LMFDB - Embedded newform 216.3.h.b.53.1

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			53.1
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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -7.48528 q^{5} +13.4853 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} -7.48528 q^{5} +13.4853 q^{7} +8.00000 q^{8} -14.9706 q^{10} +11.9706 q^{11} +26.9706 q^{14} +16.0000 q^{16} -29.9411 q^{20} +23.9411 q^{22} +31.0294 q^{25} +53.9411 q^{28} -50.0000 q^{29} -61.4264 q^{31} +32.0000 q^{32} -100.941 q^{35} -59.8823 q^{40} +47.8823 q^{44} +132.853 q^{49} +62.0589 q^{50} -4.57359 q^{53} -89.6030 q^{55} +107.882 q^{56} -100.000 q^{58} +10.0000 q^{59} -122.853 q^{62} +64.0000 q^{64} -201.882 q^{70} -143.794 q^{73} +161.426 q^{77} -58.0000 q^{79} -119.765 q^{80} -17.8528 q^{83} +95.7645 q^{88} +128.941 q^{97} +265.706 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−7.48528	−1.49706	−0.748528	−	0.663103i	$-0.769239\pi$
$5$	−7.48528	−1.49706	−0.748528	+	0.663103i	$0.769239\pi$
$6$	0	0
$7$	13.4853	1.92647	0.963234	−	0.268662i	$-0.0865816\pi$
$7$	13.4853	1.92647	0.963234	+	0.268662i	$0.0865816\pi$
$8$	8.00000	1.00000
$9$	0	0
$10$	−14.9706	−1.49706
$11$	11.9706	1.08823	0.544116	−	0.839010i	$-0.316865\pi$
$11$	11.9706	1.08823	0.544116	+	0.839010i	$0.316865\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	26.9706	1.92647
$15$	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			$\pi$
$20$	−29.9411	−1.49706
$21$	0	0
$22$	23.9411	1.08823
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	31.0294	1.24118
$26$	0	0
$27$	0	0
$28$	53.9411	1.92647
$29$	−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
$31$	−61.4264	−1.98150	−0.990748	−	0.135711i	$-0.956668\pi$
$31$	−61.4264	−1.98150	−0.990748	+	0.135711i	$0.956668\pi$
$32$	32.0000	1.00000
$33$	0	0
$34$	0	0
$35$	−100.941	−2.88403
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−59.8823	−1.49706
$41$	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			$\pi$
$44$	47.8823	1.08823
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	132.853	2.71128
$50$	62.0589	1.24118
$51$	0	0
$52$	0	0
$53$	−4.57359	−0.0862942	−0.0431471	−	0.999069i	$-0.513738\pi$
$53$	−4.57359	−0.0862942	−0.0431471	+	0.999069i	$0.513738\pi$
$54$	0	0
$55$	−89.6030	−1.62915
$56$	107.882	1.92647
$57$	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			$\pi$
$62$	−122.853	−1.98150
$63$	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
$70$	−201.882	−2.88403
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−143.794	−1.96978	−0.984890	−	0.173181i	$-0.944595\pi$
$73$	−143.794	−1.96978	−0.984890	+	0.173181i	$0.944595\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	161.426	2.09645
$78$	0	0
$79$	−58.0000	−0.734177	−0.367089	−	0.930186i	$-0.619645\pi$
$79$	−58.0000	−0.734177	−0.367089	+	0.930186i	$0.619645\pi$
$80$	−119.765	−1.49706
$81$	0	0
$82$	0	0
$83$	−17.8528	−0.215094	−0.107547	−	0.994200i	$-0.534300\pi$
$83$	−17.8528	−0.215094	−0.107547	+	0.994200i	$0.534300\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	95.7645	1.08823
$89$	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
$97$	128.941	1.32929	0.664645	−	0.747159i	$-0.268583\pi$
$97$	128.941	1.32929	0.664645	+	0.747159i	$0.268583\pi$
$98$	265.706	2.71128
$99$	0	0
$100$	124.118	1.24118
$101$	−154.397	−1.52868	−0.764341	−	0.644812i	$-0.776936\pi$
$101$	−154.397	−1.52868	−0.764341	+	0.644812i	$0.776936\pi$
$102$	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
$106$	−9.14719	−0.0862942
