LMFDB - Embedded newform 2736.2.k.f.2431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2431,2736)]

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([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.k (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.2431
Dual form			2736.2.k.f.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410i q^{7} +O(q^{10})$	$q+3.46410i q^{7} -3.46410i q^{11} -3.46410i q^{13} +6.00000 q^{17} +(4.00000 - 1.73205i) q^{19} -5.00000 q^{25} -6.92820i q^{29} -10.0000 q^{31} -3.46410i q^{37} +6.92820i q^{41} +10.3923i q^{43} -6.92820i q^{47} -5.00000 q^{49} -13.8564i q^{53} +12.0000 q^{59} +10.0000 q^{61} -4.00000 q^{67} +12.0000 q^{71} -2.00000 q^{73} +12.0000 q^{77} +10.0000 q^{79} +3.46410i q^{83} -6.92820i q^{89} +12.0000 q^{91} -6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 12 q^{17} + 8 q^{19} - 10 q^{25} - 20 q^{31} - 10 q^{49} + 24 q^{59} + 20 q^{61} - 8 q^{67} + 24 q^{71} - 4 q^{73} + 24 q^{77} + 20 q^{79} + 24 q^{91}+O(q^{100}) $ 2 * q + 12 * q^17 + 8 * q^19 - 10 * q^25 - 20 * q^31 - 10 * q^49 + 24 * q^59 + 20 * q^61 - 8 * q^67 + 24 * q^71 - 4 * q^73 + 24 * q^77 + 20 * q^79 + 24 * q^91

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
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$			3.46410i			1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$			3.46410i			1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.92820i		−	1.28654i	−0.765641	−	0.643268i	$-0.777578\pi$
$29$		−	6.92820i		−	1.28654i	0.765641	−	0.643268i	$-0.222422\pi$
$30$		0			0
$31$	−10.0000			−1.79605			−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000			−1.79605			−0.898027	+	0.439941i	$0.854999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	3.46410i		−	0.569495i	−0.958603	−	0.284747i	$-0.908090\pi$
$37$		−	3.46410i		−	0.569495i	0.958603	−	0.284747i	$-0.0919097\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		0			0
$43$			10.3923i			1.58481i	0.609994	+	0.792406i	$0.291172\pi$
$43$			10.3923i			1.58481i	−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.92820i		−	1.01058i	−0.862949	−	0.505291i	$-0.831385\pi$
$47$		−	6.92820i		−	1.01058i	0.862949	−	0.505291i	$-0.168615\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	13.8564i		−	1.90332i	−0.307148	−	0.951662i	$-0.599375\pi$
$53$		−	13.8564i		−	1.90332i	0.307148	−	0.951662i	$-0.400625\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		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
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.0000			1.36753
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$		−	6.92820i		−	0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−2.00000			−0.197066			−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000			−0.197066			−0.0985329	+	0.995134i	$0.531415\pi$
$104$		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
$109$		−	10.3923i		−	0.995402i	−0.867349	−	0.497701i	$-0.834178\pi$
$109$		−	10.3923i		−	0.995402i	0.867349	−	0.497701i	$-0.165822\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.92820i		−	0.651751i	−0.945413	−	0.325875i	$-0.894341\pi$
$113$		−	6.92820i		−	0.651751i	0.945413	−	0.325875i	$-0.105659\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$			20.7846i			1.90532i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.3923i			0.907980i	0.891007	+	0.453990i	$0.150000\pi$
$131$			10.3923i			0.907980i	−0.891007	+	0.453990i	$0.850000\pi$
$132$		0			0
$133$	6.00000	+	13.8564i	0.520266	+	1.20150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$			3.46410i			0.293821i	0.989150	+	0.146911i	$0.0469330\pi$
$139$			3.46410i			0.293821i	−0.989150	+	0.146911i	$0.953067\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000			−1.00349
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−24.0000			−1.96616			−0.983078	−	0.183186i	$-0.941359\pi$
$149$	−24.0000			−1.96616			−0.983078	+	0.183186i	$0.941359\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			3.46410i			0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$			3.46410i			0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	13.8564i		−	1.05348i	−0.850026	−	0.526742i	$-0.823414\pi$
$173$		−	13.8564i		−	1.05348i	0.850026	−	0.526742i	$-0.176586\pi$
$174$		0			0
$175$		−	17.3205i		−	1.30931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$			3.46410i			0.257485i	0.991678	+	0.128742i	$0.0410940\pi$
$181$			3.46410i			0.257485i	−0.991678	+	0.128742i	$0.958906\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	20.7846i		−	1.51992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.8564i		−	1.00261i	−0.865269	−	0.501307i	$-0.832853\pi$
$191$		−	13.8564i		−	1.00261i	0.865269	−	0.501307i	$-0.167147\pi$
$192$		0			0
$193$		−	13.8564i		−	0.997406i	−0.866773	−	0.498703i	$-0.833810\pi$
