LMFDB - Embedded newform 324.3.d.d.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(163,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.163");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.d (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:	yes
Analytic conductor:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			163.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	324.163

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -1.19615 q^{5} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} -1.19615 q^{5} +8.00000 q^{8} -2.39230 q^{10} +25.7846 q^{13} +16.0000 q^{16} +17.9808 q^{17} -4.78461 q^{20} -23.5692 q^{25} +51.5692 q^{26} -56.3731 q^{29} +32.0000 q^{32} +35.9615 q^{34} +55.7846 q^{37} -9.56922 q^{40} -80.0000 q^{41} +49.0000 q^{49} -47.1384 q^{50} +103.138 q^{52} -56.0000 q^{53} -112.746 q^{58} -92.9230 q^{61} +64.0000 q^{64} -30.8423 q^{65} +71.9230 q^{68} +28.1384 q^{73} +111.569 q^{74} -19.1384 q^{80} -160.000 q^{82} -21.5077 q^{85} -147.550 q^{89} -130.000 q^{97} +98.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8 + 16 * q^10 + 10 * q^13 + 32 * q^16 - 16 * q^17 + 32 * q^20 + 36 * q^25 + 20 * q^26 - 40 * q^29 + 64 * q^32 - 32 * q^34 + 70 * q^37 + 64 * q^40 - 160 * q^41 + 98 * q^49 + 72 * q^50 + 40 * q^52 - 112 * q^53 - 80 * q^58 + 22 * q^61 + 128 * q^64 - 176 * q^65 - 64 * q^68 - 110 * q^73 + 140 * q^74 + 128 * q^80 - 320 * q^82 - 334 * q^85 - 160 * q^89 - 260 * q^97 + 196 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000	1.00000
$2$	2.00000	1.00000
$3$	0	0
$3$	0	0
$4$	4.00000	1.00000
$5$	−1.19615	−0.239230	−0.119615	−	0.992820i	$-0.538166\pi$
$5$	−1.19615	−0.239230	−0.119615	+	0.992820i	$0.538166\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	8.00000	1.00000
$9$	0	0
$10$	−2.39230	−0.239230
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	25.7846	1.98343	0.991716	−	0.128452i	$-0.0410008\pi$
$13$	25.7846	1.98343	0.991716	+	0.128452i	$0.0410008\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	17.9808	1.05769	0.528846	−	0.848718i	$-0.322625\pi$
$17$	17.9808	1.05769	0.528846	+	0.848718i	$0.322625\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−4.78461	−0.239230
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−23.5692	−0.942769
$26$	51.5692	1.98343
$27$	0	0
$28$	0	0
$29$	−56.3731	−1.94390	−0.971949	−	0.235190i	$-0.924429\pi$
$29$	−56.3731	−1.94390	−0.971949	+	0.235190i	$0.924429\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000	1.00000
$33$	0	0
$34$	35.9615	1.05769
$35$	0	0
$36$	0	0
$37$	55.7846	1.50769	0.753846	−	0.657051i	$-0.228196\pi$
$37$	55.7846	1.50769	0.753846	+	0.657051i	$0.228196\pi$
$38$	0	0
$39$	0	0
$40$	−9.56922	−0.239230
$41$	−80.0000	−1.95122	−0.975610	−	0.219512i	$-0.929553\pi$
$41$	−80.0000	−1.95122	−0.975610	+	0.219512i	$0.929553\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	−47.1384	−0.942769
$51$	0	0
$52$	103.138	1.98343
$53$	−56.0000	−1.05660	−0.528302	−	0.849057i	$-0.677171\pi$
$53$	−56.0000	−1.05660	−0.528302	+	0.849057i	$0.677171\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−112.746	−1.94390
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−92.9230	−1.52333	−0.761664	−	0.647972i	$-0.775617\pi$
$61$	−92.9230	−1.52333	−0.761664	+	0.647972i	$0.775617\pi$
$62$	0	0
$63$	0	0
$64$	64.0000	1.00000
$65$	−30.8423	−0.474497
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	71.9230	1.05769
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	28.1384	0.385458	0.192729	−	0.981252i	$-0.438266\pi$
$73$	28.1384	0.385458	0.192729	+	0.981252i	$0.438266\pi$
