LMFDB - Embedded newform 216.3.h.b.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(53,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("216.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.h (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:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			53.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	216.53

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +9.48528 q^{5} -3.48528 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} +9.48528 q^{5} -3.48528 q^{7} +8.00000 q^{8} +18.9706 q^{10} -21.9706 q^{11} -6.97056 q^{14} +16.0000 q^{16} +37.9411 q^{20} -43.9411 q^{22} +64.9706 q^{25} -13.9411 q^{28} -50.0000 q^{29} +23.4264 q^{31} +32.0000 q^{32} -33.0589 q^{35} +75.8823 q^{40} -87.8823 q^{44} -36.8528 q^{49} +129.941 q^{50} -89.4264 q^{53} -208.397 q^{55} -27.8823 q^{56} -100.000 q^{58} +10.0000 q^{59} +46.8528 q^{62} +64.0000 q^{64} -66.1177 q^{70} +93.7939 q^{73} +76.5736 q^{77} -58.0000 q^{79} +151.765 q^{80} +151.853 q^{83} -175.765 q^{88} +61.0589 q^{97} -73.7056 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 2 q^{5} + 10 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 + 10 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 2 q^{5} + 10 q^{7} + 16 q^{8} + 4 q^{10} - 10 q^{11} + 20 q^{14} + 32 q^{16} + 8 q^{20} - 20 q^{22} + 96 q^{25} + 40 q^{28} - 100 q^{29} - 38 q^{31} + 64 q^{32} - 134 q^{35} + 16 q^{40} - 40 q^{44} + 96 q^{49} + 192 q^{50} - 94 q^{53} - 298 q^{55} + 80 q^{56} - 200 q^{58} + 20 q^{59} - 76 q^{62} + 128 q^{64} - 268 q^{70} - 50 q^{73} + 238 q^{77} - 116 q^{79} + 32 q^{80} + 134 q^{83} - 80 q^{88} + 190 q^{97} + 192 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 + 10 * q^7 + 16 * q^8 + 4 * q^10 - 10 * q^11 + 20 * q^14 + 32 * q^16 + 8 * q^20 - 20 * q^22 + 96 * q^25 + 40 * q^28 - 100 * q^29 - 38 * q^31 + 64 * q^32 - 134 * q^35 + 16 * q^40 - 40 * q^44 + 96 * q^49 + 192 * q^50 - 94 * q^53 - 298 * q^55 + 80 * q^56 - 200 * q^58 + 20 * q^59 - 76 * q^62 + 128 * q^64 - 268 * q^70 - 50 * q^73 + 238 * q^77 - 116 * q^79 + 32 * q^80 + 134 * q^83 - 80 * q^88 + 190 * q^97 + 192 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	2.00000	1.00000
$2$	2.00000	1.00000
$3$	0	0
$3$	0	0
$4$	4.00000	1.00000
$5$	9.48528	1.89706	0.948528	−	0.316693i	$-0.102572\pi$
$5$	9.48528	1.89706	0.948528	+	0.316693i	$0.102572\pi$
$6$	0	0
$7$	−3.48528	−0.497897	−0.248949	−	0.968517i	$-0.580085\pi$
$7$	−3.48528	−0.497897	−0.248949	+	0.968517i	$0.580085\pi$
$8$	8.00000	1.00000
$9$	0	0
$10$	18.9706	1.89706
$11$	−21.9706	−1.99732	−0.998662	−	0.0517139i	$-0.983532\pi$
$11$	−21.9706	−1.99732	−0.998662	+	0.0517139i	$0.983532\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−6.97056	−0.497897
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	37.9411	1.89706
$21$	0	0
$22$	−43.9411	−1.99732
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	64.9706	2.59882
$26$	0	0
$27$	0	0
$28$	−13.9411	−0.497897
$29$	−50.0000	−1.72414	−0.862069	−	0.506791i	$-0.830832\pi$
$29$	−50.0000	−1.72414	−0.862069	+	0.506791i	$0.830832\pi$
$30$	0	0
$31$	23.4264	0.755691	0.377845	−	0.925869i	$-0.376665\pi$
$31$	23.4264	0.755691	0.377845	+	0.925869i	$0.376665\pi$
$32$	32.0000	1.00000
$33$	0	0
$34$	0	0
$35$	−33.0589	−0.944539