$107$	212.706	1.98790	0.993952	−	0.109820i	$-0.0350273\pi$
$107$	212.706	1.98790	0.993952	+	0.109820i	$0.0350273\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−179.206	−1.62915
$111$	0	0
$112$	215.765	1.92647
$113$	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
$121$	22.2944	0.184251
$122$	0	0
$123$	0	0
$124$	−245.706	−1.98150
$125$	−45.1320	−0.361056
$126$	0	0
$127$	−208.338	−1.64046	−0.820229	−	0.572036i	$-0.806154\pi$
$127$	−208.338	−1.64046	−0.820229	+	0.572036i	$0.806154\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	0	0
$131$	−57.1177	−0.436013	−0.218007	−	0.975947i	$-0.569955\pi$
$131$	−57.1177	−0.436013	−0.218007	+	0.975947i	$0.569955\pi$
$132$	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			$\pi$
$140$	−403.765	−2.88403
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	374.264	2.58113
$146$	−287.588	−1.96978
$147$	0	0
$148$	0	0
$149$	85.6030	0.574517	0.287258	−	0.957853i	$-0.407256\pi$
$149$	85.6030	0.574517	0.287258	+	0.957853i	$0.407256\pi$
$150$	0	0
$151$	106.574	0.705785	0.352893	−	0.935664i	$-0.385198\pi$
$151$	106.574	0.705785	0.352893	+	0.935664i	$0.385198\pi$
$152$	0	0
$153$	0	0
$154$	322.853	2.09645
$155$	459.794	2.96641
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−116.000	−0.734177
$159$	0	0
$160$	−239.529	−1.49706
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−35.7056	−0.215094
$167$	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
$173$	273.985	1.58373	0.791864	−	0.610698i	$-0.209111\pi$
$173$	273.985	1.58373	0.791864	+	0.610698i	$0.209111\pi$
$174$	0	0
$175$	418.441	2.39109
$176$	191.529	1.08823
$177$	0	0
$178$	0	0
$179$	−122.588	−0.684848	−0.342424	−	0.939545i	$-0.611248\pi$
$179$	−122.588	−0.684848	−0.342424	+	0.939545i	$0.611248\pi$
$180$	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
$193$	−365.617	−1.89439	−0.947195	−	0.320658i	$-0.896096\pi$
$193$	−365.617	−1.89439	−0.947195	+	0.320658i	$0.896096\pi$
$194$	257.882	1.32929
$195$	0	0
$196$	531.411	2.71128
$197$	393.985	1.99992	0.999962	−	0.00876993i	$-0.00279159\pi$
$197$	393.985	1.99992	0.999962	+	0.00876993i	$0.00279159\pi$
$198$	0	0
$199$	−195.985	−0.984848	−0.492424	−	0.870355i	$-0.663889\pi$
$199$	−195.985	−0.984848	−0.492424	+	0.870355i	$0.663889\pi$
$200$	248.235	1.24118
$201$	0	0
$202$	−308.794	−1.52868
$203$	−674.264	−3.32150
$204$	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			$\pi$
$212$	−18.2944	−0.0862942
$213$	0	0
$214$	425.411	1.98790
$215$	0	0
$216$	0	0
$217$	−828.352	−3.81729
$218$	0	0
$219$	0	0
$220$	−358.412	−1.62915
$221$	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	$0.327534\pi$
$224$	431.529	1.92647
$225$	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	$0.791237\pi$
$242$	44.5887	0.184251
$243$	0	0
$244$	0	0
$245$	−994.441	−4.05894
$246$	0	0
$247$	0	0
$248$	−491.411	−1.98150
$249$	0	0
$250$	−90.2641	−0.361056
$251$	−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
$254$	−416.676	−1.64046
$255$	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
$262$	−114.235	−0.436013
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	34.2346	0.129187
$266$	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
$271$	495.690	1.82912	0.914558	−	0.404455i	$-0.132539\pi$
$271$	495.690	1.82912	0.914558	+	0.404455i	$0.132539\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	371.440	1.35069
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−807.529	−2.88403
$281$	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
