LMFDB - Embedded newform 1984.2.h.f.1983.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1984,2,Mod(1983,1984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1984, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1984.1983");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.8423197610$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.21717639.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 8 $ x^6 - x^3 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 124)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1983.5
Root			$-1.18073 - 0.778374i$ of defining polynomial
Character	$\chi$	$=$	1984.1983
Dual form			1984.2.h.f.1983.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.93800 q^{5} -5.23296i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+3.93800 q^{5} -5.23296i q^{7} -3.00000 q^{9} -0.994032i q^{19} +10.5079 q^{25} -5.56776i q^{31} -20.6074i q^{35} +2.36814 q^{41} -11.8140 q^{45} -11.1355i q^{47} -20.3839 q^{49} +11.4600i q^{59} +15.6989i q^{63} -11.1355i q^{67} +3.24490i q^{71} +9.00000 q^{81} -3.91450i q^{95} -10.2441 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{9}+O(q^{10}) $ 6 * q - 18 * q^9	$ 6 q - 18 q^{9} + 30 q^{25} - 42 q^{49} + 54 q^{81}+O(q^{100}) $ 6 * q - 18 * q^9 + 30 * q^25 - 42 * q^49 + 54 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times$.

$n$	$63$	$65$	$1861$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	3.93800	1.76113	0.880564	−	0.473927i	$-0.157164\pi$
$5$	3.93800	1.76113	0.880564	+	0.473927i	$0.157164\pi$
$6$	0	0
$7$	− 5.23296i	− 1.97787i	−0.148339	−	0.988937i	$-0.547393\pi$
$7$	− 5.23296i	− 1.97787i	0.148339	−	0.988937i	$-0.452607\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 0.994032i	− 0.228047i	−0.993478	−	0.114023i	$-0.963626\pi$
$19$	− 0.994032i	− 0.228047i	0.993478	−	0.114023i	$-0.0363739\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	10.5079	2.10157
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 5.56776i	− 1.00000i
$31$	− 5.56776i	− 1.00000i
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 20.6074i	− 3.48329i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.36814	0.369841	0.184920	−	0.982754i	$-0.440797\pi$
$41$	2.36814	0.369841	0.184920	+	0.982754i	$0.440797\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−11.8140	−1.76113
$46$	0	0
$47$	− 11.1355i	− 1.62428i	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	− 11.1355i	− 1.62428i	0.583460	−	0.812142i	$-0.301699\pi$
$48$	0	0
$49$	−20.3839	−2.91198
$50$	0	0
$51$	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
$59$	11.4600i	1.49196i	0.665969	+	0.745979i	$0.268018\pi$
$59$	11.4600i	1.49196i	−0.665969	+	0.745979i	$0.731982\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	15.6989i	1.97787i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 11.1355i	− 1.36042i	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	− 11.1355i	− 1.36042i	0.733017	−	0.680211i	$-0.238112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.24490i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$71$	3.24490i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	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
$95$	− 3.91450i	− 0.401620i
$96$	0	0
$97$	−10.2441	−1.04014	−0.520068	−	0.854125i	$-0.674093\pi$
$97$	−10.2441	−1.04014	−0.520068	+	0.854125i	$0.674093\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.07774	0.704261	0.352131	−	0.935951i	$-0.385457\pi$
$101$	7.07774	0.704261	0.352131	+	0.935951i	$0.385457\pi$
$102$	0	0
$103$	− 13.7108i	− 1.35097i	−0.737375	−	0.675483i	$-0.763935\pi$
$103$	− 13.7108i	− 1.35097i	0.737375	−	0.675483i	$-0.236065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.9378i	1.92746i	0.266878	+	0.963730i	$0.414008\pi$
$107$	19.9378i	1.92746i	−0.266878	+	0.963730i	$0.585992\pi$
$108$	0	0
$109$	−14.9537	−1.43231	−0.716155	−	0.697942i	$-0.754099\pi$
$109$	−14.9537	−1.43231	−0.716155	+	0.697942i	$0.754099\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	21.2599	1.99996	0.999981	−	0.00618051i	$-0.00196733\pi$
$113$	21.2599	1.99996	0.999981	+	0.00618051i	$0.00196733\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	21.6900	1.94001
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.1355i	− 0.972916i	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	− 11.1355i	− 0.972916i	0.873704	−	0.486458i	$-0.161711\pi$
$132$	0	0
$133$	−5.20173	−0.451047
$134$	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		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 21.9259i	− 1.76113i
$156$	0	0
$157$	22.8298	1.82201	0.911006	−	0.412392i	$-0.135307\pi$
$157$	22.8298	1.82201	0.911006	+	0.412392i	$0.135307\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.48382i	0.586178i	0.956085	+	0.293089i	$0.0946833\pi$
