LMFDB - Embedded newform 72.21.b.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,21,Mod(19,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 21, names="a")

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

chi := DirichletCharacter("72.19");

S:= CuspForms(chi, 21);

N := Newforms(S);

Level:	$N$	$=$	$72 = 2^{3} \cdot 3^{2}$
Weight:	$k$	$=$	$21$
Character orbit:	$[\chi]$	$=$	72.b (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:	$182.529910874$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			19.1
Character	$\chi$	$=$	72.19

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1024.00 q^{2} +1.04858e6 q^{4} -1.07374e9 q^{8} +4.23830e10 q^{11} +1.09951e12 q^{16} +3.35354e12 q^{17} -1.01465e12 q^{19} -4.34002e13 q^{22} +9.53674e13 q^{25} -1.12590e15 q^{32} -3.43402e15 q^{34} +1.03901e15 q^{38} +2.54181e16 q^{41} +2.78111e15 q^{43} +4.44418e16 q^{44} +7.97923e16 q^{49} -9.76562e16 q^{50} +1.73912e17 q^{59} +1.15292e18 q^{64} -3.56138e17 q^{67} +3.51644e18 q^{68} -6.01672e18 q^{73} -1.06394e18 q^{76} -2.60281e19 q^{82} +3.10229e19 q^{83} -2.84786e18 q^{86} -4.55084e19 q^{88} -6.12024e19 q^{89} -5.00091e19 q^{97} -8.17073e19 q^{98} +O(q^{100})

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$ .

