LMFDB - Embedded newform 324.4.e.d.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, 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:	$19.1166188419$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.217
Dual form			324.4.e.d.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.50000 + 14.7224i) q^{7} +O(q^{10})$	$q+(-8.50000 + 14.7224i) q^{7} +(-44.5000 - 77.0763i) q^{13} +107.000 q^{19} +(62.5000 - 108.253i) q^{25} +(-154.000 - 266.736i) q^{31} -433.000 q^{37} +(260.000 - 450.333i) q^{43} +(27.0000 + 46.7654i) q^{49} +(450.500 - 780.289i) q^{61} +(-503.500 - 872.088i) q^{67} -271.000 q^{73} +(-251.500 + 435.611i) q^{79} +1513.00 q^{91} +(-926.500 + 1604.75i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 17 q^{7}+O(q^{10}) $ 2 * q - 17 * q^7	$ 2 q - 17 q^{7} - 89 q^{13} + 214 q^{19} + 125 q^{25} - 308 q^{31} - 866 q^{37} + 520 q^{43} + 54 q^{49} + 901 q^{61} - 1007 q^{67} - 542 q^{73} - 503 q^{79} + 3026 q^{91} - 1853 q^{97}+O(q^{100}) $ 2 * q - 17 * q^7 - 89 * q^13 + 214 * q^19 + 125 * q^25 - 308 * q^31 - 866 * q^37 + 520 * q^43 + 54 * q^49 + 901 * q^61 - 1007 * q^67 - 542 * q^73 - 503 * q^79 + 3026 * q^91 - 1853 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−8.50000	+	14.7224i	−0.458957	+	0.794937i	−0.998906	−	0.0467610i	$-0.985110\pi$
$7$	−8.50000	+	14.7224i	−0.458957	+	0.794937i	0.539949	+	0.841698i	$0.318443\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−44.5000	−	77.0763i	−0.949391	−	1.64439i	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−44.5000	−	77.0763i	−0.949391	−	1.64439i	−0.202679	−	0.979245i	$-0.564965\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	107.000			1.29197			0.645986	−	0.763349i	$-0.276446\pi$
$19$	107.000			1.29197			0.645986	+	0.763349i	$0.276446\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	62.5000	−	108.253i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.837192	−	0.546908i	$-0.815805\pi$
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.0550403	−	0.998484i	$-0.517529\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−433.000			−1.92391			−0.961956	−	0.273204i	$-0.911917\pi$
$37$	−433.000			−1.92391			−0.961956	+	0.273204i	$0.911917\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	260.000	−	450.333i	0.922084	−	1.59710i	0.125900	−	0.992043i	$-0.459818\pi$
$43$	260.000	−	450.333i	0.922084	−	1.59710i	0.796184	−	0.605054i	$-0.206849\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	27.0000	+	46.7654i	0.0787172	+	0.136342i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	450.500	−	780.289i	0.945584	−	1.63780i	0.191006	−	0.981589i	$-0.438825\pi$
$61$	450.500	−	780.289i	0.945584	−	1.63780i	0.754578	−	0.656210i	$-0.227842\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−503.500	−	872.088i	−0.918094	−	1.59019i	−0.802307	−	0.596912i	$-0.796394\pi$
$67$	−503.500	−	872.088i	−0.918094	−	1.59019i	−0.115787	−	0.993274i	$-0.536939\pi$
$68$		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
$73$	−271.000			−0.434495			−0.217248	−	0.976117i	$-0.569708\pi$
$73$	−271.000			−0.434495			−0.217248	+	0.976117i	$0.569708\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−251.500	+	435.611i	−0.358177	+	0.620380i	−0.987656	−	0.156637i	$-0.949935\pi$
$79$	−251.500	+	435.611i	−0.358177	+	0.620380i	0.629480	+	0.777017i	$0.283268\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	1513.00			1.74292
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−926.500	+	1604.75i	−0.969813	+	1.67976i	−0.273725	+	0.961808i	$0.588256\pi$
$97$	−926.500	+	1604.75i	−0.969813	+	1.67976i	−0.696088	+	0.717957i	$0.745078\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	9.50000	+	16.4545i	0.00908799	+	0.0157409i	0.870534	−	0.492109i	$-0.163774\pi$
