LMFDB - Embedded newform 2160.2.f.m.1729.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(1729,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2160.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.4
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2160.1729
Dual form			2160.2.f.m.1729.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.17945 + 0.500000i) q^{5} -4.35890i q^{7} +O(q^{10})$	$q+(2.17945 + 0.500000i) q^{5} -4.35890i q^{7} -4.35890 q^{11} -4.00000i q^{17} +6.00000 q^{19} +2.00000i q^{23} +(4.50000 + 2.17945i) q^{25} -7.00000 q^{31} +(2.17945 - 9.50000i) q^{35} -8.71780i q^{37} -8.71780 q^{41} -8.71780i q^{43} +2.00000i q^{47} -12.0000 q^{49} -3.00000i q^{53} +(-9.50000 - 2.17945i) q^{55} +8.71780 q^{59} -4.00000 q^{61} -8.71780i q^{67} -4.35890i q^{73} +19.0000i q^{77} +5.00000i q^{83} +(2.00000 - 8.71780i) q^{85} +8.71780 q^{89} +(13.0767 + 3.00000i) q^{95} -4.35890i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{19} + 18 q^{25} - 28 q^{31} - 48 q^{49} - 38 q^{55} - 16 q^{61} + 8 q^{85}+O(q^{100}) $ 4 * q + 24 * q^19 + 18 * q^25 - 28 * q^31 - 48 * q^49 - 38 * q^55 - 16 * q^61 + 8 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	2.17945	+	0.500000i	0.974679	+	0.223607i
$5$	2.17945	+	0.500000i	0.974679	+	0.223607i
$6$		0			0
$7$		−	4.35890i		−	1.64751i	−0.566947	−	0.823754i	$-0.691875\pi$
$7$		−	4.35890i		−	1.64751i	0.566947	−	0.823754i	$-0.308125\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.35890			−1.31426			−0.657129	−	0.753778i	$-0.728229\pi$
$11$	−4.35890			−1.31426			−0.657129	+	0.753778i	$0.728229\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	4.50000	+	2.17945i	0.900000	+	0.435890i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.17945	−	9.50000i	0.368394	−	1.60579i
$36$		0			0
$37$		−	8.71780i		−	1.43320i	−0.697486	−	0.716599i	$-0.745698\pi$
$37$		−	8.71780i		−	1.43320i	0.697486	−	0.716599i	$-0.254302\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.71780			−1.36149			−0.680746	−	0.732520i	$-0.738344\pi$
$41$	−8.71780			−1.36149			−0.680746	+	0.732520i	$0.738344\pi$
$42$		0			0
$43$		−	8.71780i		−	1.32945i	−0.747087	−	0.664726i	$-0.768548\pi$
$43$		−	8.71780i		−	1.32945i	0.747087	−	0.664726i	$-0.231452\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	−12.0000			−1.71429
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.00000i		−	0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$		−	3.00000i		−	0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$		0			0
$55$	−9.50000	−	2.17945i	−1.28098	−	0.293877i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.71780			1.13496			0.567480	−	0.823387i	$-0.307918\pi$
$59$	8.71780			1.13496			0.567480	+	0.823387i	$0.307918\pi$
$60$		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
$67$		−	8.71780i		−	1.06505i	−0.846415	−	0.532524i	$-0.821244\pi$
$67$		−	8.71780i		−	1.06505i	0.846415	−	0.532524i	$-0.178756\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	4.35890i		−	0.510171i	−0.966919	−	0.255085i	$-0.917896\pi$
$73$		−	4.35890i		−	0.510171i	0.966919	−	0.255085i	$-0.0821035\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			19.0000i			2.16525i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.00000i			0.548821i	0.961613	+	0.274411i	$0.0884828\pi$
$83$			5.00000i			0.548821i	−0.961613	+	0.274411i	$0.911517\pi$
$84$		0			0
$85$	2.00000	−	8.71780i	0.216930	−	0.945578i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.71780			0.924085			0.462042	−	0.886858i	$-0.347117\pi$
$89$	8.71780			0.924085			0.462042	+	0.886858i	$0.347117\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.0767	+	3.00000i	1.34164	+	0.307794i
$96$		0			0
$97$		−	4.35890i		−	0.442579i	−0.975208	−	0.221290i	$-0.928973\pi$
$97$		−	4.35890i		−	0.442579i	0.975208	−	0.221290i	$-0.0710266\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.35890			−0.433727			−0.216863	−	0.976202i	$-0.569583\pi$
$101$	−4.35890			−0.433727			−0.216863	+	0.976202i	$0.569583\pi$
$102$		0			0
$103$			8.71780i			0.858990i	0.903069	+	0.429495i	$0.141308\pi$
$103$			8.71780i			0.858990i	−0.903069	+	0.429495i	$0.858692\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.0000i			1.45010i	0.688694	+	0.725052i	$0.258184\pi$