$74$	111.569	1.50769
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−19.1384	−0.239230
$81$	0	0
$82$	−160.000	−1.95122
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−21.5077	−0.253032
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−147.550	−1.65786	−0.828932	−	0.559349i	$-0.811051\pi$
$89$	−147.550	−1.65786	−0.828932	+	0.559349i	$0.811051\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−130.000	−1.34021	−0.670103	−	0.742268i	$-0.733750\pi$
$97$	−130.000	−1.34021	−0.670103	+	0.742268i	$0.733750\pi$
$98$	98.0000	1.00000
$99$	0	0
$100$	−94.2769	−0.942769
$101$	40.0000	0.396040	0.198020	−	0.980198i	$-0.436549\pi$
$101$	40.0000	0.396040	0.198020	+	0.980198i	$0.436549\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	206.277	1.98343
$105$	0	0
$106$	−112.000	−1.05660
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−194.923	−1.78828	−0.894142	−	0.447783i	$-0.852214\pi$
$109$	−194.923	−1.78828	−0.894142	+	0.447783i	$0.852214\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	137.981	1.22107	0.610534	−	0.791990i	$-0.290955\pi$
$113$	137.981	1.22107	0.610534	+	0.791990i	$0.290955\pi$
$114$	0	0
$115$	0	0
$116$	−225.492	−1.94390
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	−185.846	−1.52333
$123$	0	0
$124$	0	0
$125$	58.0962	0.464770
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	−61.6846	−0.474497
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	143.846	1.05769
$137$	269.865	1.96982	0.984910	−	0.173067i	$-0.0553679\pi$
$137$	269.865	1.96982	0.984910	+	0.173067i	$0.0553679\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	67.4308	0.465040
$146$	56.2769	0.385458
$147$	0	0
$148$	223.138	1.50769
$149$	−51.6654	−0.346748	−0.173374	−	0.984856i	$-0.555467\pi$
$149$	−51.6654	−0.346748	−0.173374	+	0.984856i	$0.555467\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−313.631	−1.99765	−0.998824	−	0.0484851i	$-0.984561\pi$
$157$	−313.631	−1.99765	−0.998824	+	0.0484851i	$0.984561\pi$
$158$	0	0
$159$	0	0
$160$	−38.2769	−0.239230
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−320.000	−1.95122
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	495.846	2.93400
$170$	−43.0155	−0.253032
$171$	0	0
$172$	0	0
$173$	−233.788	−1.35138	−0.675689	−	0.737187i	$-0.736154\pi$
$173$	−233.788	−1.35138	−0.675689	+	0.737187i	$0.736154\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−295.100	−1.65786
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	38.0000	0.209945	0.104972	−	0.994475i	$-0.466525\pi$
$181$	38.0000	0.209945	0.104972	+	0.994475i	$0.466525\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−66.7269	−0.360686
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−195.985	−1.01546	−0.507732	−	0.861515i	$-0.669516\pi$
$193$	−195.985	−1.01546	−0.507732	+	0.861515i	$0.669516\pi$
$194$	−260.000	−1.34021
$195$	0	0
$196$	196.000	1.00000
$197$	365.750	1.85660	0.928299	−	0.371834i	$-0.121271\pi$
$197$	365.750	1.85660	0.928299	+	0.371834i	$0.121271\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−188.554	−0.942769
$201$	0	0
$202$	80.0000	0.396040
$203$	0	0
$204$	0	0
$205$	95.6922	0.466791
$206$	0	0
$207$	0	0
$208$	412.554	1.98343
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−224.000	−1.05660
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−389.846	−1.78828
$219$	0	0
$220$	0	0
$221$	463.627	2.09786
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	275.962	1.22107
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	117.077	0.511253	0.255627	−	0.966776i	$-0.417718\pi$
$229$	117.077	0.511253	0.255627	+	0.966776i	$0.417718\pi$