$193$		−	13.8564i		−	0.997406i	0.866773	−	0.498703i	$-0.166190\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$		−	3.46410i		−	0.245564i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$		−	3.46410i		−	0.245564i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.0000			1.68447
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000	−	13.8564i	−0.415029	−	0.958468i
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	34.6410i		−	2.35159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	20.7846i		−	1.39812i
$222$		0			0
$223$	−2.00000			−0.133930			−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000			−0.133930			−0.0669650	+	0.997755i	$0.521332\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		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
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.92820i		−	0.448148i	−0.974572	−	0.224074i	$-0.928064\pi$
$239$		−	6.92820i		−	0.448148i	0.974572	−	0.224074i	$-0.0719358\pi$
$240$		0			0
$241$			27.7128i			1.78514i	0.450910	+	0.892570i	$0.351100\pi$
$241$			27.7128i			1.78514i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.00000	−	13.8564i	−0.381771	−	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			31.1769i			1.96787i	0.178529	+	0.983935i	$0.442866\pi$
$251$			31.1769i			1.96787i	−0.178529	+	0.983935i	$0.557134\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.7846i		−	1.29651i	−0.761424	−	0.648254i	$-0.775499\pi$
$257$		−	20.7846i		−	1.29651i	0.761424	−	0.648254i	$-0.224501\pi$
$258$		0			0
$259$	12.0000			0.745644
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6.92820i			0.427211i	0.976920	+	0.213606i	$0.0685208\pi$
$263$			6.92820i			0.427211i	−0.976920	+	0.213606i	$0.931479\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$			24.2487i			1.47300i	0.676435	+	0.736502i	$0.263524\pi$
$271$			24.2487i			1.47300i	−0.676435	+	0.736502i	$0.736476\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			17.3205i			1.04447i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			20.7846i			1.23991i	0.784639	+	0.619953i	$0.212848\pi$
$281$			20.7846i			1.23991i	−0.784639	+	0.619953i	$0.787152\pi$
$282$		0			0
$283$		−	10.3923i		−	0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$		−	10.3923i		−	0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.0000			−1.41668
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$			13.8564i			0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$			13.8564i			0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−36.0000			−2.07501
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.8564i		−	0.785725i	−0.919597	−	0.392862i	$-0.871485\pi$
$311$		−	13.8564i		−	0.785725i	0.919597	−	0.392862i	$-0.128515\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000	−	10.3923i	1.33540	−	0.578243i
$324$		0			0
$325$			17.3205i			0.960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.0000			1.32316
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			6.92820i			0.377403i	0.982034	+	0.188702i	$0.0604279\pi$
$337$			6.92820i			0.377403i	−0.982034	+	0.188702i	$0.939572\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			34.6410i			1.87592i
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$		0			0
$346$		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
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	20.7846i		−	1.09697i	−0.836160	−	0.548485i	$-0.815205\pi$
$359$		−	20.7846i		−	1.09697i	0.836160	−	0.548485i	$-0.184795\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	3.46410i		−	0.180825i	−0.995904	−	0.0904123i	$-0.971182\pi$
$367$		−	3.46410i		−	0.180825i	0.995904	−	0.0904123i	$-0.0288185\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	48.0000			2.49204
$372$		0			0
$373$			38.1051i			1.97301i	0.163737	+	0.986504i	$0.447645\pi$
$373$			38.1051i			1.97301i	−0.163737	+	0.986504i	$0.552355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000			−0.608424			−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000			−0.608424			−0.304212	+	0.952604i	$0.598393\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	14.0000			0.702640			0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000			0.702640			0.351320	+	0.936255i	$0.385733\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			20.7846i			1.03793i	0.854794	+	0.518967i	$0.173683\pi$
$401$			20.7846i			1.03793i	−0.854794	+	0.518967i	$0.826317\pi$
$402$		0			0
$403$			34.6410i			1.72559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$			34.6410i			1.71289i	0.516240	+	0.856444i	$0.327331\pi$
$409$			34.6410i			1.71289i	−0.516240	+	0.856444i	$0.672669\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			41.5692i			2.04549i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	17.3205i		−	0.846162i	−0.906092	−	0.423081i	$-0.860949\pi$
$419$		−	17.3205i		−	0.846162i	0.906092	−	0.423081i	$-0.139051\pi$
$420$		0			0
$421$		−	24.2487i		−	1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$		−	24.2487i		−	1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−30.0000			−1.45521
$426$		0			0