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1024.00	−1.00000
$2$	−1024.00	−1.00000
$3$	0	0
$3$	0	0
$4$	1.04858e6	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1.07374e9	−1.00000
$9$	0	0
$10$	0	0
$11$	4.23830e10	1.63405	0.817025	−	0.576603i	$-0.195622\pi$
$11$	4.23830e10	1.63405	0.817025	+	0.576603i	$0.195622\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.09951e12	1.00000
$17$	3.35354e12	1.66347	0.831733	−	0.555176i	$-0.187349\pi$
$17$	3.35354e12	1.66347	0.831733	+	0.555176i	$0.187349\pi$
$18$	0	0
$19$	−1.01465e12	−0.165494	−0.0827470	−	0.996571i	$-0.526369\pi$
$19$	−1.01465e12	−0.165494	−0.0827470	+	0.996571i	$0.526369\pi$
$20$	0	0
$21$	0	0
$22$	−4.34002e13	−1.63405
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	9.53674e13	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.12590e15	−1.00000
$33$	0	0
$34$	−3.43402e15	−1.66347
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.03901e15	0.165494
$39$	0	0
$40$	0	0
$41$	2.54181e16	1.89367	0.946834	−	0.321721i	$-0.104261\pi$
$41$	2.54181e16	1.89367	0.946834	+	0.321721i	$0.104261\pi$
$42$	0	0
$43$	2.78111e15	0.128687	0.0643434	−	0.997928i	$-0.479505\pi$
$43$	2.78111e15	0.128687	0.0643434	+	0.997928i	$0.479505\pi$
$44$	4.44418e16	1.63405
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	7.97923e16	1.00000
$50$	−9.76562e16	−1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.73912e17	0.340259	0.170130	−	0.985422i	$-0.445581\pi$
$59$	1.73912e17	0.340259	0.170130	+	0.985422i	$0.445581\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.15292e18	1.00000
$65$	0	0
$66$	0	0
$67$	−3.56138e17	−0.195375	−0.0976877	−	0.995217i	$-0.531145\pi$
$67$	−3.56138e17	−0.195375	−0.0976877	+	0.995217i	$0.531145\pi$
$68$	3.51644e18	1.66347
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−6.01672e18	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$73$	−6.01672e18	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$74$	0	0
$75$	0	0
$76$	−1.06394e18	−0.165494
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	−2.60281e19	−1.89367
$83$	3.10229e19	1.99941	0.999703	−	0.0243827i	$-0.00776202\pi$
$83$	3.10229e19	1.99941	0.999703	+	0.0243827i	$0.00776202\pi$
$84$	0	0
$85$	0	0
$86$	−2.84786e18	−0.128687
$87$	0	0
$88$	−4.55084e19	−1.63405
$89$	−6.12024e19	−1.96277	−0.981383	−	0.192059i	$-0.938484\pi$
$89$	−6.12024e19	−1.96277	−0.981383	+	0.192059i	$0.938484\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.00091e19	−0.678160	−0.339080	−	0.940758i	$-0.610116\pi$
$97$	−5.00091e19	−0.678160	−0.339080	+	0.940758i	$0.610116\pi$
$98$	−8.17073e19	−1.00000
$99$	0	0
$100$	1.00000e20	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.62647e20	−1.84351	−0.921756	−	0.387770i	$-0.873246\pi$
$107$	−3.62647e20	−1.84351	−0.921756	+	0.387770i	$0.873246\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.52948e20	−0.450567	−0.225284	−	0.974293i	$-0.572331\pi$
$113$	−1.52948e20	−0.450567	−0.225284	+	0.974293i	$0.572331\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.78086e20	−0.340259
$119$	0	0
$120$	0	0
$121$	1.12357e21	1.67012
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.18059e21	−1.00000
$129$	0	0
$130$	0	0
$131$	−2.17294e21	−1.45994	−0.729970	−	0.683479i	$-0.760466\pi$
$131$	−2.17294e21	−1.45994	−0.729970	+	0.683479i	$0.760466\pi$
$132$	0	0
$133$	0	0
$134$	3.64685e20	0.195375
$135$	0	0
$136$	−3.60083e21	−1.66347
$137$	−2.11640e21	−0.908642	−0.454321	−	0.890838i	$-0.650118\pi$
$137$	−2.11640e21	−0.908642	−0.454321	+	0.890838i	$0.650118\pi$
$138$	0	0
$139$	2.54239e21	0.944266	0.472133	−	0.881527i	$-0.343484\pi$