$103$	9.50000	+	16.4545i	0.00908799	+	0.0157409i	−0.861446	+	0.507850i	$0.830440\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−646.000			−0.567666			−0.283833	−	0.958874i	$-0.591606\pi$
$109$	−646.000			−0.567666			−0.283833	+	0.958874i	$0.591606\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	665.500	+	1152.68i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	380.000			0.265508			0.132754	−	0.991149i	$-0.457618\pi$
$127$	380.000			0.265508			0.132754	+	0.991149i	$0.457618\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−909.500	+	1575.30i	−0.592960	+	1.02704i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	1521.50	+	2635.32i	0.928431	+	1.60809i	0.785948	+	0.618293i	$0.212175\pi$
$139$	1521.50	+	2635.32i	0.928431	+	1.60809i	0.142484	+	0.989797i	$0.454491\pi$
$140$		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
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	1854.50	−	3212.09i	0.999451	−	1.73110i	0.471027	−	0.882119i	$-0.343883\pi$
$151$	1854.50	−	3212.09i	0.999451	−	1.73110i	0.528424	−	0.848981i	$-0.322783\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.667699	+	0.744432i	$0.267279\pi$
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.310847	+	0.950460i	$0.399387\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−379.000			−0.182120			−0.0910600	−	0.995845i	$-0.529026\pi$
$163$	−379.000			−0.182120			−0.0910600	+	0.995845i	$0.529026\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−2862.00	+	4957.13i	−1.30269	+	2.25632i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	1062.50	+	1840.30i	0.458957	+	0.794937i
$176$		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
$181$	−4699.00			−1.92969			−0.964845	−	0.262819i	$-0.915348\pi$
$181$	−4699.00			−1.92969			−0.964845	+	0.262819i	$0.915348\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−2555.50	−	4426.26i	−0.953103	−	1.65082i	−0.738650	−	0.674089i	$-0.764536\pi$
$193$	−2555.50	−	4426.26i	−0.953103	−	1.65082i	−0.214453	−	0.976734i	$-0.568797\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	4373.00			1.55776			0.778879	−	0.627175i	$-0.215789\pi$
$199$	4373.00			1.55776			0.778879	+	0.627175i	$0.215789\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1035.50	+	1793.54i	0.337852	+	0.585177i	0.984028	−	0.178011i	$-0.0569664\pi$
$211$	1035.50	+	1793.54i	0.337852	+	0.585177i	−0.646177	+	0.763188i	$0.723633\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	5236.00			1.63799
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1610.00	−	2788.60i	0.483469	−	0.837393i	−0.516351	−	0.856377i	$-0.672710\pi$
$223$	1610.00	−	2788.60i	0.483469	−	0.837393i	0.999820	+	0.0189844i	$0.00604328\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−2233.00	−	3867.67i	−0.644370	−	1.11608i	−0.984447	−	0.175684i	$-0.943786\pi$
$229$	−2233.00	−	3867.67i	−0.644370	−	1.11608i	0.340076	−	0.940398i	$-0.389547\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−2384.50	+	4130.08i	−0.637341	+	1.10391i	0.348673	+	0.937244i	$0.386632\pi$
$241$	−2384.50	+	4130.08i	−0.637341	+	1.10391i	−0.986014	+	0.166662i	$0.946701\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4761.50	−	8247.16i	−1.22659	−	2.12451i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	3680.50	−	6374.81i	0.882993	−	1.52939i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		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
$271$	−8101.00			−1.81587			−0.907935	−	0.419111i	$-0.862342\pi$
$271$	−8101.00			−1.81587			−0.907935	+	0.419111i	$0.862342\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2015.00	−	3490.08i	0.437074	−	0.757035i	−0.560388	−	0.828230i	$-0.689348\pi$