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	75.8823	1.89706
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−87.8823	−1.99732
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−36.8528	−0.752098
$50$	129.941	2.59882
$51$	0	0
$52$	0	0
$53$	−89.4264	−1.68729	−0.843645	−	0.536901i	$-0.819595\pi$
$53$	−89.4264	−1.68729	−0.843645	+	0.536901i	$0.819595\pi$
$54$	0	0
$55$	−208.397	−3.78904
$56$	−27.8823	−0.497897
$57$	0	0
$58$	−100.000	−1.72414
$59$	10.0000	0.169492	0.0847458	−	0.996403i	$-0.472992\pi$
$59$	10.0000	0.169492	0.0847458	+	0.996403i	$0.472992\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	46.8528	0.755691
$63$	0	0
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−66.1177	−0.944539
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	93.7939	1.28485	0.642424	−	0.766349i	$-0.277929\pi$
$73$	93.7939	1.28485	0.642424	+	0.766349i	$0.277929\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	76.5736	0.994462
$78$	0	0
$79$	−58.0000	−0.734177	−0.367089	−	0.930186i	$-0.619645\pi$
$79$	−58.0000	−0.734177	−0.367089	+	0.930186i	$0.619645\pi$
$80$	151.765	1.89706
$81$	0	0
$82$	0	0
$83$	151.853	1.82955	0.914776	−	0.403962i	$-0.132367\pi$
$83$	151.853	1.82955	0.914776	+	0.403962i	$0.132367\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−175.765	−1.99732
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	61.0589	0.629473	0.314736	−	0.949179i	$-0.398084\pi$
$97$	61.0589	0.629473	0.314736	+	0.949179i	$0.398084\pi$
$98$	−73.7056	−0.752098
$99$	0	0
$100$	259.882	2.59882
$101$	−35.6030	−0.352505	−0.176253	−	0.984345i	$-0.556398\pi$
$101$	−35.6030	−0.352505	−0.176253	+	0.984345i	$0.556398\pi$
$102$	0	0
$103$	−10.0000	−0.0970874	−0.0485437	−	0.998821i	$-0.515458\pi$
$103$	−10.0000	−0.0970874	−0.0485437	+	0.998821i	$0.515458\pi$
$104$	0	0
$105$	0	0
$106$	−178.853	−1.68729
$107$	−126.706	−1.18416	−0.592082	−	0.805877i	$-0.701694\pi$
$107$	−126.706	−1.18416	−0.592082	+	0.805877i	$0.701694\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−416.794	−3.78904
$111$	0	0
$112$	−55.7645	−0.497897
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−200.000	−1.72414
$117$	0	0
$118$	20.0000	0.169492
$119$	0	0
$120$	0	0
$121$	361.706	2.98930
$122$	0	0
$123$	0	0
$124$	93.7056	0.755691
$125$	379.132	3.03306
$126$	0	0
$127$	−21.6619	−0.170566	−0.0852831	−	0.996357i	$-0.527179\pi$
$127$	−21.6619	−0.170566	−0.0852831	+	0.996357i	$0.527179\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	0	0
$131$	−192.882	−1.47238	−0.736192	−	0.676773i	$-0.763378\pi$
$131$	−192.882	−1.47238	−0.736192	+	0.676773i	$0.763378\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−132.235	−0.944539
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−474.264	−3.27079
$146$	187.588	1.28485
$147$	0	0
$148$	0	0
$149$	204.397	1.37179	0.685896	−	0.727700i	$-0.259411\pi$
$149$	204.397	1.37179	0.685896	+	0.727700i	$0.259411\pi$
$150$	0	0
$151$	191.426	1.26772	0.633862	−	0.773446i	$-0.281469\pi$
$151$	191.426	1.26772	0.633862	+	0.773446i	$0.281469\pi$
$152$	0	0
$153$	0	0
$154$	153.147	0.994462
$155$	222.206	1.43359
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−116.000	−0.734177
$159$	0	0
$160$	303.529	1.89706
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	303.706	1.82955
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−319.985	−1.84962	−0.924812	−	0.380425i	$-0.875778\pi$