$107$			15.0000i			1.45010i	−0.688694	+	0.725052i	$0.741816\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$	−1.00000	+	4.35890i	−0.0932505	+	0.406469i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.4356			−1.59832
$120$		0			0
$121$	8.00000			0.727273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.71780	+	7.00000i	0.779744	+	0.626099i
$126$		0			0
$127$		−	4.35890i		−	0.386790i	−0.981121	−	0.193395i	$-0.938050\pi$
$127$		−	4.35890i		−	0.386790i	0.981121	−	0.193395i	$-0.0619498\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.0767			−1.14252			−0.571258	−	0.820770i	$-0.693544\pi$
$131$	−13.0767			−1.14252			−0.571258	+	0.820770i	$0.693544\pi$
$132$		0			0
$133$		−	26.1534i		−	2.26779i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.35890			0.357095			0.178547	−	0.983931i	$-0.442860\pi$
$149$	4.35890			0.357095			0.178547	+	0.983931i	$0.442860\pi$
$150$		0			0
$151$	11.0000			0.895167			0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000			0.895167			0.447584	+	0.894242i	$0.352285\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.2561	−	3.50000i	−1.22540	−	0.281127i
$156$		0			0
$157$		−	17.4356i		−	1.39151i	−0.718278	−	0.695756i	$-0.755069\pi$
$157$		−	17.4356i		−	1.39151i	0.718278	−	0.695756i	$-0.244931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.71780			0.687059
$162$		0			0
$163$		−	17.4356i		−	1.36566i	−0.730577	−	0.682831i	$-0.760749\pi$
$163$		−	17.4356i		−	1.36566i	0.730577	−	0.682831i	$-0.239251\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	19.0000i		−	1.44454i	−0.691609	−	0.722272i	$-0.743098\pi$
$173$		−	19.0000i		−	1.44454i	0.691609	−	0.722272i	$-0.256902\pi$
$174$		0			0
$175$	9.50000	−	19.6150i	0.718132	−	1.48276i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.7945			1.62900			0.814499	−	0.580166i	$-0.197012\pi$
$179$	21.7945			1.62900			0.814499	+	0.580166i	$0.197012\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.35890	−	19.0000i	0.320473	−	1.39691i
$186$		0			0
$187$			17.4356i			1.27502i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			21.7945i			1.56880i	0.620254	+	0.784401i	$0.287030\pi$
$193$			21.7945i			1.56880i	−0.620254	+	0.784401i	$0.712970\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.00000i		−	0.356235i	−0.984009	−	0.178118i	$-0.942999\pi$
$197$		−	5.00000i		−	0.356235i	0.984009	−	0.178118i	$-0.0570008\pi$
$198$		0			0
$199$	−3.00000			−0.212664			−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000			−0.212664			−0.106332	+	0.994331i	$0.533911\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−19.0000	−	4.35890i	−1.32702	−	0.304439i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−26.1534			−1.80907
$210$		0			0
$211$	−6.00000			−0.413057			−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000			−0.413057			−0.206529	+	0.978441i	$0.566217\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.35890	−	19.0000i	0.297274	−	1.29579i
$216$		0			0
$217$			30.5123i			2.07131i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	8.71780i		−	0.583787i	−0.956451	−	0.291893i	$-0.905715\pi$
$223$		−	8.71780i		−	0.583787i	0.956451	−	0.291893i	$-0.0942853\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			4.00000i			0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$			4.00000i			0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.0000i			0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$			14.0000i			0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$		0			0
$235$	−1.00000	+	4.35890i	−0.0652328	+	0.284343i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.71780			−0.563907			−0.281954	−	0.959428i	$-0.590982\pi$
$239$	−8.71780			−0.563907			−0.281954	+	0.959428i	$0.590982\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−26.1534	−	6.00000i	−1.67088	−	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	26.1534			1.65079			0.825394	−	0.564557i	$-0.190953\pi$
$251$	26.1534			1.65079			0.825394	+	0.564557i	$0.190953\pi$
$252$		0			0
$253$		−	8.71780i		−	0.548083i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			12.0000i			0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$			12.0000i			0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$		0			0