$290$	748.528	2.58113
$291$	0	0
$292$	−575.176	−1.96978
$293$	−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
$295$	−74.8528	−0.253738
$296$	0	0
$297$	0	0
$298$	171.206	0.574517
$299$	0	0
$300$	0	0
$301$	0	0
$302$	213.147	0.705785
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	645.706	2.09645
$309$	0	0
$310$	919.588	2.96641
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	23.4996	0.0750785	0.0375392	−	0.999295i	$-0.488048\pi$
$313$	23.4996	0.0750785	0.0375392	+	0.999295i	$0.488048\pi$
$314$	0	0
$315$	0	0
$316$	−232.000	−0.734177
$317$	−633.690	−1.99902	−0.999512	−	0.0312439i	$-0.990053\pi$
$317$	−633.690	−1.99902	−0.999512	+	0.0312439i	$0.990053\pi$
$318$	0	0
$319$	−598.528	−1.87626
$320$	−479.058	−1.49706
$321$	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			$\pi$
$332$	−71.4113	−0.215094
$333$	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
$341$	−735.309	−2.15633
$342$	0	0
$343$	1130.78	3.29673
$344$	0	0
$345$	0	0
$346$	547.970	1.58373
$347$	−646.970	−1.86447	−0.932233	−	0.361859i	$-0.882142\pi$
$347$	−646.970	−1.86447	−0.932233	+	0.361859i	$0.882142\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	836.881	2.39109
$351$	0	0
$352$	383.058	1.08823
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−245.176	−0.684848
$359$	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
$365$	1076.34	2.94887
$366$	0	0
$367$	−193.780	−0.528010	−0.264005	−	0.964521i	$-0.585044\pi$
$367$	−193.780	−0.528010	−0.264005	+	0.964521i	$0.585044\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−61.6762	−0.166243
$372$	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
$385$	−1208.32	−3.13850
$386$	−731.235	−1.89439
$387$	0	0
$388$	515.765	1.32929
$389$	558.367	1.43539	0.717695	−	0.696358i	$-0.245197\pi$
$389$	558.367	1.43539	0.717695	+	0.696358i	$0.245197\pi$
$390$	0	0
$391$	0	0
$392$	1062.82	2.71128
$393$	0	0
$394$	787.970	1.99992
$395$	434.146	1.09910
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−391.970	−0.984848
$399$	0	0
$400$	496.471	1.24118
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−617.588	−1.52868
$405$	0	0
$406$	−1348.53	−3.32150
$407$	0	0
$408$	0	0
$409$	698.411	1.70761	0.853803	−	0.520595i	$-0.174290\pi$
$409$	698.411	1.70761	0.853803	+	0.520595i	$0.174290\pi$
$410$	0	0
$411$	0	0
$412$	−40.0000	−0.0970874
$413$	134.853	0.326520
$414$	0	0
$415$	133.633	0.322008
$416$	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
$424$	−36.5887	−0.0862942
$425$	0	0
$426$	0	0
$427$	0	0
$428$	850.823	1.98790
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−140.176	−0.323732	−0.161866	−	0.986813i	$-0.551751\pi$
$433$	−140.176	−0.323732	−0.161866	+	0.986813i	$0.551751\pi$
$434$	−1656.70	−3.81729
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	582.249	1.32631	0.663154	−	0.748483i	$-0.269218\pi$
$439$	582.249	1.32631	0.663154	+	0.748483i	$0.269218\pi$
$440$	−716.824	−1.62915
$441$	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
$448$	863.058	1.92647
$449$	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
$457$	379.881	0.831250	0.415625	−	0.909536i	$-0.363563\pi$
$457$	379.881	0.831250	0.415625	+	0.909536i	$0.363563\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	918.367	1.99212	0.996059	−	0.0886899i	$-0.0282680\pi$
$461$	918.367	1.99212	0.996059	+	0.0886899i	$0.0282680\pi$
$462$	0	0
$463$	897.161	1.93771	0.968856	−	0.247625i	$-0.0796501\pi$
$463$	897.161	1.93771	0.968856	+	0.247625i	$0.0796501\pi$
$464$	−800.000	−1.72414
$465$	0	0
