LMFDB - Embedded newform 784.4.f.e.783.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (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:	no
Analytic conductor:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			783.3
Root			$1.84776i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.e.783.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.72352i q^{5} -27.0000 q^{9} +O(q^{10})$	$q+4.72352i q^{5} -27.0000 q^{9} -18.5770i q^{13} -47.0456i q^{17} +102.688 q^{25} +108.894 q^{29} +128.693 q^{37} +348.054i q^{41} -127.535i q^{45} +572.000 q^{53} +945.916i q^{61} +87.7490 q^{65} -967.123i q^{73} +729.000 q^{81} +222.221 q^{85} +1475.53i q^{89} +1450.45i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} - 500 q^{25} + 2288 q^{53} + 4192 q^{65} + 2916 q^{81} - 5328 q^{85}+O(q^{100}) $ 4 * q - 108 * q^9 - 500 * q^25 + 2288 * q^53 + 4192 * q^65 + 2916 * q^81 - 5328 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	4.72352i	0.422484i	0.977434	+	0.211242i	$0.0677509\pi$
$5$	4.72352i	0.422484i	−0.977434	+	0.211242i	$0.932249\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−27.0000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 18.5770i	− 0.396334i	−0.980168	−	0.198167i	$-0.936501\pi$
$13$	− 18.5770i	− 0.396334i	0.980168	−	0.198167i	$-0.0634989\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 47.0456i	− 0.671190i	−0.942006	−	0.335595i	$-0.891063\pi$
$17$	− 47.0456i	− 0.671190i	0.942006	−	0.335595i	$-0.108937\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	102.688	0.821507
$26$	0	0
$27$	0	0
$28$	0	0
$29$	108.894	0.697282	0.348641	−	0.937256i	$-0.386643\pi$
$29$	108.894	0.697282	0.348641	+	0.937256i	$0.386643\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	128.693	0.571813	0.285906	−	0.958258i	$-0.407705\pi$
$37$	128.693	0.571813	0.285906	+	0.958258i	$0.407705\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	348.054i	1.32578i	0.748718	+	0.662889i	$0.230670\pi$
$41$	348.054i	1.32578i	−0.748718	+	0.662889i	$0.769330\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 127.535i	− 0.422484i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	572.000	1.48246	0.741229	−	0.671253i	$-0.234243\pi$
$53$	572.000	1.48246	0.741229	+	0.671253i	$0.234243\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	945.916i	1.98544i	0.120426	+	0.992722i	$0.461574\pi$
$61$	945.916i	1.98544i	−0.120426	+	0.992722i	$0.538426\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	87.7490	0.167445
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 967.123i	− 1.55059i	−0.631598	−	0.775296i	$-0.717601\pi$
$73$	− 967.123i	− 1.55059i	0.631598	−	0.775296i	$-0.282399\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	222.221	0.283567
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1475.53i	1.75737i	0.477406	+	0.878683i	$0.341577\pi$
$89$	1475.53i	1.75737i	−0.477406	+	0.878683i	$0.658423\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1450.45i	1.51826i	0.650940	+	0.759129i	$0.274375\pi$
$97$	1450.45i	1.51826i	−0.650940	+	0.759129i	$0.725625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1294.89i	− 1.27570i	−0.770160	−	0.637851i	$-0.779823\pi$
$101$	− 1294.89i	− 1.27570i	0.770160	−	0.637851i	$-0.220177\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	2266.98	1.99209	0.996045	−	0.0888549i	$-0.0283207\pi$
$109$	2266.98	1.99209	0.996045	+	0.0888549i	$0.0283207\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1328.00	1.10556	0.552778	−	0.833329i	$-0.313568\pi$
$113$	1328.00	1.10556	0.552778	+	0.833329i	$0.313568\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	501.580i	0.396334i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1331.00	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1075.49i	0.769558i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3098.54	1.93231	0.966154	−	0.257965i	$-0.0830518\pi$
$137$	3098.54	1.93231	0.966154	+	0.257965i	$0.0830518\pi$
$138$	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
$145$	514.365i	0.294591i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−940.000	−0.516831	−0.258415	−	0.966034i	$-0.583200\pi$
$149$	−940.000	−0.516831	−0.258415	+	0.966034i	$0.583200\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	1270.23i	0.671190i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3734.75i	− 1.89851i	−0.314512	−	0.949253i	$-0.601841\pi$
$157$	− 3734.75i	− 1.89851i	0.314512	−	0.949253i	$-0.398159\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1851.89	0.842919
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3420.96i	1.50341i	0.659497	+	0.751707i	$0.270769\pi$
$173$	3420.96i	1.50341i	−0.659497	+	0.751707i	$0.729231\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 1133.76i	− 0.465589i	−0.972526	−	0.232794i	$-0.925213\pi$
$181$	− 1133.76i	− 0.465589i	0.972526	−	0.232794i	$-0.0747869\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	607.886i	0.241582i
$186$	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
$193$	−72.0000	−0.0268532	−0.0134266	−	0.999910i	$-0.504274\pi$