$259$	−38.0000			−2.36121
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.00000i			0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$			2.00000i			0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$		0			0
$265$	1.50000	−	6.53835i	0.0921443	−	0.401648i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	5.00000			0.303728			0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000			0.303728			0.151864	+	0.988401i	$0.451472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.6150	−	9.50000i	−1.18283	−	0.572872i
$276$		0			0
$277$			17.4356i			1.04760i	0.851840	+	0.523802i	$0.175487\pi$
$277$			17.4356i			1.04760i	−0.851840	+	0.523802i	$0.824513\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$			8.71780i			0.518219i	0.965848	+	0.259110i	$0.0834291\pi$
$283$			8.71780i			0.518219i	−0.965848	+	0.259110i	$0.916571\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			38.0000i			2.24307i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	19.0000	+	4.35890i	1.10622	+	0.253785i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−38.0000			−2.19028
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.71780	−	2.00000i	−0.499180	−	0.114520i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	26.1534			1.48302			0.741511	−	0.670940i	$-0.234109\pi$
$311$	26.1534			1.48302			0.741511	+	0.670940i	$0.234109\pi$
$312$		0			0
$313$			21.7945i			1.23190i	0.787786	+	0.615949i	$0.211227\pi$
$313$			21.7945i			1.23190i	−0.787786	+	0.615949i	$0.788773\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.0000i			0.954815i	0.878682	+	0.477408i	$0.158423\pi$
$317$			17.0000i			0.954815i	−0.878682	+	0.477408i	$0.841577\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.71780			0.480628
$330$		0			0
$331$	−10.0000			−0.549650			−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000			−0.549650			−0.274825	+	0.961494i	$0.588620\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.35890	−	19.0000i	0.238152	−	1.03808i
$336$		0			0
$337$			17.4356i			0.949777i	0.880046	+	0.474889i	$0.157512\pi$
$337$			17.4356i			0.949777i	−0.880046	+	0.474889i	$0.842488\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.5123			1.65233
$342$		0			0
$343$			21.7945i			1.17679i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$		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
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.1534			−1.38032			−0.690162	−	0.723655i	$-0.742461\pi$
$359$	−26.1534			−1.38032			−0.690162	+	0.723655i	$0.742461\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.17945	−	9.50000i	0.114078	−	0.497253i
$366$		0			0
$367$		−	4.35890i		−	0.227533i	−0.993508	−	0.113766i	$-0.963708\pi$
$367$		−	4.35890i		−	0.227533i	0.993508	−	0.113766i	$-0.0362915\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.0767			−0.678908
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4.00000i			0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$			4.00000i			0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$		0			0
$385$	−9.50000	+	41.4095i	−0.484165	+	2.11043i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.35890			−0.221005			−0.110502	−	0.993876i	$-0.535246\pi$
$389$	−4.35890			−0.221005			−0.110502	+	0.993876i	$0.535246\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	34.8712i		−	1.75013i	−0.484001	−	0.875067i	$-0.660817\pi$
$397$		−	34.8712i		−	1.75013i	0.484001	−	0.875067i	$-0.339183\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.8712			1.74138			0.870692	−	0.491828i	$-0.163671\pi$
$401$	34.8712			1.74138			0.870692	+	0.491828i	$0.163671\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			38.0000i			1.88359i
$408$		0			0
$409$	31.0000			1.53285			0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000			1.53285			0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	38.0000i		−	1.86986i
$414$		0			0
$415$	−2.50000	+	10.8972i	−0.122720	+	0.534925i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.71780			−0.425892			−0.212946	−	0.977064i	$-0.568306\pi$
$419$	−8.71780			−0.425892			−0.212946	+	0.977064i	$0.568306\pi$
$420$		0			0