$139$	2.54239e21	0.944266	0.472133	+	0.881527i	$0.343484\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	6.16112e21	1.40001
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.08948e21	0.165494
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.06792e19	0.00533845	0.00266923	−	0.999996i	$-0.499150\pi$
$163$	7.06792e19	0.00533845	0.00266923	+	0.999996i	$0.499150\pi$
$164$	2.66528e22	1.89367
$165$	0	0
$166$	−3.17674e22	−1.99941
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.90050e22	1.00000
$170$	0	0
$171$	0	0
$172$	2.91621e21	0.128687
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	4.66006e22	1.63405
$177$	0	0
$178$	6.26713e22	1.96277
$179$	3.92234e22	1.16149	0.580744	−	0.814086i	$-0.302762\pi$
$179$	3.92234e22	1.16149	0.580744	+	0.814086i	$0.302762\pi$
$180$	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
$187$	1.42133e23	2.71818
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	1.20384e23	1.67878	0.839391	−	0.543528i	$-0.182912\pi$
$193$	1.20384e23	1.67878	0.839391	+	0.543528i	$0.182912\pi$
$194$	5.12093e22	0.678160
$195$	0	0
$196$	8.36683e22	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.02400e23	−1.00000
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.30041e22	−0.270425
$210$	0	0
$211$	−2.82329e23	−1.61410	−0.807051	−	0.590482i	$-0.798938\pi$
$211$	−2.82329e23	−1.61410	−0.807051	+	0.590482i	$0.798938\pi$
$212$	0	0
$213$	0	0
$214$	3.71350e23	1.84351
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	1.56619e23	0.450567
$227$	2.21143e23	0.608716	0.304358	−	0.952558i	$-0.401558\pi$
$227$	2.21143e23	0.608716	0.304358	+	0.952558i	$0.401558\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.85874e23	1.66645	0.833227	−	0.552931i	$-0.186491\pi$
$233$	7.85874e23	1.66645	0.833227	+	0.552931i	$0.186491\pi$
$234$	0	0
$235$	0	0
$236$	1.82360e23	0.340259
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	7.13387e23	1.07933	0.539666	−	0.841879i	$-0.318551\pi$
$241$	7.13387e23	1.07933	0.539666	+	0.841879i	$0.318551\pi$
$242$	−1.15054e24	−1.67012
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.55766e24	1.56941	0.784703	−	0.619872i	$-0.212815\pi$
$251$	1.55766e24	1.56941	0.784703	+	0.619872i	$0.212815\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.20893e24	1.00000
$257$	−1.50407e24	−1.19657	−0.598284	−	0.801284i	$-0.704151\pi$
$257$	−1.50407e24	−1.19657	−0.598284	+	0.801284i	$0.704151\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	2.22509e24	1.45994
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−3.73437e23	−0.195375
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	3.68725e24	1.66347
$273$	0	0
$274$	2.16720e24	0.908642
$275$	4.04196e24	1.63405
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−2.60341e24	−0.944266
$279$	0	0
$280$	0	0
$281$	−2.04420e24	−0.665979	−0.332989	−	0.942931i	$-0.608057\pi$
$281$	−2.04420e24	−0.665979	−0.332989	+	0.942931i	$0.608057\pi$
$282$	0	0
$283$	−7.68742e23	−0.233301	−0.116650	−	0.993173i	$-0.537216\pi$
$283$	−7.68742e23	−0.233301	−0.116650	+	0.993173i	$0.537216\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.18197e24	1.76712
$290$	0	0
$291$	0	0
$292$	−6.30899e24	−1.40001
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−1.11562e24	−0.165494
$305$	0	0
$306$	0	0
$307$	2.64867e24	0.356160	0.178080	−	0.984016i	$-0.443011\pi$
$307$	2.64867e24	0.356160	0.178080	+	0.984016i	$0.443011\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.53618e25	1.70215	0.851075	−	0.525044i	$-0.175951\pi$
$313$	1.53618e25	1.70215	0.851075	+	0.525044i	$0.175951\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.40268e24	−0.275293
