi$
$503$		−	18.3848i		−	0.819737i	0.912145	−	0.409868i	$-0.134425\pi$
$504$		0			0
$505$	22.0000			0.978987
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	20.0000			0.884748
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.0000			−0.528783
$516$		0			0
$517$	10.0000			0.439799
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$		−	8.48528i		−	0.371035i	−0.982641	−	0.185518i	$-0.940604\pi$
$523$		−	8.48528i		−	0.371035i	0.982641	−	0.185518i	$-0.0593962\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			16.9706i			0.733701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			4.24264i			0.182743i
$540$		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
$543$		0			0
$544$		0			0
$545$	−24.0000			−1.02805
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.00000	−	25.4558i	−0.255609	−	1.08446i
$552$		0			0
$553$		−	16.9706i		−	0.721662i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	15.5563i		−	0.659144i	−0.944131	−	0.329572i	$-0.893096\pi$
$557$		−	15.5563i		−	0.659144i	0.944131	−	0.329572i	$-0.106904\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		−	8.48528i		−	0.356978i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.24264i		−	0.176930i
$576$		0			0
$577$	−4.00000			−0.166522			−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000			−0.166522			−0.0832611	+	0.996528i	$0.526534\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	31.1127i		−	1.29077i
$582$		0			0
$583$		−	8.48528i		−	0.351424i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	35.3553i		−	1.45927i	−0.683836	−	0.729636i	$-0.739690\pi$
$587$		−	35.3553i		−	1.45927i	0.683836	−	0.729636i	$-0.260310\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	32.5269i		−	1.33572i	−0.744287	−	0.667860i	$-0.767210\pi$
$593$		−	32.5269i		−	1.33572i	0.744287	−	0.667860i	$-0.232790\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$		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
$601$		−	33.9411i		−	1.38449i	−0.721664	−	0.692244i	$-0.756622\pi$
$601$		−	33.9411i		−	1.38449i	0.721664	−	0.692244i	$-0.243378\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			12.7279i			0.517464i
$606$		0			0
$607$		−	16.9706i		−	0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$		−	16.9706i		−	0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−28.0000			−1.13091			−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000			−1.13091			−0.565455	+	0.824779i	$0.691299\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			43.8406i			1.76496i	0.470353	+	0.882478i	$0.344127\pi$
$617$			43.8406i			1.76496i	−0.470353	+	0.882478i	$0.655873\pi$
$618$		0			0
$619$	10.0000			0.401934			0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000			0.401934			0.200967	+	0.979598i	$0.435592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−26.0000			−1.03504			−0.517522	−	0.855670i	$-0.673145\pi$
$631$	−26.0000			−1.03504			−0.517522	+	0.855670i	$0.673145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	−20.0000			−0.788723			−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000			−0.788723			−0.394362	+	0.918955i	$0.629034\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			41.0122i			1.61236i	0.591673	+	0.806178i	$0.298468\pi$
$647$			41.0122i			1.61236i	−0.591673	+	0.806178i	$0.701532\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	7.07107i		−	0.276712i	−0.990383	−	0.138356i	$-0.955818\pi$
$653$		−	7.07107i		−	0.276712i	0.990383	−	0.138356i	$-0.0441819\pi$
$654$		0			0
$655$	14.0000			0.547025
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$			8.48528i			0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$			8.48528i			0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000	+	2.82843i	−0.465340	+	0.109682i
$666$		0			0
$667$		−	8.48528i		−	0.328551i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			5.65685i			0.218380i
$672$		0			0
$673$		−	33.9411i		−	1.30833i	−0.756350	−	0.654167i	$-0.773019\pi$
$673$		−	33.9411i		−	1.30833i	0.756350	−	0.654167i	$-0.226981\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$	10.0000			0.382080
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	19.7990i		−	0.751018i
$696$		0			0
$697$			8.48528i			0.321403i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7.07107i		−	0.267071i	−0.991044	−	0.133535i	$-0.957367\pi$
$701$		−	7.07107i		−	0.267071i	0.991044	−	0.133535i	$-0.0426329\pi$
$702$		0			0
$703$	36.0000	−	8.48528i	1.35777	−	0.320028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			31.1127i			1.17011i
$708$		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
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			32.5269i			1.21305i	0.795065	+	0.606525i	$0.207437\pi$
$719$			32.5269i			1.21305i	−0.795065	+	0.606525i	$0.792563\pi$
$720$		0			0
$721$		−	16.9706i		−	0.632017i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.0000			0.668503
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.65685i			0.209226i
$732$		0			0
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000			0.442026
$738$		0			0
$739$	34.0000			1.25071			0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000			1.25071			0.625355	+	0.780340i	$0.284954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0000			0.440237			0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000			0.440237			0.220119	+	0.975473i	$0.429356\pi$
$744$		0			0
$745$	−26.0000			−0.952566
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−24.0000			−0.876941
$750$		0			0
$751$			8.48528i			0.309632i	0.987943	+	0.154816i	$0.0494785\pi$
$751$			8.48528i			0.309632i	−0.987943	+	0.154816i	$0.950521\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	49.4975i		−	1.79428i	−0.441744	−	0.897141i	$-0.645640\pi$
$761$		−	49.4975i		−	1.79428i	0.441744	−	0.897141i	$-0.354360\pi$
$762$		0			0
$763$		−	33.9411i		−	1.22875i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−28.0000			−1.00971			−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000			−1.00971			−0.504853	+	0.863205i	$0.668453\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.00000	−	25.4558i	−0.214972	−	0.912050i
$780$		0			0
$781$		−	16.9706i		−	0.607254i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	5.65685i		−	0.201902i
$786$		0			0
$787$			25.4558i			0.907403i	0.891154	+	0.453701i	$0.149897\pi$
$787$			25.4558i			0.907403i	−0.891154	+	0.453701i	$0.850103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	−10.0000			−0.353775
$800$		0			0
$801$		0			0
$802$		0			0
$803$			14.1421i			0.499065i
$804$		0			0

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{5} -2.00000 q^{7} +O(q^{10})\)	\(q+1.41421i q^{5} -2.00000 q^{7} -1.41421i q^{11} +1.41421i q^{17} +(-1.00000 - 4.24264i) q^{19} -1.41421i q^{23} +3.00000 q^{25} +6.00000 q^{29} -2.82843i q^{35} +8.48528i q^{37} +6.00000 q^{41} +4.00000 q^{43} +7.07107i q^{47} -3.00000 q^{49} +6.00000 q^{53} +2.00000 q^{55} -4.00000 q^{61} +8.48528i q^{67} +12.0000 q^{71} -10.0000 q^{73} +2.82843i q^{77} +8.48528i q^{79} +15.5563i q^{83} -2.00000 q^{85} +6.00000 q^{89} +(6.00000 - 1.41421i) q^{95} -8.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^7	\( 2 q - 4 q^{7} - 2 q^{19} + 6 q^{25} + 12 q^{29} + 12 q^{41} + 8 q^{43} - 6 q^{49} + 12 q^{53} + 4 q^{55} - 8 q^{61} + 24 q^{71} - 20 q^{73} - 4 q^{85} + 12 q^{89} + 12 q^{95}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^19 + 6 * q^25 + 12 * q^29 + 12 * q^41 + 8 * q^43 - 6 * q^49 + 12 * q^53 + 4 * q^55 - 8 * q^61 + 24 * q^71 - 20 * q^73 - 4 * q^85 + 12 * q^89 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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