$466$	0	0
$467$	−910.970	−1.95068	−0.975342	−	0.220698i	$-0.929167\pi$
$467$	−910.970	−1.95068	−0.975342	+	0.220698i	$0.929167\pi$
$468$	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
$484$	89.1775	0.184251
$485$	−965.161	−1.99002
$486$	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
$490$	−1988.88	−4.05894
$491$	−511.705	−1.04217	−0.521084	−	0.853505i	$-0.674472\pi$
$491$	−511.705	−1.04217	−0.521084	+	0.853505i	$0.674472\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−982.823	−1.98150
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−180.528	−0.361056
$501$	0	0
$502$	−940.000	−1.87251
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	1155.70	2.28852
$506$	0	0
$507$	0	0
$508$	−833.352	−1.64046
$509$	394.691	0.775425	0.387713	−	0.921780i	$-0.373265\pi$
$509$	394.691	0.775425	0.387713	+	0.921780i	$0.373265\pi$
$510$	0	0
$511$	−1939.10	−3.79472
$512$	512.000	1.00000
$513$	0	0
$514$	0	0
$515$	74.8528	0.145345
$516$	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			$\pi$
$524$	−228.471	−0.436013
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	68.4693	0.129187
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1592.16	−2.97600
$536$	0	0
$537$	0	0
$538$	860.000	1.59851
$539$	1590.32	2.95051
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	991.381	1.82912
$543$	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
$550$	742.880	1.35069
$551$	0	0
$552$	0	0
$553$	−782.146	−1.41437
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1008.54	1.81067	0.905335	−	0.424698i	$-0.139620\pi$
$557$	1008.54	1.81067	0.905335	+	0.424698i	$0.139620\pi$
$558$	0	0
$559$	0	0
$560$	−1615.06	−2.88403
$561$	0	0
$562$	0	0
$563$	1096.38	1.94739	0.973695	−	0.227854i	$-0.0731709\pi$
$563$	1096.38	1.94739	0.973695	+	0.227854i	$0.0731709\pi$
$564$	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
$580$	1497.06	2.58113
$581$	−240.750	−0.414372
$582$	0	0
$583$	−54.7485	−0.0939082
$584$	−1150.35	−1.96978
$585$	0	0
$586$	−772.000	−1.31741
$587$	241.823	0.411963	0.205982	−	0.978556i	$-0.433961\pi$
$587$	241.823	0.411963	0.205982	+	0.978556i	$0.433961\pi$
$588$	0	0
$589$	0	0
$590$	−149.706	−0.253738
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	342.412	0.574517
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	683.853	1.13786	0.568929	−	0.822387i	$-0.307358\pi$
$601$	683.853	1.13786	0.568929	+	0.822387i	$0.307358\pi$
$602$	0	0
$603$	0	0
$604$	426.294	0.705785
$605$	−166.880	−0.275834
$606$	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
$616$	1291.41	2.09645
$617$	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			$\pi$
$620$	1839.18	2.96641
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−437.910	−0.700656
$626$	46.9991	0.0750785
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−406.868	−0.644799	−0.322399	−	0.946604i	$-0.604489\pi$
$631$	−406.868	−0.644799	−0.322399	+	0.946604i	$0.604489\pi$
$632$	−464.000	−0.734177
$633$	0	0
$634$	−1267.38	−1.99902
$635$	1559.47	2.45586
$636$	0	0
$637$	0	0
$638$	−1197.06	−1.87626
$639$	0	0
$640$	−958.116	−1.49706
$641$	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
$649$	119.706	0.184446
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1224.25	−1.87481	−0.937403	−	0.348245i	$-0.886778\pi$
$653$	−1224.25	−1.87481	−0.937403	+	0.348245i	$0.886778\pi$
$654$	0	0
$655$	427.542	0.652737
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1315.35	−1.99598	−0.997991	−	0.0633633i	$-0.979817\pi$