$324$	0	0
$325$	0	0
$326$	−7.23755e22	−0.00533845
$327$	0	0
$328$	−2.72924e25	−1.89367
$329$	0	0
$330$	0	0
$331$	−3.08691e25	−1.95544	−0.977718	−	0.209923i	$-0.932679\pi$
$331$	−3.08691e25	−1.95544	−0.977718	+	0.209923i	$0.932679\pi$
$332$	3.25298e25	1.99941
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.11942e24	0.0592508	0.0296254	−	0.999561i	$-0.490569\pi$
$337$	1.11942e24	0.0592508	0.0296254	+	0.999561i	$0.490569\pi$
$338$	−1.94611e25	−1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−2.98620e24	−0.128687
$345$	0	0
$346$	0	0
$347$	5.03994e25	1.99127	0.995635	−	0.0933286i	$-0.0297507\pi$
$347$	5.03994e25	1.99127	0.995635	+	0.0933286i	$0.0297507\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−4.77190e25	−1.63405
$353$	−5.62790e25	−1.87327	−0.936633	−	0.350313i	$-0.886075\pi$
$353$	−5.62790e25	−1.87327	−0.936633	+	0.350313i	$0.886075\pi$
$354$	0	0
$355$	0	0
$356$	−6.41754e25	−1.96277
$357$	0	0
$358$	−4.01647e25	−1.16149
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−3.65604e25	−0.972612
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−1.45544e26	−2.71818
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.83008e25	−0.626347	−0.313174	−	0.949696i	$-0.601392\pi$
$379$	−3.83008e25	−0.626347	−0.313174	+	0.949696i	$0.601392\pi$
$380$	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
$386$	−1.23273e26	−1.67878
$387$	0	0
$388$	−5.24384e25	−0.678160
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−8.56763e25	−1.00000
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.04858e26	1.00000
$401$	2.13090e26	1.98207	0.991034	−	0.133612i	$-0.0426576\pi$
$401$	2.13090e26	1.98207	0.991034	+	0.133612i	$0.0426576\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	1.27053e26	0.969955	0.484977	−	0.874527i	$-0.338828\pi$
$409$	1.27053e26	0.969955	0.484977	+	0.874527i	$0.338828\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	4.40362e25	0.270425
$419$	3.16629e26	1.89850	0.949249	−	0.314527i	$-0.101846\pi$
$419$	3.16629e26	1.89850	0.949249	+	0.314527i	$0.101846\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	2.89105e26	1.61410
$423$	0	0
$424$	0	0
$425$	3.19818e26	1.66347
$426$	0	0
$427$	0	0
$428$	−3.80263e26	−1.84351
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−2.40752e26	−1.03918	−0.519591	−	0.854415i	$-0.673916\pi$
$433$	−2.40752e26	−1.03918	−0.519591	+	0.854415i	$0.673916\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.30434e26	−0.448078	−0.224039	−	0.974580i	$-0.571924\pi$
$443$	−1.30434e26	−0.448078	−0.224039	+	0.974580i	$0.571924\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.95620e26	1.78857	0.894285	−	0.447499i	$-0.147685\pi$
$449$	5.95620e26	1.78857	0.894285	+	0.447499i	$0.147685\pi$
$450$	0	0
$451$	1.07729e27	3.09435
$452$	−1.60378e26	−0.450567
$453$	0	0
$454$	−2.26450e26	−0.608716
$455$	0	0
$456$	0	0
$457$	6.94600e26	1.74813	0.874064	−	0.485812i	$-0.161476\pi$
$457$	6.94600e26	1.74813	0.874064	+	0.485812i	$0.161476\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−8.04735e26	−1.66645
$467$	9.22043e26	1.86888	0.934442	−	0.356114i	$-0.115899\pi$
$467$	9.22043e26	1.86888	0.934442	+	0.356114i	$0.115899\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.86737e26	−0.340259
$473$	1.17872e26	0.210281
$474$	0	0
$475$	−9.67650e25	−0.165494
$476$	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
$482$	−7.30509e26	−1.07933
$483$	0	0
$484$	1.17815e27	1.67012
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.59202e27	−1.95494	−0.977469	−	0.211079i	$-0.932302\pi$