$421$	16.0000			0.779792			0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000			0.779792			0.389896	+	0.920859i	$0.372511\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.71780	−	18.0000i	0.422875	−	0.873128i
$426$		0			0
$427$			17.4356i			0.843768i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.71780			0.419922			0.209961	−	0.977710i	$-0.432666\pi$
$431$	8.71780			0.419922			0.209961	+	0.977710i	$0.432666\pi$
$432$		0			0
$433$			21.7945i			1.04738i	0.851910	+	0.523688i	$0.175444\pi$
$433$			21.7945i			1.04738i	−0.851910	+	0.523688i	$0.824556\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			12.0000i			0.574038i
$438$		0			0
$439$	9.00000			0.429547			0.214773	−	0.976664i	$-0.431099\pi$
$439$	9.00000			0.429547			0.214773	+	0.976664i	$0.431099\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	19.0000	+	4.35890i	0.900686	+	0.206632i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.1534			1.23425			0.617127	−	0.786863i	$-0.288296\pi$
$449$	26.1534			1.23425			0.617127	+	0.786863i	$0.288296\pi$
$450$		0			0
$451$	38.0000			1.78935
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			4.35890i			0.203901i	0.994789	+	0.101950i	$0.0325083\pi$
$457$			4.35890i			0.203901i	−0.994789	+	0.101950i	$0.967492\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−39.2301			−1.82713			−0.913564	−	0.406696i	$-0.866681\pi$
$461$	−39.2301			−1.82713			−0.913564	+	0.406696i	$0.866681\pi$
$462$		0			0
$463$			13.0767i			0.607726i	0.952716	+	0.303863i	$0.0982765\pi$
$463$			13.0767i			0.607726i	−0.952716	+	0.303863i	$0.901724\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	21.0000i		−	0.971764i	−0.874024	−	0.485882i	$-0.838498\pi$
$467$		−	21.0000i		−	0.971764i	0.874024	−	0.485882i	$-0.161502\pi$
$468$		0			0
$469$	−38.0000			−1.75468
$470$		0			0
$471$		0			0
$472$		0			0
$473$			38.0000i			1.74724i
$474$		0			0
$475$	27.0000	+	13.0767i	1.23884	+	0.600000i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.71780			−0.398326			−0.199163	−	0.979966i	$-0.563822\pi$
$479$	−8.71780			−0.398326			−0.199163	+	0.979966i	$0.563822\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.17945	−	9.50000i	0.0989637	−	0.431373i
$486$		0			0
$487$			26.1534i			1.18512i	0.805525	+	0.592562i	$0.201883\pi$
$487$			26.1534i			1.18512i	−0.805525	+	0.592562i	$0.798117\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.0767			−0.590143			−0.295072	−	0.955475i	$-0.595343\pi$
$491$	−13.0767			−0.590143			−0.295072	+	0.955475i	$0.595343\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	30.0000			1.34298			0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000			1.34298			0.671492	+	0.741012i	$0.265654\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.0000i		−	1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$		−	30.0000i		−	1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$		0			0
$505$	−9.50000	−	2.17945i	−0.422744	−	0.0969842i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.0767			−0.579614			−0.289807	−	0.957085i	$-0.593591\pi$
$509$	−13.0767			−0.579614			−0.289807	+	0.957085i	$0.593591\pi$
$510$		0			0
$511$	−19.0000			−0.840511
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.35890	+	19.0000i	−0.192076	+	0.837240i
$516$		0			0
$517$		−	8.71780i		−	0.383408i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.4356			−0.763867			−0.381934	−	0.924190i	$-0.624742\pi$
$521$	−17.4356			−0.763867			−0.381934	+	0.924190i	$0.624742\pi$
$522$		0			0
$523$			34.8712i			1.52481i	0.647100	+	0.762405i	$0.275982\pi$
$523$			34.8712i			1.52481i	−0.647100	+	0.762405i	$0.724018\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			28.0000i			1.21970i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−7.50000	+	32.6917i	−0.324253	+	1.41339i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	52.3068			2.25301
$540$		0			0
$541$	40.0000			1.71973			0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000			1.71973			0.859867	+	0.510518i	$0.170546\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−21.7945	−	5.00000i	−0.933574	−	0.214176i
$546$		0			0
$547$		−	17.4356i		−	0.745492i	−0.927933	−	0.372746i	$-0.878416\pi$