$277$	2015.00	−	3490.08i	0.437074	−	0.757035i	0.997462	+	0.0711951i	$0.0226813\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$	−2800.00	−	4849.74i	−0.588137	−	1.01868i	−0.994476	−	0.104961i	$-0.966528\pi$
$283$	−2800.00	−	4849.74i	−0.588137	−	1.01868i	0.406340	−	0.913722i	$-0.366805\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4913.00			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	4420.00	+	7655.66i	0.846394	+	1.46600i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10640.0			1.97804			0.989018	−	0.147797i	$-0.0472182\pi$
$307$	10640.0			1.97804			0.989018	+	0.147797i	$0.0472182\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	4554.50	−	7888.63i	0.822478	−	1.42457i	−0.0813539	−	0.996685i	$-0.525924\pi$
$313$	4554.50	−	7888.63i	0.822478	−	1.42457i	0.903832	−	0.427888i	$-0.140742\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−11125.0			−1.89878
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5445.50	−	9431.88i	0.904265	−	1.56623i	0.0823644	−	0.996602i	$-0.473753\pi$
$331$	5445.50	−	9431.88i	0.904265	−	1.56623i	0.821901	−	0.569631i	$-0.192914\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6146.50	−	10646.1i	−0.993535	−	1.72085i	−0.595086	−	0.803662i	$-0.702882\pi$
$337$	−6146.50	−	10646.1i	−0.993535	−	1.72085i	−0.398448	−	0.917191i	$-0.630451\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6749.00			−1.06242
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	−683.500	+	1183.86i	−0.104834	+	0.181577i	−0.913670	−	0.406456i	$-0.866764\pi$
$349$	−683.500	+	1183.86i	−0.104834	+	0.181577i	0.808837	+	0.588033i	$0.200098\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	4590.00			0.669194
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−4706.50	+	8151.90i	−0.669420	+	1.15947i	0.308646	+	0.951177i	$0.400124\pi$
$367$	−4706.50	+	8151.90i	−0.669420	+	1.15947i	−0.978066	+	0.208293i	$0.933209\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6300.50	+	10912.8i	0.874605	+	1.51486i	0.857183	+	0.515011i	$0.172212\pi$
$373$	6300.50	+	10912.8i	0.874605	+	1.51486i	0.0174213	+	0.999848i	$0.494454\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	14687.0			1.99056			0.995278	−	0.0970683i	$-0.0309465\pi$
$379$	14687.0			1.99056			0.995278	+	0.0970683i	$0.0309465\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1190.00			0.150439			0.0752196	−	0.997167i	$-0.476034\pi$
$397$	1190.00			0.150439			0.0752196	+	0.997167i	$0.476034\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−13706.0	+	23739.5i	−1.69416	+	2.93436i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4148.50	−	7185.41i	−0.501541	−	0.868694i	−0.999998	−	0.00177990i	$-0.999433\pi$
$409$	−4148.50	−	7185.41i	−0.501541	−	0.868694i	0.498458	−	0.866914i	$-0.333900\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	3339.50	−	5784.18i	0.386597	−	0.669605i	−0.605392	−	0.795927i	$-0.706984\pi$
$421$	3339.50	−	5784.18i	0.386597	−	0.669605i	0.991989	+	0.126322i	$0.0403172\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7658.50	+	13264.9i	0.867964	+	1.50336i
$428$		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
$433$	−2590.00			−0.287454			−0.143727	−	0.989617i	$-0.545909\pi$
$433$	−2590.00			−0.287454			−0.143727	+	0.989617i	$0.545909\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−7462.00	+	12924.6i	−0.811257	+	1.40514i	0.100728	+	0.994914i	$0.467883\pi$
$439$	−7462.00	+	12924.6i	−0.811257	+	1.40514i	−0.911985	+	0.410224i	$0.865450\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6355.00	+	11007.2i	−0.650491	+	1.12668i	0.332513	+	0.943099i	$0.392103\pi$