$173$	−319.985	−1.84962	−0.924812	+	0.380425i	$0.875778\pi$
$174$	0	0
$175$	−226.441	−1.29395
$176$	−351.529	−1.99732
$177$	0	0
$178$	0	0
$179$	352.588	1.96976	0.984882	−	0.173225i	$-0.0554187\pi$
$179$	352.588	1.96976	0.984882	+	0.173225i	$0.0554187\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	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
$193$	75.6173	0.391800	0.195900	−	0.980624i	$-0.437237\pi$
$193$	75.6173	0.391800	0.195900	+	0.980624i	$0.437237\pi$
$194$	122.118	0.629473
$195$	0	0
$196$	−147.411	−0.752098
$197$	−199.985	−1.01515	−0.507576	−	0.861607i	$-0.669458\pi$
$197$	−199.985	−1.01515	−0.507576	+	0.861607i	$0.669458\pi$
$198$	0	0
$199$	397.985	1.99992	0.999962	−	0.00872575i	$-0.00277753\pi$
$199$	397.985	1.99992	0.999962	+	0.00872575i	$0.00277753\pi$
$200$	519.765	2.59882
$201$	0	0
$202$	−71.2061	−0.352505
$203$	174.264	0.858444
$204$	0	0
$205$	0	0
$206$	−20.0000	−0.0970874
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−357.706	−1.68729
$213$	0	0
$214$	−253.411	−1.18416
$215$	0	0
$216$	0	0
$217$	−81.6476	−0.376256
$218$	0	0
$219$	0	0
$220$	−833.588	−3.78904
$221$	0	0
$222$	0	0
$223$	230.000	1.03139	0.515695	−	0.856772i	$-0.327534\pi$
$223$	230.000	1.03139	0.515695	+	0.856772i	$0.327534\pi$
$224$	−111.529	−0.497897
$225$	0	0
$226$	0	0
$227$	346.000	1.52423	0.762115	−	0.647442i	$-0.224161\pi$
$227$	346.000	1.52423	0.762115	+	0.647442i	$0.224161\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−400.000	−1.72414
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	40.0000	0.169492
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−382.000	−1.58506	−0.792531	−	0.609831i	$-0.791237\pi$
$241$	−382.000	−1.58506	−0.792531	+	0.609831i	$0.791237\pi$
$242$	723.411	2.98930
$243$	0	0
$244$	0	0
$245$	−349.559	−1.42677
$246$	0	0
$247$	0	0
$248$	187.411	0.755691
$249$	0	0
$250$	758.264	3.03306
$251$	−470.000	−1.87251	−0.936255	−	0.351321i	$-0.885733\pi$
$251$	−470.000	−1.87251	−0.936255	+	0.351321i	$0.885733\pi$
$252$	0	0
$253$	0	0
$254$	−43.3238	−0.170566
$255$	0	0
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−385.765	−1.47238
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−848.235	−3.20089
$266$	0	0
$267$	0	0
$268$	0	0
$269$	430.000	1.59851	0.799257	−	0.600990i	$-0.205227\pi$
$269$	430.000	1.59851	0.799257	+	0.600990i	$0.205227\pi$
$270$	0	0
$271$	−437.690	−1.61509	−0.807547	−	0.589803i	$-0.799205\pi$
$271$	−437.690	−1.61509	−0.807547	+	0.589803i	$0.799205\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1427.44	−5.19069
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−264.471	−0.944539
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	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
$289$	289.000	1.00000
$290$	−948.528	−3.27079
$291$	0	0
$292$	375.176	1.28485
$293$	−386.000	−1.31741	−0.658703	−	0.752403i	$-0.728895\pi$
$293$	−386.000	−1.31741	−0.658703	+	0.752403i	$0.728895\pi$
$294$	0	0
$295$	94.8528	0.321535
$296$	0	0
$297$	0	0
$298$	408.794	1.37179
$299$	0	0
$300$	0	0
$301$	0	0
$302$	382.853	1.26772
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	306.294	0.994462
$309$	0	0
$310$	444.412	1.43359
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−553.500	−1.76837	−0.884185	−	0.467138i	$-0.845285\pi$