$427$			34.6410i			1.67640i
$428$		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
$433$		−	20.7846i		−	0.998845i	−0.866359	−	0.499422i	$-0.833546\pi$
$433$		−	20.7846i		−	0.998845i	0.866359	−	0.499422i	$-0.166454\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−2.00000			−0.0954548			−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000			−0.0954548			−0.0477274	+	0.998860i	$0.515198\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	17.3205i		−	0.822922i	−0.911427	−	0.411461i	$-0.865019\pi$
$443$		−	17.3205i		−	0.822922i	0.911427	−	0.411461i	$-0.134981\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	34.6410i		−	1.63481i	−0.576063	−	0.817405i	$-0.695412\pi$
$449$		−	34.6410i		−	1.63481i	0.576063	−	0.817405i	$-0.304588\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−38.0000			−1.77757			−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000			−1.77757			−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$		−	3.46410i		−	0.160990i	−0.996755	−	0.0804952i	$-0.974350\pi$
$463$		−	3.46410i		−	0.160990i	0.996755	−	0.0804952i	$-0.0256502\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	38.1051i		−	1.76329i	−0.471909	−	0.881647i	$-0.656435\pi$
$467$		−	38.1051i		−	1.76329i	0.471909	−	0.881647i	$-0.343565\pi$
$468$		0			0
$469$		−	13.8564i		−	0.639829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	36.0000			1.65528
$474$		0			0
$475$	−20.0000	+	8.66025i	−0.917663	+	0.397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			20.7846i			0.949673i	0.880074	+	0.474837i	$0.157493\pi$
$479$			20.7846i			0.949673i	−0.880074	+	0.474837i	$0.842507\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−34.0000			−1.54069			−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000			−1.54069			−0.770344	+	0.637629i	$0.779915\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.3205i		−	0.781664i	−0.920462	−	0.390832i	$-0.872187\pi$
$491$		−	17.3205i		−	0.781664i	0.920462	−	0.390832i	$-0.127813\pi$
$492$		0			0
$493$		−	41.5692i		−	1.87218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			41.5692i			1.86463i
$498$		0			0
$499$		−	31.1769i		−	1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$		−	31.1769i		−	1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13.8564i		−	0.617827i	−0.951090	−	0.308913i	$-0.900035\pi$
$503$		−	13.8564i		−	0.617827i	0.951090	−	0.308913i	$-0.0999653\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			6.92820i			0.307087i	0.988142	+	0.153544i	$0.0490686\pi$
$509$			6.92820i			0.307087i	−0.988142	+	0.153544i	$0.950931\pi$
$510$		0			0
$511$		−	6.92820i		−	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−24.0000			−1.05552
$518$		0			0
$519$		0			0
$520$		0			0
$521$			20.7846i			0.910590i	0.890341	+	0.455295i	$0.150466\pi$
$521$			20.7846i			0.910590i	−0.890341	+	0.455295i	$0.849534\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−60.0000			−2.61364
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			17.3205i			0.746047i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−32.0000			−1.36822			−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000			−1.36822			−0.684111	+	0.729378i	$0.739809\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000	−	27.7128i	−0.511217	−	1.18061i
$552$		0			0
$553$			34.6410i			1.47309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.0000			−0.508456			−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000			−0.508456			−0.254228	+	0.967144i	$0.581821\pi$
$558$		0			0
$559$	36.0000			1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			20.7846i			0.871336i	0.900107	+	0.435668i	$0.143488\pi$
$569$			20.7846i			0.871336i	−0.900107	+	0.435668i	$0.856512\pi$
$570$		0			0
$571$			24.2487i			1.01478i	0.861717	+	0.507388i	$0.169389\pi$
$571$			24.2487i			1.01478i	−0.861717	+	0.507388i	$0.830611\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	10.0000			0.416305			0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000			0.416305			0.208153	+	0.978096i	$0.433255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$	−48.0000			−1.98796
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	10.3923i		−	0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$		−	10.3923i		−	0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$		0			0
$589$	−40.0000	+	17.3205i	−1.64817	+	0.713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$			20.7846i			0.847822i	0.905704	+	0.423911i	$0.139343\pi$
$601$			20.7846i			0.847822i	−0.905704	+	0.423911i	$0.860657\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000			−0.970936
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−42.0000			−1.69086			−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000			−1.69086			−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$		−	45.0333i		−	1.81004i	−0.425367	−	0.905021i	$-0.639855\pi$
$619$		−	45.0333i		−	1.81004i	0.425367	−	0.905021i	$-0.360145\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.0000			0.961540