$163$	7.48382i	0.586178i	−0.956085	+	0.293089i	$0.905317\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	2.98210i	0.228047i
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	− 54.9873i	− 4.15665i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	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
$191$	24.1767i	1.74937i	0.484694	+	0.874684i	$0.338931\pi$
$191$	24.1767i	1.74937i	−0.484694	+	0.874684i	$0.661069\pi$
$192$	0	0
$193$	18.1201	1.30432	0.652158	−	0.758083i	$-0.273864\pi$
$193$	18.1201	1.30432	0.652158	+	0.758083i	$0.273864\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.32573	0.651337
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 17.9497i	− 1.23571i	−0.786291	−	0.617856i	$-0.788002\pi$
$211$	− 17.9497i	− 1.23571i	0.786291	−	0.617856i	$-0.211998\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−29.1359	−1.97787
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−31.5236	−2.10157
$226$	0	0
$227$	− 11.1355i	− 0.739091i	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	− 11.1355i	− 0.739091i	0.929213	−	0.369546i	$-0.120487\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.64761	0.566524	0.283262	−	0.959043i	$-0.408583\pi$
$233$	8.64761	0.566524	0.283262	+	0.959043i	$0.408583\pi$
$234$	0	0
$235$	− 43.8518i	− 2.86057i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−80.2718	−5.12838
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.3996	1.52201	0.761003	−	0.648748i	$-0.224707\pi$
$257$	24.3996	1.52201	0.761003	+	0.648748i	$0.224707\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	16.7033i	1.00000i
$280$	0	0
$281$	−0.771601	−0.0460298	−0.0230149	−	0.999735i	$-0.507327\pi$
$281$	−0.771601	−0.0460298	−0.0230149	+	0.999735i	$0.507327\pi$
$282$	0	0
$283$	33.4066i	1.98582i	0.118888	+	0.992908i	$0.462067\pi$
$283$	33.4066i	1.98582i	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.3924i	− 0.731498i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	45.1293i	2.62753i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.3918i	1.84870i	0.381549	+	0.924349i	$0.375391\pi$
$307$	32.3918i	1.84870i	−0.381549	+	0.924349i	$0.624609\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.1529i	1.59640i	0.602391	+	0.798201i	$0.294215\pi$
$311$	28.1529i	1.59640i	−0.602391	+	0.798201i	$0.705785\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	61.8223i	3.48329i
$316$	0	0
$317$	16.5503	0.929556	0.464778	−	0.885427i	$-0.346134\pi$
$317$	16.5503	0.929556	0.464778	+	0.885427i	$0.346134\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−58.2718	−3.21263
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 43.8518i	− 2.39588i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	70.0373i	3.78166i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	12.7784i	0.678208i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 22.1887i	− 1.17107i	−0.810646	−	0.585537i	$-0.800884\pi$
$359$	− 22.1887i	− 1.17107i	0.810646	−	0.585537i	$-0.199116\pi$
$360$	0	0
$361$	18.0119	0.947995
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−7.10441	−0.369841
$370$	0	0
$371$	0	0
$372$	0	0
$373$	38.5818	1.99769	0.998844	−	0.0480672i	$-0.0153062\pi$
$373$	38.5818	1.99769	0.998844	+	0.0480672i	$0.0153062\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	33.4066i	1.71598i	0.513665	+	0.857991i	$0.328287\pi$
$379$	33.4066i	1.71598i	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.0935	−0.908086	−0.454043	−	0.890980i	$-0.650019\pi$
$397$	−18.0935	−0.908086	−0.454043	+	0.890980i	$0.650019\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	35.4420	1.76113
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	59.9695	2.95091
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	40.8697i	1.99661i	0.0581712	+	0.998307i	$0.481473\pi$
$419$	40.8697i	1.99661i	−0.0581712	+	0.998307i	$0.518527\pi$
$420$	0	0
$421$	10.2175	0.497969	0.248985	−	0.968507i	$-0.419903\pi$
$421$	10.2175	0.497969	0.248985	+	0.968507i	$0.419903\pi$
$422$	0	0
$423$	33.4066i	1.62428i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 11.1355i	− 0.536380i	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	− 11.1355i	− 0.536380i	0.963366	−	0.268190i	$-0.0864254\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	19.6750i	0.939037i	0.882923	+	0.469519i	$0.155572\pi$
$439$	19.6750i	0.939037i	−0.882923	+	0.469519i	$0.844428\pi$
$440$	0	0
$441$	61.1516	2.91198
$442$	0	0
$443$	− 5.49576i	− 0.261111i	−0.991441	−	0.130556i	$-0.958324\pi$
$443$	− 5.49576i	− 0.261111i	0.991441	−	0.130556i	$-0.0416761\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	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		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.9617i	0.738619i	0.929307	+	0.369309i	$0.120406\pi$