$313$	−553.500	−1.76837	−0.884185	+	0.467138i	$0.845285\pi$
$314$	0	0
$315$	0	0
$316$	−232.000	−0.734177
$317$	299.690	0.945396	0.472698	−	0.881225i	$-0.343280\pi$
$317$	299.690	0.945396	0.472698	+	0.881225i	$0.343280\pi$
$318$	0	0
$319$	1098.53	3.44366
$320$	607.058	1.89706
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	607.411	1.82955
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−190.000	−0.563798	−0.281899	−	0.959444i	$-0.590964\pi$
$337$	−190.000	−0.563798	−0.281899	+	0.959444i	$0.590964\pi$
$338$	338.000	1.00000
$339$	0	0
$340$	0	0
$341$	−514.691	−1.50936
$342$	0	0
$343$	299.221	0.872365
$344$	0	0
$345$	0	0
$346$	−639.970	−1.84962
$347$	540.970	1.55899	0.779495	−	0.626408i	$-0.215476\pi$
$347$	540.970	1.55899	0.779495	+	0.626408i	$0.215476\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−452.881	−1.29395
$351$	0	0
$352$	−703.058	−1.99732
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	705.176	1.96976
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	889.662	2.43743
$366$	0	0
$367$	−516.220	−1.40659	−0.703297	−	0.710896i	$-0.748290\pi$
$367$	−516.220	−1.40659	−0.703297	+	0.710896i	$0.748290\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	311.676	0.840098
$372$	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
$379$	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
$385$	726.322	1.88655
$386$	151.235	0.391800
$387$	0	0
$388$	244.235	0.629473
$389$	−748.367	−1.92382	−0.961911	−	0.273363i	$-0.911864\pi$
$389$	−748.367	−1.92382	−0.961911	+	0.273363i	$0.911864\pi$
$390$	0	0
$391$	0	0
$392$	−294.823	−0.752098
$393$	0	0
$394$	−399.970	−1.01515
$395$	−550.146	−1.39278
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	795.970	1.99992
$399$	0	0
$400$	1039.53	2.59882
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−142.412	−0.352505
$405$	0	0
$406$	348.528	0.858444
$407$	0	0
$408$	0	0
$409$	19.5887	0.0478942	0.0239471	−	0.999713i	$-0.492377\pi$
$409$	19.5887	0.0478942	0.0239471	+	0.999713i	$0.492377\pi$
$410$	0	0
$411$	0	0
$412$	−40.0000	−0.0970874
$413$	−34.8528	−0.0843894
$414$	0	0
$415$	1440.37	3.47076
$416$	0	0
$417$	0	0
$418$	0	0
$419$	730.000	1.74224	0.871122	−	0.491067i	$-0.163393\pi$
$419$	730.000	1.74224	0.871122	+	0.491067i	$0.163393\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−715.411	−1.68729
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−506.823	−1.18416
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	810.176	1.87108	0.935538	−	0.353227i	$-0.114916\pi$
$433$	810.176	1.87108	0.935538	+	0.353227i	$0.114916\pi$
$434$	−163.295	−0.376256
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−860.249	−1.95956	−0.979782	−	0.200066i	$-0.935884\pi$
$439$	−860.249	−1.95956	−0.979782	+	0.200066i	$0.935884\pi$
$440$	−1667.18	−3.78904
$441$	0	0
$442$	0	0
$443$	−86.0000	−0.194131	−0.0970655	−	0.995278i	$-0.530946\pi$
$443$	−86.0000	−0.194131	−0.0970655	+	0.995278i	$0.530946\pi$
$444$	0	0
$445$	0	0
$446$	460.000	1.03139
$447$	0	0
$448$	−223.058	−0.497897
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	692.000	1.52423
$455$	0	0
$456$	0	0
$457$	−909.881	−1.99099	−0.995494	−	0.0948261i	$-0.969771\pi$
$457$	−909.881	−1.99099	−0.995494	+	0.0948261i	$0.969771\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−388.367	−0.842444	−0.421222	−	0.906958i	$-0.638399\pi$