$547$		−	17.4356i		−	0.745492i	0.927933	−	0.372746i	$-0.121584\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			33.0000i			1.39825i	0.714997	+	0.699127i	$0.246428\pi$
$557$			33.0000i			1.39825i	−0.714997	+	0.699127i	$0.753572\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	11.0000i		−	0.463595i	−0.972764	−	0.231797i	$-0.925539\pi$
$563$		−	11.0000i		−	0.463595i	0.972764	−	0.231797i	$-0.0744606\pi$
$564$		0			0
$565$	9.00000	−	39.2301i	0.378633	−	1.65042i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.4356			−0.730938			−0.365469	−	0.930823i	$-0.619091\pi$
$569$	−17.4356			−0.730938			−0.365469	+	0.930823i	$0.619091\pi$
$570$		0			0
$571$	−16.0000			−0.669579			−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000			−0.669579			−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.35890	+	9.00000i	−0.181779	+	0.375326i
$576$		0			0
$577$			17.4356i			0.725853i	0.931818	+	0.362927i	$0.118222\pi$
$577$			17.4356i			0.725853i	−0.931818	+	0.362927i	$0.881778\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21.7945			0.904188
$582$		0			0
$583$			13.0767i			0.541581i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			3.00000i			0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$			3.00000i			0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$		0			0
$589$	−42.0000			−1.73058
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		0			0
$595$	−38.0000	−	8.71780i	−1.55785	−	0.357395i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.1534			1.06860			0.534299	−	0.845295i	$-0.320576\pi$
$599$	26.1534			1.06860			0.534299	+	0.845295i	$0.320576\pi$
$600$		0			0
$601$	−13.0000			−0.530281			−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000			−0.530281			−0.265141	+	0.964210i	$0.585418\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	17.4356	+	4.00000i	0.708858	+	0.162623i
$606$		0			0
$607$		−	8.71780i		−	0.353845i	−0.984225	−	0.176922i	$-0.943386\pi$
$607$		−	8.71780i		−	0.353845i	0.984225	−	0.176922i	$-0.0566141\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			8.71780i			0.352109i	0.984380	+	0.176054i	$0.0563334\pi$
$613$			8.71780i			0.352109i	−0.984380	+	0.176054i	$0.943667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$	32.0000			1.28619			0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000			1.28619			0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	38.0000i		−	1.52244i
$624$		0			0
$625$	15.5000	+	19.6150i	0.620000	+	0.784602i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−34.8712			−1.39041
$630$		0			0
$631$	−5.00000			−0.199047			−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000			−0.199047			−0.0995234	+	0.995035i	$0.531732\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.17945	−	9.50000i	0.0864888	−	0.376996i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.71780			−0.344332			−0.172166	−	0.985068i	$-0.555077\pi$
$641$	−8.71780			−0.344332			−0.172166	+	0.985068i	$0.555077\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		0			0
$649$	−38.0000			−1.49163
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	35.0000i		−	1.36966i	−0.728705	−	0.684828i	$-0.759877\pi$
$653$		−	35.0000i		−	1.36966i	0.728705	−	0.684828i	$-0.240123\pi$
$654$		0			0
$655$	−28.5000	−	6.53835i	−1.11359	−	0.255474i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.35890			−0.169799			−0.0848993	−	0.996390i	$-0.527057\pi$
$659$	−4.35890			−0.169799			−0.0848993	+	0.996390i	$0.527057\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	13.0767	−	57.0000i	0.507093	−	2.21037i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	17.4356			0.673094
$672$		0			0
$673$			13.0767i			0.504070i	0.967718	+	0.252035i	$0.0810997\pi$
$673$			13.0767i			0.504070i	−0.967718	+	0.252035i	$0.918900\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			26.0000i			0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$			26.0000i			0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$		0			0
$679$	−19.0000			−0.729153
$680$		0			0
$681$		0			0
$682$		0			0
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$		0			0
$685$	−9.00000	+	39.2301i	−0.343872	+	1.49890i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.71780	+	2.00000i	0.330685	+	0.0758643i