$659$	−1315.35	−1.99598	−0.997991	+	0.0633633i	$0.979817\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	−142.823	−0.215094
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1249.00	1.85587	0.927933	−	0.372746i	$-0.121584\pi$
$673$	1249.00	1.85587	0.927933	+	0.372746i	$0.121584\pi$
$674$	−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
$679$	1738.81	2.56084
$680$	0	0
$681$	0	0
$682$	−1470.62	−2.15633
$683$	−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
$686$	2261.56	3.29673
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1095.94	1.58373
$693$	0	0
$694$	−1293.94	−1.86447
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1673.76	2.39109
$701$	1238.40	1.76661	0.883306	−	0.468797i	$-0.155312\pi$
$701$	1238.40	1.76661	0.883306	+	0.468797i	$0.155312\pi$
$702$	0	0
$703$	0	0
$704$	766.116	1.08823
$705$	0	0
$706$	0	0
$707$	−2082.09	−2.94496
$708$	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
$716$	−490.352	−0.684848
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−134.853	−0.187036
$722$	722.000	1.00000
$723$	0	0
$724$	0	0
$725$	−1551.47	−2.13996
$726$	0	0
$727$	631.662	0.868861	0.434430	−	0.900705i	$-0.356950\pi$
$727$	631.662	0.868861	0.434430	+	0.900705i	$0.356950\pi$
$728$	0	0
$729$	0	0
$730$	2152.68	2.94887
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−387.559	−0.528010
$735$	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
$742$	−123.352	−0.166243
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−640.763	−0.860084
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2868.40	3.82963
$750$	0	0
$751$	1167.69	1.55485	0.777424	−	0.628977i	$-0.216526\pi$
$751$	1167.69	1.55485	0.777424	+	0.628977i	$0.216526\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−797.733	−1.05660
$756$	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
$769$	1534.09	1.99491	0.997456	−	0.0712917i	$-0.0227121\pi$
$769$	1534.09	1.99491	0.997456	+	0.0712917i	$0.0227121\pi$
$770$	−2416.64	−3.13850
$771$	0	0
$772$	−1462.47	−1.89439
$773$	−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
$775$	−1906.03	−2.45939
$776$	1031.53	1.32929
$777$	0	0
$778$	1116.73	1.43539
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2125.65	2.71128
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1575.94	1.99992
$789$	0	0
$790$	868.293	1.09910
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−783.939	−0.984848
$797$	159.102	0.199626	0.0998129	−	0.995006i	$-0.468176\pi$
$797$	159.102	0.199626	0.0998129	+	0.995006i	$0.468176\pi$
$798$	0	0
$799$	0	0
$800$	992.942	1.24118
$801$	0	0
$802$	0	0
$803$	−1721.29	−2.14358
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1235.18	−1.52868
$809$	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			$\pi$
$812$	−2697.06	−3.32150
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1396.82	1.70761
$819$	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
$823$	−1601.13	−1.94548	−0.972740	−	0.231898i	$-0.925507\pi$
$823$	−1601.13	−1.94548	−0.972740	+	0.231898i	$0.925507\pi$
$824$	−80.0000	−0.0970874
$825$	0	0
$826$	269.706	0.326520
$827$	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			$\pi$
$830$	267.267	0.322008
$831$	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
$845$	−1265.01	−1.49706
$846$	0	0
$847$	300.646	0.354954
$848$	−73.1775	−0.0862942
$849$	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
$856$	1701.65	1.98790
$857$	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
$865$	−2050.85	−2.37093
$866$	−280.352	−0.323732