$461$	−388.367	−0.842444	−0.421222	+	0.906958i	$0.638399\pi$
$462$	0	0
$463$	−647.161	−1.39776	−0.698878	−	0.715241i	$-0.746317\pi$
$463$	−647.161	−1.39776	−0.698878	+	0.715241i	$0.746317\pi$
$464$	−800.000	−1.72414
$465$	0	0
$466$	0	0
$467$	276.970	0.593083	0.296541	−	0.955020i	$-0.404167\pi$
$467$	276.970	0.593083	0.296541	+	0.955020i	$0.404167\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	80.0000	0.169492
$473$	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
$481$	0	0
$482$	−764.000	−1.58506
$483$	0	0
$484$	1446.82	2.98930
$485$	579.161	1.19415
$486$	0	0
$487$	−970.000	−1.99179	−0.995893	−	0.0905356i	$-0.971142\pi$
$487$	−970.000	−1.99179	−0.995893	+	0.0905356i	$0.971142\pi$
$488$	0	0
$489$	0	0
$490$	−699.119	−1.42677
$491$	981.705	1.99940	0.999699	−	0.0245196i	$-0.00780562\pi$
$491$	981.705	1.99940	0.999699	+	0.0245196i	$0.00780562\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	374.823	0.755691
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1516.53	3.03306
$501$	0	0
$502$	−940.000	−1.87251
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−337.705	−0.668722
$506$	0	0
$507$	0	0
$508$	−86.6476	−0.170566
$509$	615.309	1.20886	0.604429	−	0.796659i	$-0.293401\pi$
$509$	615.309	1.20886	0.604429	+	0.796659i	$0.293401\pi$
$510$	0	0
$511$	−326.898	−0.639723
$512$	512.000	1.00000
$513$	0	0
$514$	0	0
$515$	−94.8528	−0.184180
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−771.529	−1.47238
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	−1696.47	−3.20089
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1201.84	−2.24643
$536$	0	0
$537$	0	0
$538$	860.000	1.59851
$539$	809.677	1.50218
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−875.381	−1.61509
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−2854.88	−5.19069
$551$	0	0
$552$	0	0
$553$	202.146	0.365545
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−94.5433	−0.169737	−0.0848683	−	0.996392i	$-0.527047\pi$
$557$	−94.5433	−0.169737	−0.0848683	+	0.996392i	$0.527047\pi$
$558$	0	0
$559$	0	0
$560$	−528.942	−0.944539
$561$	0	0
$562$	0	0
$563$	−770.381	−1.36835	−0.684175	−	0.729318i	$-0.739838\pi$
$563$	−770.381	−1.36835	−0.684175	+	0.729318i	$0.739838\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	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
$577$	290.000	0.502600	0.251300	−	0.967909i	$-0.419142\pi$
$577$	290.000	0.502600	0.251300	+	0.967909i	$0.419142\pi$
$578$	578.000	1.00000
$579$	0	0
$580$	−1897.06	−3.27079
$581$	−529.250	−0.910929
$582$	0	0
$583$	1964.75	3.37007
$584$	750.352	1.28485
$585$	0	0
$586$	−772.000	−1.31741
$587$	−1115.82	−1.90089	−0.950445	−	0.310893i	$-0.899372\pi$
$587$	−1115.82	−1.90089	−0.950445	+	0.310893i	$0.899372\pi$
$588$	0	0
$589$	0	0
$590$	189.706	0.321535
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	817.588	1.37179
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	514.147	0.855486	0.427743	−	0.903900i	$-0.359309\pi$
$601$	514.147	0.855486	0.427743	+	0.903900i	$0.359309\pi$
$602$	0	0
$603$	0	0
$604$	765.706	1.26772
$605$	3430.88	5.67088
$606$	0	0
$607$	−730.000	−1.20264	−0.601318	−	0.799010i	$-0.705357\pi$
$607$	−730.000	−1.20264	−0.601318	+	0.799010i	$0.705357\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	612.589	0.994462