$230$	0	0
$231$	0	0
$232$	−450.985	−1.94390
$233$	389.865	1.67324	0.836621	−	0.547782i	$-0.184528\pi$
$233$	389.865	1.67324	0.836621	+	0.547782i	$0.184528\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	416.846	1.72965	0.864826	−	0.502072i	$-0.167429\pi$
$241$	416.846	1.72965	0.864826	+	0.502072i	$0.167429\pi$
$242$	242.000	1.00000
$243$	0	0
$244$	−371.692	−1.52333
$245$	−58.6115	−0.239230
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	116.192	0.464770
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	−473.673	−1.84309	−0.921543	−	0.388277i	$-0.873070\pi$
$257$	−473.673	−1.84309	−0.921543	+	0.388277i	$0.873070\pi$
$258$	0	0
$259$	0	0
$260$	−123.369	−0.474497
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	66.9845	0.252772
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−140.488	−0.522262	−0.261131	−	0.965303i	$-0.584095\pi$
$269$	−140.488	−0.522262	−0.261131	+	0.965303i	$0.584095\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	287.692	1.05769
$273$	0	0
$274$	539.731	1.96982
$275$	0	0
$276$	0	0
$277$	230.000	0.830325	0.415162	−	0.909747i	$-0.363725\pi$
$277$	230.000	0.830325	0.415162	+	0.909747i	$0.363725\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	560.104	1.99325	0.996626	−	0.0820785i	$-0.0261558\pi$
$281$	560.104	1.99325	0.996626	+	0.0820785i	$0.0261558\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	34.3078	0.118712
$290$	134.862	0.465040
$291$	0	0
$292$	112.554	0.385458
$293$	425.634	1.45268	0.726339	−	0.687337i	$-0.241220\pi$
$293$	425.634	1.45268	0.726339	+	0.687337i	$0.241220\pi$
$294$	0	0
$295$	0	0
$296$	446.277	1.50769
$297$	0	0
$298$	−103.331	−0.346748
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	111.150	0.364427
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−565.400	−1.80639	−0.903195	−	0.429231i	$-0.858785\pi$
$313$	−565.400	−1.80639	−0.903195	+	0.429231i	$0.858785\pi$
$314$	−627.261	−1.99765
$315$	0	0
$316$	0	0
$317$	−437.904	−1.38140	−0.690700	−	0.723141i	$-0.742697\pi$
$317$	−437.904	−1.38140	−0.690700	+	0.723141i	$0.742697\pi$
$318$	0	0
$319$	0	0
$320$	−76.5538	−0.239230
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−607.723	−1.86992
$326$	0	0
$327$	0	0
$328$	−640.000	−1.95122
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	350.000	1.03858	0.519288	−	0.854599i	$-0.326197\pi$
$337$	350.000	1.03858	0.519288	+	0.854599i	$0.326197\pi$
$338$	991.692	2.93400
$339$	0	0
$340$	−86.0309	−0.253032
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−467.577	−1.35138
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−598.000	−1.71347	−0.856734	−	0.515759i	$-0.827510\pi$
$349$	−598.000	−1.71347	−0.856734	+	0.515759i	$0.827510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	544.000	1.54108	0.770538	−	0.637394i	$-0.219988\pi$
$353$	544.000	1.54108	0.770538	+	0.637394i	$0.219988\pi$
$354$	0	0
$355$	0	0
$356$	−590.200	−1.65786
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	76.0000	0.209945
$363$	0	0
$364$	0	0
$365$	−33.6579	−0.0922133
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	−133.454	−0.360686
$371$	0	0
$372$	0	0
$373$	−550.000	−1.47453	−0.737265	−	0.675603i	$-0.763883\pi$
$373$	−550.000	−1.47453	−0.737265	+	0.675603i	$0.763883\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1453.56	−3.85559
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−391.969	−1.01546
$387$	0	0
$388$	−520.000	−1.34021
$389$	−680.000	−1.74807	−0.874036	−	0.485861i	$-0.838506\pi$
$389$	−680.000	−1.74807	−0.874036	+	0.485861i	$0.838506\pi$
$390$	0	0
$391$	0	0
$392$	392.000	1.00000