$457$	−6355.00	+	11007.2i	−0.650491	+	1.12668i	−0.983004	+	0.183585i	$0.941230\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	−5984.50	−	10365.5i	−0.600698	−	1.04044i	−0.992716	−	0.120482i	$-0.961556\pi$
$463$	−5984.50	−	10365.5i	−0.600698	−	1.04044i	0.392017	−	0.919958i	$-0.371777\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	17119.0			1.68546
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6687.50	−	11583.1i	0.645986	−	1.11888i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	19268.5	+	33374.0i	1.82654	+	3.16367i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−6103.00			−0.567871			−0.283936	−	0.958843i	$-0.591640\pi$
$487$	−6103.00			−0.567871			−0.283936	+	0.958843i	$0.591640\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	7568.00	+	13108.2i	0.678938	+	1.17596i	0.975301	+	0.220880i	$0.0708930\pi$
$499$	7568.00	+	13108.2i	0.678938	+	1.17596i	−0.296363	+	0.955075i	$0.595774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	2303.50	−	3989.78i	0.199415	−	0.345396i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−11881.0			−0.993346			−0.496673	−	0.867938i	$-0.665445\pi$
$523$	−11881.0			−0.993346			−0.496673	+	0.867938i	$0.665445\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6083.50	−	10536.9i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1889.00			0.150119			0.0750596	−	0.997179i	$-0.476085\pi$
$541$	1889.00			0.150119			0.0750596	+	0.997179i	$0.476085\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−10646.5	+	18440.3i	−0.832196	+	1.44141i	0.0640963	+	0.997944i	$0.479584\pi$
$547$	−10646.5	+	18440.3i	−0.832196	+	1.44141i	−0.896293	+	0.443463i	$0.853750\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−4275.50	−	7405.38i	−0.328775	−	0.569456i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	−46280.0			−3.50167
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	11970.5	+	20733.5i	0.877320	+	1.51956i	0.854270	+	0.519829i	$0.174004\pi$
$571$	11970.5	+	20733.5i	0.877320	+	1.51956i	0.0230498	+	0.999734i	$0.492662\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	27323.0			1.97135			0.985677	−	0.168644i	$-0.0539387\pi$
$577$	27323.0			1.97135			0.985677	+	0.168644i	$0.0539387\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$	−16478.0	−	28540.7i	−1.15274	−	1.99661i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	14651.0	−	25376.3i	0.994387	−	1.72233i	0.405567	−	0.914065i	$-0.367074\pi$
$601$	14651.0	−	25376.3i	0.994387	−	1.72233i	0.588820	−	0.808264i	$-0.299593\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3068.50	−	5314.80i	−0.205184	−	0.355389i	0.745007	−	0.667056i	$-0.232446\pi$
$607$	−3068.50	−	5314.80i	−0.205184	−	0.355389i	−0.950191	+	0.311667i	$0.899113\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	12851.0			0.846732			0.423366	−	0.905959i	$-0.360848\pi$
$613$	12851.0			0.846732			0.423366	+	0.905959i	$0.360848\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$	18.5000	−	32.0429i	0.00120126	−	0.00208064i	−0.865424	−	0.501040i	$-0.832951\pi$
$619$	18.5000	−	32.0429i	0.00120126	−	0.00208064i	0.866625	+	0.498959i	$0.166284\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	26459.0			1.66928			0.834640	−	0.550795i	$-0.185675\pi$
$631$	26459.0			1.66928			0.834640	+	0.550795i	$0.185675\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2403.00	−	4162.12i	0.149467	−	0.258884i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−6580.00	−	11396.9i	−0.403561	−	0.698989i	0.590592	−	0.806971i	$-0.298894\pi$
$643$	−6580.00	−	11396.9i	−0.403561	−	0.698989i	−0.994153	+	0.107982i	$0.965561\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	−16865.5	−	29211.9i	−0.992423	−	1.71893i	−0.602615	−	0.798032i	$-0.705875\pi$