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	888.824	1.43359
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1971.91	3.15506
$626$	−1107.00	−1.76837
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−831.132	−1.31717	−0.658583	−	0.752508i	$-0.728844\pi$
$631$	−831.132	−1.31717	−0.658583	+	0.752508i	$0.728844\pi$
$632$	−464.000	−0.734177
$633$	0	0
$634$	599.381	0.945396
$635$	−205.469	−0.323574
$636$	0	0
$637$	0	0
$638$	2197.06	3.44366
$639$	0	0
$640$	1214.12	1.89706
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	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
$649$	−219.706	−0.338529
$650$	0	0
$651$	0	0
$652$	0	0
$653$	218.249	0.334225	0.167112	−	0.985938i	$-0.446556\pi$
$653$	218.249	0.334225	0.167112	+	0.985938i	$0.446556\pi$
$654$	0	0
$655$	−1829.54	−2.79319
$656$	0	0
$657$	0	0
$658$	0	0
$659$	585.352	0.888242	0.444121	−	0.895967i	$-0.353516\pi$
$659$	585.352	0.888242	0.444121	+	0.895967i	$0.353516\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	1214.82	1.82955
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1059.00	−1.57355	−0.786774	−	0.617241i	$-0.788250\pi$
$673$	−1059.00	−1.57355	−0.786774	+	0.617241i	$0.788250\pi$
$674$	−380.000	−0.563798
$675$	0	0
$676$	676.000	1.00000
$677$	−1346.00	−1.98818	−0.994092	−	0.108545i	$-0.965381\pi$
$677$	−1346.00	−1.98818	−0.994092	+	0.108545i	$0.965381\pi$
$678$	0	0
$679$	−212.807	−0.313413
$680$	0	0
$681$	0	0
$682$	−1029.38	−1.50936
$683$	−1334.00	−1.95315	−0.976574	−	0.215182i	$-0.930965\pi$
$683$	−1334.00	−1.95315	−0.976574	+	0.215182i	$0.930965\pi$
$684$	0	0
$685$	0	0
$686$	598.442	0.872365
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−1279.94	−1.84962
$693$	0	0
$694$	1081.94	1.55899
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−905.763	−1.29395
$701$	−1188.40	−1.69529	−0.847643	−	0.530567i	$-0.821979\pi$
$701$	−1188.40	−1.69529	−0.847643	+	0.530567i	$0.821979\pi$
$702$	0	0
$703$	0	0
$704$	−1406.12	−1.99732
$705$	0	0
$706$	0	0
$707$	124.087	0.175511
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1410.35	1.96976
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	34.8528	0.0483395
$722$	722.000	1.00000
$723$	0	0
$724$	0	0
$725$	−3248.53	−4.48073
$726$	0	0
$727$	818.338	1.12564	0.562818	−	0.826581i	$-0.309717\pi$
$727$	818.338	1.12564	0.562818	+	0.826581i	$0.309717\pi$
$728$	0	0
$729$	0	0
$730$	1779.32	2.43743
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1032.44	−1.40659
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	623.352	0.840098
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1938.76	2.60237
$746$	0	0
$747$	0	0
$748$	0	0
$749$	441.605	0.589592
$750$	0	0
$751$	234.310	0.311997	0.155998	−	0.987757i	$-0.450141\pi$
$751$	234.310	0.311997	0.155998	+	0.987757i	$0.450141\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1815.73	2.40494
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−672.087	−0.873975	−0.436987	−	0.899468i	$-0.643955\pi$
$769$	−672.087	−0.873975	−0.436987	+	0.899468i	$0.643955\pi$
$770$	1452.64	1.88655
$771$	0	0
$772$	302.469	0.391800
$773$	−1154.00	−1.49288	−0.746442	−	0.665450i	$-0.768240\pi$
$773$	−1154.00	−1.49288	−0.746442	+	0.665450i	$0.768240\pi$
$774$	0	0
$775$	1522.03	1.96391
$776$	488.471	0.629473
$777$	0	0
$778$	−1496.73	−1.92382