$393$	0	0
$394$	731.500	1.85660
$395$	0	0
$396$	0	0
$397$	69.9076	0.176090	0.0880448	−	0.996117i	$-0.471938\pi$
$397$	69.9076	0.176090	0.0880448	+	0.996117i	$0.471938\pi$
$398$	0	0
$399$	0	0
$400$	−377.108	−0.942769
$401$	−651.088	−1.62366	−0.811831	−	0.583893i	$-0.801529\pi$
$401$	−651.088	−1.62366	−0.811831	+	0.583893i	$0.801529\pi$
$402$	0	0
$403$	0	0
$404$	160.000	0.396040
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−183.154	−0.447809	−0.223905	−	0.974611i	$-0.571880\pi$
$409$	−183.154	−0.447809	−0.223905	+	0.974611i	$0.571880\pi$
$410$	191.384	0.466791
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	825.108	1.98343
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−698.461	−1.65905	−0.829527	−	0.558467i	$-0.811390\pi$
$421$	−698.461	−1.65905	−0.829527	+	0.558467i	$0.811390\pi$
$422$	0	0
$423$	0	0
$424$	−448.000	−1.05660
$425$	−423.793	−0.997159
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	561.677	1.29717	0.648587	−	0.761140i	$-0.275360\pi$
$433$	561.677	1.29717	0.648587	+	0.761140i	$0.275360\pi$
$434$	0	0
$435$	0	0
$436$	−779.692	−1.78828
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	927.254	2.09786
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	176.492	0.396612
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−560.000	−1.24722	−0.623608	−	0.781737i	$-0.714334\pi$
$449$	−560.000	−1.24722	−0.623608	+	0.781737i	$0.714334\pi$
$450$	0	0
$451$	0	0
$452$	551.923	1.22107
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	134.015	0.293250	0.146625	−	0.989192i	$-0.453159\pi$
$457$	134.015	0.293250	0.146625	+	0.989192i	$0.453159\pi$
$458$	234.154	0.511253
$459$	0	0
$460$	0	0
$461$	760.000	1.64859	0.824295	−	0.566161i	$-0.191572\pi$
$461$	760.000	1.64859	0.824295	+	0.566161i	$0.191572\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−901.969	−1.94390
$465$	0	0
$466$	779.731	1.67324
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	1438.38	2.99040
$482$	833.692	1.72965
$483$	0	0
$484$	484.000	1.00000
$485$	155.500	0.320618
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−743.384	−1.52333
$489$	0	0
$490$	−117.223	−0.239230
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	−1013.63	−2.05605
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	232.385	0.464770
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−47.8461	−0.0947447
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−440.000	−0.864440	−0.432220	−	0.901768i	$-0.642270\pi$
$509$	−440.000	−0.864440	−0.432220	+	0.901768i	$0.642270\pi$
$510$	0	0
$511$	0	0
$512$	512.000	1.00000
$513$	0	0
$514$	−947.346	−1.84309
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−246.739	−0.474497
$521$	880.000	1.68906	0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000	1.68906	0.844530	+	0.535509i	$0.179880\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	133.969	0.252772
$531$	0	0
$532$	0	0
$533$	−2062.77	−3.87011
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−280.977	−0.522262
$539$	0	0
$540$	0	0
$541$	−386.461	−0.714346	−0.357173	−	0.934038i	$-0.616259\pi$
$541$	−386.461	−0.714346	−0.357173	+	0.934038i	$0.616259\pi$
$542$	0	0
$543$	0	0
$544$	575.384	1.05769
$545$	233.158	0.427812
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	1079.46	1.96982
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	460.000	0.830325
$555$	0	0
$556$	0	0
$557$	246.212	0.442032	0.221016	−	0.975270i	$-0.429063\pi$
$557$	246.212	0.442032	0.221016	+	0.975270i	$0.429063\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	1120.21	1.99325