$491$	−1.59202e27	−1.95494	−0.977469	+	0.211079i	$0.932302\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.90641e27	1.99164	0.995820	−	0.0913348i	$-0.0291133\pi$
$499$	1.90641e27	1.99164	0.995820	+	0.0913348i	$0.0291133\pi$
$500$	0	0
$501$	0	0
$502$	−1.59504e27	−1.56941
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.23794e27	−1.00000
$513$	0	0
$514$	1.54017e27	1.19657
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.57885e27	1.75004	0.875021	−	0.484085i	$-0.160848\pi$
$521$	2.57885e27	1.75004	0.875021	+	0.484085i	$0.160848\pi$
$522$	0	0
$523$	2.21598e27	1.44726	0.723632	−	0.690186i	$-0.242471\pi$
$523$	2.21598e27	1.44726	0.723632	+	0.690186i	$0.242471\pi$
$524$	−2.27849e27	−1.45994
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.71616e27	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	3.82400e26	0.195375
$537$	0	0
$538$	0	0
$539$	3.38184e27	1.63405
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−3.77575e27	−1.66347
$545$	0	0
$546$	0	0
$547$	−4.19373e27	−1.74875	−0.874375	−	0.485251i	$-0.838728\pi$
$547$	−4.19373e27	−1.74875	−0.874375	+	0.485251i	$0.838728\pi$
$548$	−2.21921e27	−0.908642
$549$	0	0
$550$	−4.13897e27	−1.63405
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	2.66589e27	0.944266
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	2.09326e27	0.665979
$563$	−2.78356e27	−0.869994	−0.434997	−	0.900432i	$-0.643251\pi$
$563$	−2.78356e27	−0.869994	−0.434997	+	0.900432i	$0.643251\pi$
$564$	0	0
$565$	0	0
$566$	7.87192e26	0.233301
$567$	0	0
$568$	0	0
$569$	6.37985e27	1.79344	0.896722	−	0.442595i	$-0.145942\pi$
$569$	6.37985e27	1.79344	0.896722	+	0.442595i	$0.145942\pi$
$570$	0	0
$571$	−7.29382e27	−1.97967	−0.989837	−	0.142204i	$-0.954581\pi$
$571$	−7.29382e27	−1.97967	−0.989837	+	0.142204i	$0.954581\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.17943e27	−1.99970	−0.999850	−	0.0173302i	$-0.994483\pi$
$577$	−8.17943e27	−1.99970	−0.999850	+	0.0173302i	$0.994483\pi$
$578$	−7.35434e27	−1.76712
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	6.46040e27	1.40001
$585$	0	0
$586$	0	0
$587$	7.68162e27	1.58151	0.790755	−	0.612133i	$-0.209688\pi$
$587$	7.68162e27	1.58151	0.790755	+	0.612133i	$0.209688\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.00007e27	−0.557936	−0.278968	−	0.960300i	$-0.589992\pi$
$593$	−3.00007e27	−0.557936	−0.278968	+	0.960300i	$0.589992\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	8.69391e27	1.41407	0.707035	−	0.707179i	$-0.250033\pi$
$601$	8.69391e27	1.41407	0.707035	+	0.707179i	$0.250033\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.14240e27	0.165494
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−2.71224e27	−0.356160
$615$	0	0
$616$	0	0
$617$	−1.47242e28	−1.84153	−0.920765	−	0.390119i	$-0.872434\pi$
$617$	−1.47242e28	−1.84153	−0.920765	+	0.390119i	$0.872434\pi$
$618$	0	0
$619$	6.02524e24	0.000729572 0	0.000364786	−	1.00000i	$-0.499884\pi$
$619$	6.02524e24	0.000729572 0	0.000364786		1.00000i	$0.499884\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9.09495e27	1.00000
$626$	−1.57305e28	−1.70215
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.77175e28	−1.51294	−0.756471	−	0.654027i	$-0.773078\pi$
$641$	−1.77175e28	−1.51294	−0.756471	+	0.654027i	$0.773078\pi$
$642$	0	0
$643$	−2.35774e28	−1.95158	−0.975791	−	0.218706i	$-0.929816\pi$
$643$	−2.35774e28	−1.95158	−0.975791	+	0.218706i	$0.929816\pi$
$644$	0	0
$645$	0	0
$646$	3.48434e27	0.275293
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	7.37092e27	0.556000
$650$	0	0
$651$	0	0
$652$	7.41125e25	0.00533845
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	2.79475e28	1.89367