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−589.645	−0.752098
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−799.939	−1.01515
$789$	0	0
$790$	−1100.29	−1.39278
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	1591.94	1.99992
$797$	−1453.10	−1.82321	−0.911607	−	0.411063i	$-0.865158\pi$
$797$	−1453.10	−1.82321	−0.911607	+	0.411063i	$0.865158\pi$
$798$	0	0
$799$	0	0
$800$	2079.06	2.59882
$801$	0	0
$802$	0	0
$803$	−2060.71	−2.56626
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−284.824	−0.352505
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	697.056	0.858444
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	39.1775	0.0478942
$819$	0	0
$820$	0	0
$821$	670.000	0.816078	0.408039	−	0.912965i	$-0.366213\pi$
$821$	670.000	0.816078	0.408039	+	0.912965i	$0.366213\pi$
$822$	0	0
$823$	1131.13	1.37440	0.687199	−	0.726469i	$-0.258840\pi$
$823$	1131.13	1.37440	0.687199	+	0.726469i	$0.258840\pi$
$824$	−80.0000	−0.0970874
$825$	0	0
$826$	−69.7056	−0.0843894
$827$	1546.00	1.86941	0.934704	−	0.355428i	$-0.115665\pi$
$827$	1546.00	1.86941	0.934704	+	0.355428i	$0.115665\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	2880.73	3.47076
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	1460.00	1.74224
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1659.00	1.97265
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1603.01	1.89706
$846$	0	0
$847$	−1260.65	−1.48837
$848$	−1430.82	−1.68729
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1013.65	−1.18416
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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
$865$	−3035.15	−3.50884
$866$	1620.35	1.87108
$867$	0	0
$868$	−326.590	−0.376256
$869$	1274.29	1.46639
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1321.38	−1.51015
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1720.50	−1.95956
$879$	0	0
$880$	−3334.35	−3.78904
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−172.000	−0.194131
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	75.4978	0.0849244
$890$	0	0
$891$	0	0
$892$	920.000	1.03139
$893$	0	0
$894$	0	0
$895$	3344.40	3.73675
$896$	−446.116	−0.497897
$897$	0	0
$898$	0	0
$899$	−1171.32	−1.30291
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1384.00	1.52423
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−3336.29	−3.65421
$914$	−1819.76	−1.99099
$915$	0	0
$916$	0	0
$917$	672.249	0.733096
$918$	0	0
$919$	−1815.92	−1.97598	−0.987989	−	0.154523i	$-0.950616\pi$
$919$	−1815.92	−1.97598	−0.987989	+	0.154523i	$0.950616\pi$
$920$	0	0
$921$	0	0
$922$	−776.733	−0.842444
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−1294.32	−1.39776
$927$	0	0
$928$	−1600.00	−1.72414
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	553.939	0.593083
$935$	0	0
$936$	0	0
$937$	239.293	0.255382	0.127691	−	0.991814i	$-0.459243\pi$
$937$	239.293	0.255382	0.127691	+	0.991814i	$0.459243\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1371.75	1.45776	0.728878	−	0.684644i	$-0.240042\pi$
$941$	1371.75	1.45776	0.728878	+	0.684644i	$0.240042\pi$
$942$	0	0
$943$	0	0
$944$	160.000	0.169492
$945$	0	0
$946$	0	0
$947$	−1840.94	−1.94397	−0.971985	−	0.235043i	$-0.924477\pi$
$947$	−1840.94	−1.94397	−0.971985	+	0.235043i	$0.924477\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−412.203	−0.428932