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	−165.046	−0.292117
$566$	0	0
$567$	0	0
$568$	0	0
$569$	920.104	1.61705	0.808527	−	0.588459i	$-0.200265\pi$
$569$	920.104	1.61705	0.808527	+	0.588459i	$0.200265\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	658.138	1.14062	0.570311	−	0.821429i	$-0.306823\pi$
$577$	658.138	1.14062	0.570311	+	0.821429i	$0.306823\pi$
$578$	68.6156	0.118712
$579$	0	0
$580$	269.723	0.465040
$581$	0	0
$582$	0	0
$583$	0	0
$584$	225.108	0.385458
$585$	0	0
$586$	851.269	1.45268
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	892.554	1.50769
$593$	437.404	0.737612	0.368806	−	0.929506i	$-0.379767\pi$
$593$	437.404	0.737612	0.368806	+	0.929506i	$0.379767\pi$
$594$	0	0
$595$	0	0
$596$	−206.662	−0.346748
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	135.308	0.225138	0.112569	−	0.993644i	$-0.464092\pi$
$601$	135.308	0.225138	0.112569	+	0.993644i	$0.464092\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−144.734	−0.239230
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	222.300	0.364427
$611$	0	0
$612$	0	0
$613$	−70.0000	−0.114192	−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	−70.0000	−0.114192	−0.0570962	+	0.998369i	$0.518184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−426.135	−0.690656	−0.345328	−	0.938482i	$-0.612232\pi$
$617$	−426.135	−0.690656	−0.345328	+	0.938482i	$0.612232\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	519.739	0.831582
$626$	−1130.80	−1.80639
$627$	0	0
$628$	−1254.52	−1.99765
$629$	1003.05	1.59467
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−875.808	−1.38140
$635$	0	0
$636$	0	0
$637$	1263.45	1.98343
$638$	0	0
$639$	0	0
$640$	−153.108	−0.239230
$641$	854.819	1.33357	0.666785	−	0.745250i	$-0.267670\pi$
$641$	854.819	1.33357	0.666785	+	0.745250i	$0.267670\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	−1215.45	−1.86992
$651$	0	0
$652$	0	0
$653$	1144.00	1.75191	0.875957	−	0.482389i	$-0.160231\pi$
$653$	1144.00	1.75191	0.875957	+	0.482389i	$0.160231\pi$
$654$	0	0
$655$	0	0
$656$	−1280.00	−1.95122
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−69.3848	−0.104969	−0.0524847	−	0.998622i	$-0.516714\pi$
$661$	−69.3848	−0.104969	−0.0524847	+	0.998622i	$0.516714\pi$
$662$	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
$673$	571.092	0.848577	0.424288	−	0.905527i	$-0.360524\pi$
$673$	571.092	0.848577	0.424288	+	0.905527i	$0.360524\pi$
$674$	700.000	1.03858
$675$	0	0
$676$	1983.38	2.93400
$677$	−104.000	−0.153619	−0.0768095	−	0.997046i	$-0.524473\pi$
$677$	−104.000	−0.153619	−0.0768095	+	0.997046i	$0.524473\pi$
$678$	0	0
$679$	0	0
$680$	−172.062	−0.253032
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−322.800	−0.471241
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1443.94	−2.09570
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−935.154	−1.35138
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1438.46	−2.06379
$698$	−1196.00	−1.71347
$699$	0	0
$700$	0	0
$701$	−1387.57	−1.97941	−0.989704	−	0.143129i	$-0.954284\pi$
$701$	−1387.57	−1.97941	−0.989704	+	0.143129i	$0.954284\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	1088.00	1.54108
$707$	0	0
$708$	0	0
$709$	884.154	1.24704	0.623522	−	0.781806i	$-0.285701\pi$
$709$	884.154	1.24704	0.623522	+	0.781806i	$0.285701\pi$
$710$	0	0
$711$	0	0
$712$	−1180.40	−1.65786
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	722.000	1.00000
$723$	0	0
$724$	152.000	0.209945
$725$	1328.67	1.83265
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	−67.3157	−0.0922133
$731$	0	0
$732$	0	0
$733$	−1450.00	−1.97817	−0.989086	−	0.147340i	$-0.952929\pi$