$657$	0	0
$658$	0	0
$659$	−1.96646e28	−1.27301	−0.636504	−	0.771274i	$-0.719620\pi$
$659$	−1.96646e28	−1.27301	−0.636504	+	0.771274i	$0.719620\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	3.16099e28	1.95544
$663$	0	0
$664$	−3.33105e28	−1.99941
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.82190e28	−1.48044	−0.740221	−	0.672364i	$-0.765279\pi$
$673$	−2.82190e28	−1.48044	−0.740221	+	0.672364i	$0.765279\pi$
$674$	−1.14629e27	−0.0592508
$675$	0	0
$676$	1.99281e28	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.34265e28	−1.96584	−0.982919	−	0.184038i	$-0.941083\pi$
$683$	−4.34265e28	−1.96584	−0.982919	+	0.184038i	$0.941083\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	3.05787e27	0.128687
$689$	0	0
$690$	0	0
$691$	−3.84706e28	−1.55006	−0.775030	−	0.631924i	$-0.782265\pi$
$691$	−3.84706e28	−1.55006	−0.775030	+	0.631924i	$0.782265\pi$
$692$	0	0
$693$	0	0
$694$	−5.16089e28	−1.99127
$695$	0	0
$696$	0	0
$697$	8.52404e28	3.15005
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	4.88643e28	1.63405
$705$	0	0
$706$	5.76297e28	1.87327
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	6.57156e28	1.96277
$713$	0	0
$714$	0	0
$715$	0	0
$716$	4.11287e28	1.16149
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	3.74379e28	0.972612
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.32657e27	0.214066
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.50942e28	−0.319253
$738$	0	0
$739$	5.68807e28	1.17090	0.585451	−	0.810708i	$-0.300917\pi$
$739$	5.68807e28	1.17090	0.585451	+	0.810708i	$0.300917\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	1.49037e29	2.71818
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	3.92200e28	0.626347
$759$	0	0
$760$	0	0
$761$	7.16003e28	1.09918	0.549589	−	0.835435i	$-0.314784\pi$
$761$	7.16003e28	1.09918	0.549589	+	0.835435i	$0.314784\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4.32623e28	−0.598200	−0.299100	−	0.954222i	$-0.596686\pi$
$769$	−4.32623e28	−0.598200	−0.299100	+	0.954222i	$0.596686\pi$
$770$	0	0
$771$	0	0
$772$	1.26231e29	1.67878
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	5.36969e28	0.678160
$777$	0	0
$778$	0	0
$779$	−2.57906e28	−0.313391
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	8.77325e28	1.00000
$785$	0	0
$786$	0	0
$787$	−1.82292e29	−1.99996	−0.999979	−	0.00648819i	$-0.997935\pi$
$787$	−1.82292e29	−1.99996	−0.999979	+	0.00648819i	$0.997935\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1.07374e29	−1.00000
$801$	0	0
$802$	−2.18204e29	−1.98207
$803$	−2.55007e29	−2.28768
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.24019e29	1.03277	0.516383	−	0.856357i	$-0.327278\pi$
$809$	1.24019e29	1.03277	0.516383	+	0.856357i	$0.327278\pi$
$810$	0	0
$811$	−1.11745e29	−0.907860	−0.453930	−	0.891037i	$-0.649978\pi$
$811$	−1.11745e29	−0.907860	−0.453930	+	0.891037i	$0.649978\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.82187e27	−0.0212969
$818$	−1.30102e29	−0.969955
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.19924e29	−1.46966	−0.734830	−	0.678252i	$-0.762738\pi$
$827$	−2.19924e29	−1.46966	−0.734830	+	0.678252i	$0.762738\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.67586e29	1.66347
$834$	0	0
$835$	0	0
$836$	−4.50931e28	−0.270425
$837$	0	0
$838$	−3.24228e29	−1.89850
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.76995e29	1.00000
$842$	0	0
$843$	0	0
$844$	−2.96043e29	−1.61410
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−3.27494e29	−1.66347
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	3.89389e29	1.84351