$467$	15.9617i	0.738619i	−0.929307	+	0.369309i	$0.879594\pi$
$468$	0	0
$469$	−58.2718	−2.69074
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 10.4452i	− 0.479257i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.6546i	1.49203i	0.665931	+	0.746013i	$0.268034\pi$
$479$	32.6546i	1.49203i	−0.665931	+	0.746013i	$0.731966\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−40.3415	−1.83181
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.9804	0.761676
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 38.6188i	− 1.72193i	−0.508667	−	0.860963i	$-0.669862\pi$
$503$	− 38.6188i	− 1.72193i	0.508667	−	0.860963i	$-0.330138\pi$
$504$	0	0
$505$	27.8722	1.24029
$506$	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
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 53.9932i	− 2.37923i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 34.3799i	− 1.49196i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	78.5152i	3.39451i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−46.4578	−1.99738	−0.998688	−	0.0512107i	$-0.983692\pi$
$541$	−46.4578	−1.99738	−0.998688	+	0.0512107i	$0.983692\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−58.8879	−2.52248
$546$	0	0
$547$	− 26.4276i	− 1.12996i	−0.825104	−	0.564981i	$-0.808883\pi$
$547$	− 26.4276i	− 1.12996i	0.825104	−	0.564981i	$-0.191117\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.4361i	0.650553i	0.945619	+	0.325277i	$0.105457\pi$
$563$	15.4361i	0.650553i	−0.945619	+	0.325277i	$0.894543\pi$
$564$	0	0
$565$	83.7215	3.52219
$566$	0	0
$567$	− 47.0966i	− 1.97787i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−5.53454	−0.228047
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−48.0276	−1.97226	−0.986129	−	0.165978i	$-0.946922\pi$
$593$	−48.0276	−1.97226	−0.986129	+	0.165978i	$0.946922\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 30.1409i	− 1.23153i	−0.787932	−	0.615763i	$-0.788848\pi$
$599$	− 30.1409i	− 1.23153i	0.787932	−	0.615763i	$-0.211152\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	33.4066i	1.36042i
$604$	0	0
$605$	−43.3180	−1.76113
$606$	0	0
$607$	− 11.1355i	− 0.451977i	−0.974130	−	0.225989i	$-0.927439\pi$
$607$	− 11.1355i	− 0.451977i	0.974130	−	0.225989i	$-0.0725612\pi$
$608$	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
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	32.8760	1.31504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 9.73469i	− 0.385098i
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	− 43.8518i	− 1.71343i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 51.3356i	− 1.99975i	−0.0158077	−	0.999875i	$-0.505032\pi$
$659$	− 51.3356i	− 1.99975i	0.0158077	−	0.999875i	$-0.494968\pi$
$660$	0	0
$661$	32.3023	1.25641	0.628207	−	0.778046i	$-0.283789\pi$
$661$	32.3023	1.25641	0.628207	+	0.778046i	$0.283789\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20.4844	−0.794353
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	53.6072i	2.05726i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.8935i	1.41169i	0.708366	+	0.705846i	$0.249433\pi$
$683$	36.8935i	1.41169i	−0.708366	+	0.705846i	$0.750567\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	49.3475i	1.87727i	0.344915	+	0.938634i	$0.387908\pi$
$691$	49.3475i	1.87727i	−0.344915	+	0.938634i	$0.612092\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−49.5975	−1.87327	−0.936636	−	0.350304i	$-0.886078\pi$
$701$	−49.5975	−1.87327	−0.936636	+	0.350304i	$0.886078\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 37.0375i	− 1.39294i
$708$	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
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−71.7482	−2.67204
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.1972i	0.415280i	0.978205	+	0.207640i	$0.0665782\pi$
$727$	11.1972i	0.415280i	−0.978205	+	0.207640i	$0.933422\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.4105	0.495330	0.247665	−	0.968846i	$-0.420337\pi$
$733$	13.4105	0.495330	0.247665	+	0.968846i	$0.420337\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	39.3800	1.44277
$746$	0	0
$747$	0	0
$748$	0	0
$749$	104.334	3.81227
$750$	0	0
$751$	49.0847i	1.79113i	0.444934	+	0.895563i	$0.353227\pi$
$751$	49.0847i	1.79113i	−0.444934	+	0.895563i	$0.646773\pi$
$752$	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			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	78.2524i	2.83293i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	52.7639	1.90272	0.951358	−	0.308089i	$-0.0996893\pi$