$696$		0			0
$697$			34.8712i			1.32084i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.7945			−0.823167			−0.411583	−	0.911372i	$-0.635024\pi$
$701$	−21.7945			−0.823167			−0.411583	+	0.911372i	$0.635024\pi$
$702$		0			0
$703$		−	52.3068i		−	1.97279i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			19.0000i			0.714569i
$708$		0			0
$709$	−12.0000			−0.450669			−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000			−0.450669			−0.225335	+	0.974281i	$0.572348\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	14.0000i		−	0.524304i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.8712			−1.30048			−0.650238	−	0.759731i	$-0.725331\pi$
$719$	−34.8712			−1.30048			−0.650238	+	0.759731i	$0.725331\pi$
$720$		0			0
$721$	38.0000			1.41519
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			39.2301i			1.45496i	0.686127	+	0.727482i	$0.259309\pi$
$727$			39.2301i			1.45496i	−0.686127	+	0.727482i	$0.740691\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.8712			−1.28976
$732$		0			0
$733$		−	8.71780i		−	0.321999i	−0.986954	−	0.161000i	$-0.948528\pi$
$733$		−	8.71780i		−	0.321999i	0.986954	−	0.161000i	$-0.0514718\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			38.0000i			1.39975i
$738$		0			0
$739$	16.0000			0.588570			0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000			0.588570			0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	48.0000i		−	1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$		−	48.0000i		−	1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$		0			0
$745$	9.50000	+	2.17945i	0.348053	+	0.0798489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	65.3835			2.38906
$750$		0			0
$751$	−35.0000			−1.27717			−0.638584	−	0.769552i	$-0.720480\pi$
$751$	−35.0000			−1.27717			−0.638584	+	0.769552i	$0.720480\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	23.9739	+	5.50000i	0.872501	+	0.200165i
$756$		0			0
$757$		−	8.71780i		−	0.316854i	−0.987371	−	0.158427i	$-0.949358\pi$
$757$		−	8.71780i		−	0.316854i	0.987371	−	0.158427i	$-0.0506422\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−52.3068			−1.89612			−0.948060	−	0.318092i	$-0.896958\pi$
$761$	−52.3068			−1.89612			−0.948060	+	0.318092i	$0.896958\pi$
$762$		0			0
$763$			43.5890i			1.57803i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−9.00000			−0.324548			−0.162274	−	0.986746i	$-0.551883\pi$
$769$	−9.00000			−0.324548			−0.162274	+	0.986746i	$0.551883\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	−31.5000	−	15.2561i	−1.13151	−	0.548017i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−52.3068			−1.87409
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.71780	−	38.0000i	0.311152	−	1.35628i
$786$		0			0
$787$		−	17.4356i		−	0.621512i	−0.950490	−	0.310756i	$-0.899418\pi$
$787$		−	17.4356i		−	0.621512i	0.950490	−	0.310756i	$-0.100582\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−78.4602			−2.78972
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	37.0000i		−	1.31061i	−0.755366	−	0.655304i	$-0.772541\pi$
$797$		−	37.0000i		−	1.31061i	0.755366	−	0.655304i	$-0.227459\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$		0			0
$802$		0			0
$803$

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.17945 + 0.500000i) q^{5} -4.35890i q^{7} +O(q^{10})\)	\(q+(2.17945 + 0.500000i) q^{5} -4.35890i q^{7} -4.35890 q^{11} -4.00000i q^{17} +6.00000 q^{19} +2.00000i q^{23} +(4.50000 + 2.17945i) q^{25} -7.00000 q^{31} +(2.17945 - 9.50000i) q^{35} -8.71780i q^{37} -8.71780 q^{41} -8.71780i q^{43} +2.00000i q^{47} -12.0000 q^{49} -3.00000i q^{53} +(-9.50000 - 2.17945i) q^{55} +8.71780 q^{59} -4.00000 q^{61} -8.71780i q^{67} -4.35890i q^{73} +19.0000i q^{77} +5.00000i q^{83} +(2.00000 - 8.71780i) q^{85} +8.71780 q^{89} +(13.0767 + 3.00000i) q^{95} -4.35890i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{19} + 18 q^{25} - 28 q^{31} - 48 q^{49} - 38 q^{55} - 16 q^{61} + 8 q^{85}+O(q^{100}) \) 4 * q + 24 * q^19 + 18 * q^25 - 28 * q^31 - 48 * q^49 - 38 * q^55 - 16 * q^61 + 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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