$661$	−16865.5	−	29211.9i	−0.992423	−	1.71893i	−0.389808	−	0.920896i	$-0.627459\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−4949.50	+	8572.79i	−0.283491	+	0.491020i	−0.972242	−	0.233977i	$-0.924826\pi$
$673$	−4949.50	+	8572.79i	−0.283491	+	0.491020i	0.688751	+	0.724998i	$0.258159\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−15750.5	−	27280.7i	−0.890204	−	1.54188i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	8036.00	−	13918.8i	0.442408	−	0.766273i	−0.555460	−	0.831543i	$-0.687458\pi$
$691$	8036.00	−	13918.8i	0.442408	−	0.766273i	0.997868	+	0.0652705i	$0.0207910\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−46331.0			−2.48564
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4311.50	−	7467.74i	0.228381	−	0.395567i	−0.728948	−	0.684569i	$-0.759990\pi$
$709$	4311.50	−	7467.74i	0.228381	−	0.395567i	0.957328	+	0.289003i	$0.0933236\pi$
$710$		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
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−323.000			−0.0166840
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	5390.00	−	9335.75i	0.274971	−	0.476264i	−0.695157	−	0.718858i	$-0.744665\pi$
$727$	5390.00	−	9335.75i	0.274971	−	0.476264i	0.970128	+	0.242594i	$0.0779984\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−7525.00	−	13033.7i	−0.379184	−	0.656767i	0.611759	−	0.791044i	$-0.290462\pi$
$733$	−7525.00	−	13033.7i	−0.379184	−	0.656767i	−0.990944	+	0.134277i	$0.957129\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	31376.0			1.56182			0.780910	−	0.624644i	$-0.214756\pi$
$739$	31376.0			1.56182			0.780910	+	0.624644i	$0.214756\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8784.50	+	15215.2i	0.426832	+	0.739295i	0.996590	−	0.0825179i	$-0.0262962\pi$
$751$	8784.50	+	15215.2i	0.426832	+	0.739295i	−0.569757	+	0.821813i	$0.692963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24137.0			1.15888			0.579441	−	0.815014i	$-0.303271\pi$
$757$	24137.0			1.15888			0.579441	+	0.815014i	$0.303271\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	5491.00	−	9510.69i	0.260534	−	0.451258i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	17208.5	+	29806.0i	0.806963	+	1.39770i	0.914958	+	0.403549i	$0.132224\pi$
$769$	17208.5	+	29806.0i	0.806963	+	1.39770i	−0.107995	+	0.994151i	$0.534443\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−38500.0			−1.78447
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	7326.50	+	12689.9i	0.331844	+	0.574771i	0.982874	−	0.184281i	$-0.0589958\pi$
$787$	7326.50	+	12689.9i	0.331844	+	0.574771i	−0.651029	+	0.759053i	$0.725662\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−80189.0			−3.59091
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.50000 + 14.7224i) q^{7} +O(q^{10})\)	\(q+(-8.50000 + 14.7224i) q^{7} +(-44.5000 - 77.0763i) q^{13} +107.000 q^{19} +(62.5000 - 108.253i) q^{25} +(-154.000 - 266.736i) q^{31} -433.000 q^{37} +(260.000 - 450.333i) q^{43} +(27.0000 + 46.7654i) q^{49} +(450.500 - 780.289i) q^{61} +(-503.500 - 872.088i) q^{67} -271.000 q^{73} +(-251.500 + 435.611i) q^{79} +1513.00 q^{91} +(-926.500 + 1604.75i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 17 q^{7}+O(q^{10}) \) 2 * q - 17 * q^7	\( 2 q - 17 q^{7} - 89 q^{13} + 214 q^{19} + 125 q^{25} - 308 q^{31} - 866 q^{37} + 520 q^{43} + 54 q^{49} + 901 q^{61} - 1007 q^{67} - 542 q^{73} - 503 q^{79} + 3026 q^{91} - 1853 q^{97}+O(q^{100}) \) 2 * q - 17 * q^7 - 89 * q^13 + 214 * q^19 + 125 * q^25 - 308 * q^31 - 866 * q^37 + 520 * q^43 + 54 * q^49 + 901 * q^61 - 1007 * q^67 - 542 * q^73 - 503 * q^79 + 3026 * q^91 - 1853 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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