$867$	0	0
$868$	−3313.41	−3.81729
$869$	−694.293	−0.798956
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−608.618	−0.695564
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1164.50	1.32631
$879$	0	0
$880$	−1433.65	−1.62915
$881$	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
$889$	−2809.50	−3.16029
$890$	0	0
$891$	0	0
$892$	920.000	1.03139
$893$	0	0
$894$	0	0
$895$	917.605	1.02526
$896$	1726.12	1.92647
$897$	0	0
$898$	0	0
$899$	3071.32	3.41637
$900$	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
$913$	−213.708	−0.234073
$914$	759.763	0.831250
$915$	0	0
$916$	0	0
$917$	−770.249	−0.839966
$918$	0	0
$919$	1153.92	1.25563	0.627815	−	0.778362i	$-0.283949\pi$
$919$	1153.92	1.25563	0.627815	+	0.778362i	$0.283949\pi$
$920$	0	0
$921$	0	0
$922$	1836.73	1.99212
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1794.32	1.93771
$927$	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
$934$	−1821.94	−1.95068
$935$	0	0
$936$	0	0
$937$	−1729.29	−1.84556	−0.922782	−	0.385323i	$-0.874090\pi$
$937$	−1729.29	−1.84556	−0.922782	+	0.385323i	$0.874090\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1801.75	−1.91472	−0.957358	−	0.288904i	$-0.906709\pi$
$941$	−1801.75	−1.91472	−0.957358	+	0.288904i	$0.906709\pi$
$942$	0	0
$943$	0	0
$944$	160.000	0.169492
$945$	0	0
$946$	0	0
$947$	534.939	0.564878	0.282439	−	0.959285i	$-0.408857\pi$
$947$	534.939	0.564878	0.282439	+	0.959285i	$0.408857\pi$
$948$	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
$961$	2812.20	2.92633
$962$	0	0
$963$	0	0
$964$	−1528.00	−1.58506
$965$	2736.75	2.83601
$966$	0	0
$967$	−691.956	−0.715570	−0.357785	−	0.933804i	$-0.616468\pi$
$967$	−691.956	−0.715570	−0.357785	+	0.933804i	$0.616468\pi$
$968$	178.355	0.184251
$969$	0	0
$970$	−1930.32	−1.99002
$971$	−1151.68	−1.18607	−0.593036	−	0.805176i	$-0.702071\pi$
$971$	−1151.68	−1.18607	−0.593036	+	0.805176i	$0.702071\pi$
$972$	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
$980$	−3977.76	−4.05894
$981$	0	0
$982$	−1023.41	−1.04217
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−2949.09	−2.99400
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	539.366	0.544264	0.272132	−	0.962260i	$-0.412271\pi$
$991$	539.366	0.544264	0.272132	+	0.962260i	$0.412271\pi$
$992$	−1965.65	−1.98150
$993$	0	0
$994$	0	0
$995$	1467.00	1.47437
$996$	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.53.1	yes	2
3.2	odd	2		216.3.h.a.53.2	✓	2
4.3	odd	2		864.3.h.b.593.1		2
8.3	odd	2		864.3.h.a.593.2		2
8.5	even	2		216.3.h.a.53.2	✓	2
12.11	even	2		864.3.h.a.593.2		2
24.5	odd	2	CM	216.3.h.b.53.1	yes	2
24.11	even	2		864.3.h.b.593.1		2

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -7.48528 q^{5} +13.4853 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -7.48528 q^{5} +13.4853 q^{7} +8.00000 q^{8} -14.9706 q^{10} +11.9706 q^{11} +26.9706 q^{14} +16.0000 q^{16} -29.9411 q^{20} +23.9411 q^{22} +31.0294 q^{25} +53.9411 q^{28} -50.0000 q^{29} -61.4264 q^{31} +32.0000 q^{32} -100.941 q^{35} -59.8823 q^{40} +47.8823 q^{44} +132.853 q^{49} +62.0589 q^{50} -4.57359 q^{53} -89.6030 q^{55} +107.882 q^{56} -100.000 q^{58} +10.0000 q^{59} -122.853 q^{62} +64.0000 q^{64} -201.882 q^{70} -143.794 q^{73} +161.426 q^{77} -58.0000 q^{79} -119.765 q^{80} -17.8528 q^{83} +95.7645 q^{88} +128.941 q^{97} +265.706 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 2 q^{5} + 10 q^{7} + 16 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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