$769$	52.7639	1.90272	0.951358	+	0.308089i	$0.0996893\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 58.5054i	− 2.10157i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.35400i	− 0.0843409i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	89.9036	3.20880
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 111.252i	− 3.95567i
$792$	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
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 55.6776i	− 1.95511i	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	− 55.6776i	− 1.95511i	0.210688	−	0.977553i	$-0.432429\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	29.4713i	1.03234i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 55.6776i	− 1.92221i	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	− 55.6776i	− 1.92221i	0.276191	−	0.961103i	$-0.410928\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	51.1940	1.76113
$846$	0	0
$847$	57.5626i	1.97787i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	11.7435i	0.401620i
$856$	0	0
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−55.1320	−1.87455
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	30.7324	1.04014
$874$	0	0
$875$	− 113.503i	− 3.83710i
$876$	0	0
$877$	57.4735	1.94074	0.970371	−	0.241618i	$-0.0776781\pi$
$877$	57.4735	1.94074	0.970371	+	0.241618i	$0.0776781\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	53.5864i	1.79926i	0.436657	+	0.899628i	$0.356162\pi$
$887$	53.5864i	1.79926i	−0.436657	+	0.899628i	$0.643838\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.0691	−0.370413
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.3679i	1.20758i	0.797144	+	0.603789i	$0.206343\pi$
$907$	36.3679i	1.20758i	−0.797144	+	0.603789i	$0.793657\pi$
$908$	0	0
$909$	−21.2332	−0.704261
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−58.2718	−1.92430
$918$	0	0
$919$	− 55.6776i	− 1.83664i	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	− 55.6776i	− 1.83664i	0.395843	−	0.918318i	$-0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	41.1325i	1.35097i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	20.2622i	0.664068i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	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
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	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
$955$	95.2081i	3.08086i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 59.8134i	− 1.92746i
$964$	0	0
$965$	71.3572	2.29707
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 55.6776i	− 1.78678i	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	− 55.6776i	− 1.78678i	0.449281	−	0.893390i	$-0.351680\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.6242	1.58762	0.793809	−	0.608167i	$-0.208095\pi$
$977$	49.6242	1.58762	0.793809	+	0.608167i	$0.208095\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	44.8612	1.43231
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−62.2098	−1.97020	−0.985102	−	0.171972i	$-0.944986\pi$
$997$	−62.2098	−1.97020	−0.985102	+	0.171972i	$0.944986\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.h.f.1983.5		6
4.3	odd	2	inner	1984.2.h.f.1983.6		6
8.3	odd	2		124.2.d.c.123.2	yes	6
8.5	even	2		124.2.d.c.123.1	✓	6
24.5	odd	2		1116.2.g.f.991.6		6
24.11	even	2		1116.2.g.f.991.5		6
31.30	odd	2	CM	1984.2.h.f.1983.5		6
124.123	even	2	inner	1984.2.h.f.1983.6		6
248.61	odd	2		124.2.d.c.123.1	✓	6
248.123	even	2		124.2.d.c.123.2	yes	6
744.371	odd	2		1116.2.g.f.991.5		6
744.557	even	2		1116.2.g.f.991.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.d.c.123.1	✓	6	8.5	even	2
124.2.d.c.123.1	✓	6	248.61	odd	2
124.2.d.c.123.2	yes	6	8.3	odd	2
124.2.d.c.123.2	yes	6	248.123	even	2
1116.2.g.f.991.5		6	24.11	even	2
1116.2.g.f.991.5		6	744.371	odd	2
1116.2.g.f.991.6		6	24.5	odd	2
1116.2.g.f.991.6		6	744.557	even	2
1984.2.h.f.1983.5		6	1.1	even	1	trivial
1984.2.h.f.1983.5		6	31.30	odd	2	CM
1984.2.h.f.1983.6		6	4.3	odd	2	inner
1984.2.h.f.1983.6		6	124.123	even	2	inner

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:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.93800 q^{5} -5.23296i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+3.93800 q^{5} -5.23296i q^{7} -3.00000 q^{9} -0.994032i q^{19} +10.5079 q^{25} -5.56776i q^{31} -20.6074i q^{35} +2.36814 q^{41} -11.8140 q^{45} -11.1355i q^{47} -20.3839 q^{49} +11.4600i q^{59} +15.6989i q^{63} -11.1355i q^{67} +3.24490i q^{71} +9.00000 q^{81} -3.91450i q^{95} -10.2441 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{9}+O(q^{10}) \) 6 * q - 18 * q^9	\( 6 q - 18 q^{9} + 30 q^{25} - 42 q^{49} + 54 q^{81}+O(q^{100}) \) 6 * q - 18 * q^9 + 30 * q^25 - 42 * q^49 + 54 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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