$962$	0	0
$963$	0	0
$964$	−1528.00	−1.58506
$965$	717.252	0.743266
$966$	0	0
$967$	−1218.04	−1.25961	−0.629805	−	0.776753i	$-0.716865\pi$
$967$	−1218.04	−1.25961	−0.629805	+	0.776753i	$0.716865\pi$
$968$	2893.65	2.98930
$969$	0	0
$970$	1158.32	1.19415
$971$	−778.324	−0.801569	−0.400785	−	0.916172i	$-0.631262\pi$
$971$	−778.324	−0.801569	−0.400785	+	0.916172i	$0.631262\pi$
$972$	0	0
$973$	0	0
$974$	−1940.00	−1.99179
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	−1398.24	−1.42677
$981$	0	0
$982$	1963.41	1.99940
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−1896.91	−1.92580
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1921.37	−1.93882	−0.969408	−	0.245457i	$-0.921062\pi$
$991$	−1921.37	−1.93882	−0.969408	+	0.245457i	$0.921062\pi$
$992$	749.645	0.755691
$993$	0	0
$994$	0	0
$995$	3775.00	3.79397
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.h.b.53.2	yes	2
3.2	odd	2		216.3.h.a.53.1	✓	2
4.3	odd	2		864.3.h.b.593.2		2
8.3	odd	2		864.3.h.a.593.1		2
8.5	even	2		216.3.h.a.53.1	✓	2
12.11	even	2		864.3.h.a.593.1		2
24.5	odd	2	CM	216.3.h.b.53.2	yes	2
24.11	even	2		864.3.h.b.593.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.h.a.53.1	✓	2	3.2	odd	2
216.3.h.a.53.1	✓	2	8.5	even	2
216.3.h.b.53.2	yes	2	1.1	even	1	trivial
216.3.h.b.53.2	yes	2	24.5	odd	2	CM
864.3.h.a.593.1		2	8.3	odd	2
864.3.h.a.593.1		2	12.11	even	2
864.3.h.b.593.2		2	4.3	odd	2
864.3.h.b.593.2		2	24.11	even	2

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +9.48528 q^{5} -3.48528 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +9.48528 q^{5} -3.48528 q^{7} +8.00000 q^{8} +18.9706 q^{10} -21.9706 q^{11} -6.97056 q^{14} +16.0000 q^{16} +37.9411 q^{20} -43.9411 q^{22} +64.9706 q^{25} -13.9411 q^{28} -50.0000 q^{29} +23.4264 q^{31} +32.0000 q^{32} -33.0589 q^{35} +75.8823 q^{40} -87.8823 q^{44} -36.8528 q^{49} +129.941 q^{50} -89.4264 q^{53} -208.397 q^{55} -27.8823 q^{56} -100.000 q^{58} +10.0000 q^{59} +46.8528 q^{62} +64.0000 q^{64} -66.1177 q^{70} +93.7939 q^{73} +76.5736 q^{77} -58.0000 q^{79} +151.765 q^{80} +151.853 q^{83} -175.765 q^{88} +61.0589 q^{97} -73.7056 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 2 q^{5} + 10 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 + 10 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 2 q^{5} + 10 q^{7} + 16 q^{8} + 4 q^{10} - 10 q^{11} + 20 q^{14} + 32 q^{16} + 8 q^{20} - 20 q^{22} + 96 q^{25} + 40 q^{28} - 100 q^{29} - 38 q^{31} + 64 q^{32} - 134 q^{35} + 16 q^{40} - 40 q^{44} + 96 q^{49} + 192 q^{50} - 94 q^{53} - 298 q^{55} + 80 q^{56} - 200 q^{58} + 20 q^{59} - 76 q^{62} + 128 q^{64} - 268 q^{70} - 50 q^{73} + 238 q^{77} - 116 q^{79} + 32 q^{80} + 134 q^{83} - 80 q^{88} + 190 q^{97} + 192 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 + 10 * q^7 + 16 * q^8 + 4 * q^10 - 10 * q^11 + 20 * q^14 + 32 * q^16 + 8 * q^20 - 20 * q^22 + 96 * q^25 + 40 * q^28 - 100 * q^29 - 38 * q^31 + 64 * q^32 - 134 * q^35 + 16 * q^40 - 40 * q^44 + 96 * q^49 + 192 * q^50 - 94 * q^53 - 298 * q^55 + 80 * q^56 - 200 * q^58 + 20 * q^59 - 76 * q^62 + 128 * q^64 - 268 * q^70 - 50 * q^73 + 238 * q^77 - 116 * q^79 + 32 * q^80 + 134 * q^83 - 80 * q^88 + 190 * q^97 + 192 * q^98

Introduction

L-functions

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