$857$	3.50153e28	0.163851	0.0819256	−	0.996638i	$-0.473893\pi$
$857$	3.50153e28	0.163851	0.0819256	+	0.996638i	$0.473893\pi$
$858$	0	0
$859$	−4.25332e29	−1.94445	−0.972224	−	0.234053i	$-0.924801\pi$
$859$	−4.25332e29	−1.94445	−0.972224	+	0.234053i	$0.924801\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	2.46530e29	1.03918
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.59188e29	−1.98517	−0.992587	−	0.121534i	$-0.961219\pi$
$881$	−5.59188e29	−1.98517	−0.992587	+	0.121534i	$0.961219\pi$
$882$	0	0
$883$	4.50564e29	1.56369	0.781843	−	0.623476i	$-0.214280\pi$
$883$	4.50564e29	1.56369	0.781843	+	0.623476i	$0.214280\pi$
$884$	0	0
$885$	0	0
$886$	1.33564e29	0.448078
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−6.09915e29	−1.78857
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−1.10315e30	−3.09435
$903$	0	0
$904$	1.64227e29	0.450567
$905$	0	0
$906$	0	0
$907$	−7.39355e29	−1.96237	−0.981183	−	0.193079i	$-0.938153\pi$
$907$	−7.39355e29	−1.96237	−0.981183	+	0.193079i	$0.938153\pi$
$908$	2.31885e29	0.608716
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	1.31484e30	3.26713
$914$	−7.11271e29	−1.74813
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.33455e29	1.94956	0.974779	−	0.223173i	$-0.0716415\pi$
$929$	9.33455e29	1.94956	0.974779	+	0.223173i	$0.0716415\pi$
$930$	0	0
$931$	−8.09616e28	−0.165494
$932$	8.24048e29	1.66645
$933$	0	0
$934$	−9.44172e29	−1.86888
$935$	0	0
$936$	0	0
$937$	7.34116e29	1.40724	0.703621	−	0.710575i	$-0.251565\pi$
$937$	7.34116e29	1.40724	0.703621	+	0.710575i	$0.251565\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.91218e29	0.340259
$945$	0	0
$946$	−1.20701e29	−0.210281
$947$	8.42382e29	1.45214	0.726071	−	0.687619i	$-0.241344\pi$
$947$	8.42382e29	1.45214	0.726071	+	0.687619i	$0.241344\pi$
$948$	0	0
$949$	0	0
$950$	9.90873e28	0.165494
$951$	0	0
$952$	0	0
$953$	9.07852e29	1.46922	0.734609	−	0.678491i	$-0.237366\pi$
$953$	9.07852e29	1.46922	0.734609	+	0.678491i	$0.237366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	6.71791e29	1.00000
$962$	0	0
$963$	0	0
$964$	7.48041e29	1.07933
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−1.20642e30	−1.67012
$969$	0	0
$970$	0	0
$971$	7.67973e29	1.03075	0.515376	−	0.856964i	$-0.327652\pi$
$971$	7.67973e29	1.03075	0.515376	+	0.856964i	$0.327652\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.46005e29	0.689051	0.344525	−	0.938777i	$-0.388040\pi$
$977$	5.46005e29	0.689051	0.344525	+	0.938777i	$0.388040\pi$
$978$	0	0
$979$	−2.59394e30	−3.20726
$980$	0	0
$981$	0	0
$982$	1.63023e30	1.95494
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	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
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.95216e30	−1.99164
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.21.b.a.19.1		1
3.2	odd	2		8.21.d.a.3.1	✓	1
8.3	odd	2	CM	72.21.b.a.19.1		1
12.11	even	2		32.21.d.a.15.1		1
24.5	odd	2		32.21.d.a.15.1		1
24.11	even	2		8.21.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.21.d.a.3.1	✓	1	3.2	odd	2
8.21.d.a.3.1	✓	1	24.11	even	2
32.21.d.a.15.1		1	12.11	even	2
32.21.d.a.15.1		1	24.5	odd	2
72.21.b.a.19.1		1	1.1	even	1	trivial
72.21.b.a.19.1		1	8.3	odd	2	CM

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

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Modular forms

Varieties

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Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	72.21.b.a.19.1

Level	$72$
Weight	$21$
Character	72.19
Self dual	yes
Analytic conductor	$182.530$
Analytic rank	$0$
Dimension	$1$
CM discriminant	-8
Inner twists	$2$