$733$	−1450.00	−1.97817	−0.989086	+	0.147340i	$0.952929\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−266.908	−0.360686
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	61.7997	0.0829526
$746$	−1100.00	−1.47453
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	−2907.12	−3.85559
$755$	0	0
$756$	0	0
$757$	1190.00	1.57199	0.785997	−	0.618230i	$-0.212150\pi$
$757$	1190.00	1.57199	0.785997	+	0.618230i	$0.212150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	692.450	0.909921	0.454961	−	0.890512i	$-0.349653\pi$
$761$	692.450	0.909921	0.454961	+	0.890512i	$0.349653\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1520.23	−1.97689	−0.988446	−	0.151571i	$-0.951567\pi$
$769$	−1520.23	−1.97689	−0.988446	+	0.151571i	$0.951567\pi$
$770$	0	0
$771$	0	0
$772$	−783.938	−1.01546
$773$	1085.75	1.40459	0.702296	−	0.711885i	$-0.252158\pi$
$773$	1085.75	1.40459	0.702296	+	0.711885i	$0.252158\pi$
$774$	0	0
$775$	0	0
$776$	−1040.00	−1.34021
$777$	0	0
$778$	−1360.00	−1.74807
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	784.000	1.00000
$785$	375.150	0.477898
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1463.00	1.85660
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2395.98	−3.02142
$794$	139.815	0.176090
$795$	0	0
$796$	0	0
$797$	389.288	0.488442	0.244221	−	0.969720i	$-0.421468\pi$
$797$	389.288	0.488442	0.244221	+	0.969720i	$0.421468\pi$
$798$	0	0
$799$	0	0
$800$	−754.215	−0.942769
$801$	0	0
$802$	−1302.18	−1.62366
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	320.000	0.396040
$809$	−1034.63	−1.27890	−0.639448	−	0.768835i	$-0.720837\pi$
$809$	−1034.63	−1.27890	−0.639448	+	0.768835i	$0.720837\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−366.308	−0.447809
$819$	0	0
$820$	382.769	0.466791
$821$	1443.05	1.75767	0.878837	−	0.477123i	$-0.158320\pi$
$821$	1443.05	1.75767	0.878837	+	0.477123i	$0.158320\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1258.00	−1.51749	−0.758745	−	0.651387i	$-0.774187\pi$
$829$	−1258.00	−1.51749	−0.758745	+	0.651387i	$0.774187\pi$
$830$	0	0
$831$	0	0
$832$	1650.22	1.98343
$833$	881.057	1.05769
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2336.92	2.77874
$842$	−1396.92	−1.65905
$843$	0	0
$844$	0	0
$845$	−593.108	−0.701902
$846$	0	0
$847$	0	0
$848$	−896.000	−1.05660
$849$	0	0
$850$	−847.585	−0.997159
$851$	0	0
$852$	0	0
$853$	410.000	0.480657	0.240328	−	0.970692i	$-0.422745\pi$
$853$	410.000	0.480657	0.240328	+	0.970692i	$0.422745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1660.94	1.93809	0.969044	−	0.246887i	$-0.0794076\pi$
$857$	1660.94	1.93809	0.969044	+	0.246887i	$0.0794076\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	279.647	0.323291
$866$	1123.35	1.29717
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1559.38	−1.78828
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1407.75	−1.60519	−0.802596	−	0.596523i	$-0.796549\pi$
$877$	−1407.75	−1.60519	−0.802596	+	0.596523i	$0.796549\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1600.00	1.81612	0.908059	−	0.418842i	$-0.137564\pi$
$881$	1600.00	1.81612	0.908059	+	0.418842i	$0.137564\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	1854.51	2.09786
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	352.985	0.396612
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1120.00	−1.24722
$899$	0	0
$900$	0	0
$901$	−1006.92	−1.11756
$902$	0	0
$903$	0	0
$904$	1103.85	1.22107
$905$	−45.4538	−0.0502252
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	268.031	0.293250
$915$	0	0
$916$	468.308	0.511253