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	20.7846i		−	0.828737i
$630$		0			0
$631$			17.3205i			0.689519i	0.938691	+	0.344759i	$0.112039\pi$
$631$			17.3205i			0.689519i	−0.938691	+	0.344759i	$0.887961\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			17.3205i			0.686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	6.92820i		−	0.273648i	−0.990595	−	0.136824i	$-0.956311\pi$
$641$		−	6.92820i		−	0.273648i	0.990595	−	0.136824i	$-0.0436894\pi$
$642$		0			0
$643$			31.1769i			1.22950i	0.788723	+	0.614749i	$0.210743\pi$
$643$			31.1769i			1.22950i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.8564i			0.544752i	0.962191	+	0.272376i	$0.0878094\pi$
$647$			13.8564i			0.544752i	−0.962191	+	0.272376i	$0.912191\pi$
$648$		0			0
$649$		−	41.5692i		−	1.63173i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$			24.2487i			0.943166i	0.881822	+	0.471583i	$0.156317\pi$
$661$			24.2487i			0.943166i	−0.881822	+	0.471583i	$0.843683\pi$
$662$		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
$671$		−	34.6410i		−	1.33730i
$672$		0			0
$673$			34.6410i			1.33531i	0.744469	+	0.667657i	$0.232703\pi$
$673$			34.6410i			1.33531i	−0.744469	+	0.667657i	$0.767297\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−48.0000			−1.82865
$690$		0			0
$691$		−	31.1769i		−	1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$		−	31.1769i		−	1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			41.5692i			1.57455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	36.0000			1.35970			0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000			1.35970			0.679851	+	0.733351i	$0.262045\pi$
$702$		0			0
$703$	−6.00000	−	13.8564i	−0.226294	−	0.522604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	22.0000			0.826227			0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000			0.826227			0.413114	+	0.910679i	$0.364441\pi$
$710$		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
$719$			34.6410i			1.29189i	0.763383	+	0.645946i	$0.223537\pi$
$719$			34.6410i			1.29189i	−0.763383	+	0.645946i	$0.776463\pi$
$720$		0			0
$721$		−	6.92820i		−	0.258020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			34.6410i			1.28654i
$726$		0			0
$727$			10.3923i			0.385429i	0.981255	+	0.192715i	$0.0617292\pi$
$727$			10.3923i			0.385429i	−0.981255	+	0.192715i	$0.938271\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			62.3538i			2.30624i
$732$		0			0
$733$	−26.0000			−0.960332			−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000			−0.960332			−0.480166	+	0.877178i	$0.659424\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			13.8564i			0.510407i
$738$		0			0
$739$		−	10.3923i		−	0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$		−	10.3923i		−	0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			41.5692i			1.51891i
$750$		0			0
$751$	−38.0000			−1.38664			−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000			−1.38664			−0.693320	+	0.720630i	$0.743853\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$	36.0000			1.30329
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	41.5692i		−	1.50098i
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	34.6410i		−	1.24595i	−0.782241	−	0.622975i	$-0.785924\pi$
$773$		−	34.6410i		−	1.24595i	0.782241	−	0.622975i	$-0.214076\pi$
$774$		0			0
$775$	50.0000			1.79605
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000	+	27.7128i	0.429945	+	0.992915i
$780$		0			0
$781$		−	41.5692i		−	1.48746i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$		−	34.6410i		−	1.23014i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			48.4974i			1.71787i	0.512087	+	0.858933i	$0.328872\pi$
$797$			48.4974i			1.71787i	−0.512087	+	0.858933i	$0.671128\pi$
$798$		0			0
$799$		−	41.5692i		−	1.47061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			6.92820i			0.244491i
$804$		0			0
$805$		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.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.46410i q^{7} +O(q^{10})\)	\(q+3.46410i q^{7} -3.46410i q^{11} -3.46410i q^{13} +6.00000 q^{17} +(4.00000 - 1.73205i) q^{19} -5.00000 q^{25} -6.92820i q^{29} -10.0000 q^{31} -3.46410i q^{37} +6.92820i q^{41} +10.3923i q^{43} -6.92820i q^{47} -5.00000 q^{49} -13.8564i q^{53} +12.0000 q^{59} +10.0000 q^{61} -4.00000 q^{67} +12.0000 q^{71} -2.00000 q^{73} +12.0000 q^{77} +10.0000 q^{79} +3.46410i q^{83} -6.92820i q^{89} +12.0000 q^{91} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 12 q^{17} + 8 q^{19} - 10 q^{25} - 20 q^{31} - 10 q^{49} + 24 q^{59} + 20 q^{61} - 8 q^{67} + 24 q^{71} - 4 q^{73} + 24 q^{77} + 20 q^{79} + 24 q^{91}+O(q^{100}) \) 2 * q + 12 * q^17 + 8 * q^19 - 10 * q^25 - 20 * q^31 - 10 * q^49 + 24 * q^59 + 20 * q^61 - 8 * q^67 + 24 * q^71 - 4 * q^73 + 24 * q^77 + 20 * q^79 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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