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1520.00	1.64859
$923$	0	0
$924$	0	0
$925$	−1314.80	−1.42141
$926$	0	0
$927$	0	0
$928$	−1803.94	−1.94390
$929$	−1143.43	−1.23082	−0.615411	−	0.788206i	$-0.711010\pi$
$929$	−1143.43	−1.23082	−0.615411	+	0.788206i	$0.711010\pi$
$930$	0	0
$931$	0	0
$932$	1559.46	1.67324
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1794.63	1.91529	0.957647	−	0.287945i	$-0.0929721\pi$
$937$	1794.63	1.91529	0.957647	+	0.287945i	$0.0929721\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−703.450	−0.747555	−0.373778	−	0.927518i	$-0.621938\pi$
$941$	−703.450	−0.747555	−0.373778	+	0.927518i	$0.621938\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	725.539	0.764530
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1793.21	−1.88165	−0.940824	−	0.338895i	$-0.889947\pi$
$953$	−1793.21	−1.88165	−0.940824	+	0.338895i	$0.889947\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	2876.77	2.99040
$963$	0	0
$964$	1667.38	1.72965
$965$	234.427	0.242930
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	968.000	1.00000
$969$	0	0
$970$	311.000	0.320618
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−1486.77	−1.52333
$977$	496.000	0.507677	0.253838	−	0.967247i	$-0.418307\pi$
$977$	496.000	0.507677	0.253838	+	0.967247i	$0.418307\pi$
$978$	0	0
$979$	0	0
$980$	−234.446	−0.239230
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−437.493	−0.444155
$986$	−2027.26	−2.05605
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−280.677	−0.281522	−0.140761	−	0.990044i	$-0.544955\pi$
$997$	−280.677	−0.281522	−0.140761	+	0.990044i	$0.544955\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.d.d.163.1	yes	2
3.2	odd	2		324.3.d.a.163.2	✓	2
4.3	odd	2	CM	324.3.d.d.163.1	yes	2
9.2	odd	6		324.3.f.n.271.1		4
9.4	even	3		324.3.f.k.55.2		4
9.5	odd	6		324.3.f.n.55.1		4
9.7	even	3		324.3.f.k.271.2		4
12.11	even	2		324.3.d.a.163.2	✓	2
36.7	odd	6		324.3.f.k.271.2		4
36.11	even	6		324.3.f.n.271.1		4
36.23	even	6		324.3.f.n.55.1		4
36.31	odd	6		324.3.f.k.55.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.d.a.163.2	✓	2	3.2	odd	2
324.3.d.a.163.2	✓	2	12.11	even	2
324.3.d.d.163.1	yes	2	1.1	even	1	trivial
324.3.d.d.163.1	yes	2	4.3	odd	2	CM
324.3.f.k.55.2		4	9.4	even	3
324.3.f.k.55.2		4	36.31	odd	6
324.3.f.k.271.2		4	9.7	even	3
324.3.f.k.271.2		4	36.7	odd	6
324.3.f.n.55.1		4	9.5	odd	6
324.3.f.n.55.1		4	36.23	even	6
324.3.f.n.271.1		4	9.2	odd	6
324.3.f.n.271.1		4	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -1.19615 q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -1.19615 q^{5} +8.00000 q^{8} -2.39230 q^{10} +25.7846 q^{13} +16.0000 q^{16} +17.9808 q^{17} -4.78461 q^{20} -23.5692 q^{25} +51.5692 q^{26} -56.3731 q^{29} +32.0000 q^{32} +35.9615 q^{34} +55.7846 q^{37} -9.56922 q^{40} -80.0000 q^{41} +49.0000 q^{49} -47.1384 q^{50} +103.138 q^{52} -56.0000 q^{53} -112.746 q^{58} -92.9230 q^{61} +64.0000 q^{64} -30.8423 q^{65} +71.9230 q^{68} +28.1384 q^{73} +111.569 q^{74} -19.1384 q^{80} -160.000 q^{82} -21.5077 q^{85} -147.550 q^{89} -130.000 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8 + 16 * q^10 + 10 * q^13 + 32 * q^16 - 16 * q^17 + 32 * q^20 + 36 * q^25 + 20 * q^26 - 40 * q^29 + 64 * q^32 - 32 * q^34 + 70 * q^37 + 64 * q^40 - 160 * q^41 + 98 * q^49 + 72 * q^50 + 40 * q^52 - 112 * q^53 - 80 * q^58 + 22 * q^61 + 128 * q^64 - 176 * q^65 - 64 * q^68 - 110 * q^73 + 140 * q^74 + 128 * q^80 - 320 * q^82 - 334 * q^85 - 160 * q^89 - 260 * q^97 + 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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