$193$	−72.0000	−0.0268532	−0.0134266	+	0.999910i	$0.504274\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1174.00	−0.424589	−0.212295	−	0.977206i	$-0.568094\pi$
$197$	−1174.00	−0.424589	−0.212295	+	0.977206i	$0.568094\pi$
$198$	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
$205$	−1644.04	−0.560120
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−873.968	−0.266016
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−2772.59	−0.821507
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	34.3455i	0.00991099i	0.999988	+	0.00495549i	$0.00157739\pi$
$229$	34.3455i	0.00991099i	−0.999988	+	0.00495549i	$0.998423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5434.82	1.52810	0.764050	−	0.645158i	$-0.223208\pi$
$233$	5434.82	1.52810	0.764050	+	0.645158i	$0.223208\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	2888.29i	0.771997i	0.922499	+	0.385999i	$0.126143\pi$
$241$	2888.29i	0.771997i	−0.922499	+	0.385999i	$0.873857\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	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
$257$	− 6892.69i	− 1.67297i	−0.547987	−	0.836487i	$-0.684606\pi$
$257$	− 6892.69i	− 1.67297i	0.547987	−	0.836487i	$-0.315394\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2940.15	−0.697282
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	2701.85i	0.626315i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6261.78i	1.41928i	0.704563	+	0.709642i	$0.251143\pi$
$269$	6261.78i	1.41928i	−0.704563	+	0.709642i	$0.748857\pi$
$270$	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
$277$	−9126.00	−1.97952	−0.989762	−	0.142727i	$-0.954413\pi$
$277$	−9126.00	−1.97952	−0.989762	+	0.142727i	$0.954413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1158.24	0.245889	0.122945	−	0.992414i	$-0.460766\pi$
$281$	1158.24	0.245889	0.122945	+	0.992414i	$0.460766\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2699.71	0.549504
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 10022.1i	− 1.99829i	−0.0413610	−	0.999144i	$-0.513169\pi$
$293$	− 10022.1i	− 1.99829i	0.0413610	−	0.999144i	$-0.486831\pi$
$294$	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
$304$	0	0
$305$	−4468.05	−0.838819
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 10665.6i	− 1.92606i	−0.269393	−	0.963030i	$-0.586823\pi$
$313$	− 10665.6i	− 1.92606i	0.269393	−	0.963030i	$-0.413177\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10274.0	−1.82033	−0.910166	−	0.414243i	$-0.864046\pi$
$317$	−10274.0	−1.82033	−0.910166	+	0.414243i	$0.864046\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 1907.65i	− 0.325591i
$326$	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			$\pi$
$332$	0	0
$333$	−3474.72	−0.571813
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9038.24	−1.46096	−0.730481	−	0.682933i	$-0.760704\pi$
$337$	−9038.24	−1.46096	−0.730481	+	0.682933i	$0.760704\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	− 11905.7i	− 1.82606i	−0.407890	−	0.913031i	$-0.633735\pi$
$349$	− 11905.7i	− 1.82606i	0.407890	−	0.913031i	$-0.366265\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13132.1i	1.98003i	0.140959	+	0.990015i	$0.454981\pi$
$353$	13132.1i	1.98003i	−0.140959	+	0.990015i	$0.545019\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4568.22	0.655101
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	− 9397.46i	− 1.32578i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6372.00	−0.884530	−0.442265	−	0.896884i	$-0.645825\pi$
$373$	−6372.00	−0.884530	−0.442265	+	0.896884i	$0.645825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2022.94i	− 0.276357i
$378$	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		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10582.6	−1.37932	−0.689662	−	0.724131i	$-0.742241\pi$
$389$	−10582.6	−1.37932	−0.689662	+	0.724131i	$0.742241\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4074.14i	− 0.515051i	−0.966271	−	0.257526i	$-0.917093\pi$
$397$	− 4074.14i	− 0.515051i	0.966271	−	0.257526i	$-0.0829072\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9533.21	1.18720	0.593598	−	0.804761i	$-0.297707\pi$
$401$	9533.21	1.18720	0.593598	+	0.804761i	$0.297707\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3443.44i	0.422484i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	11049.6i	1.33587i	0.744222	+	0.667933i	$0.232821\pi$
$409$	11049.6i	1.33587i	−0.744222	+	0.667933i	$0.767179\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−10890.0	−1.26068	−0.630340	−	0.776319i	$-0.717084\pi$
$421$	−10890.0	−1.26068	−0.630340	+	0.776319i	$0.717084\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4831.04i	− 0.551388i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	11132.2i	1.23552i	0.786366	+	0.617761i	$0.211960\pi$
$433$	11132.2i	1.23552i	−0.786366	+	0.617761i	$0.788040\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−6969.68	−0.742459
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10120.0	1.06368	0.531840	−	0.846845i	$-0.321501\pi$
$449$	10120.0	1.06368	0.531840	+	0.846845i	$0.321501\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10456.0	1.07026	0.535132	−	0.844768i	$-0.320262\pi$
$457$	10456.0	1.07026	0.535132	+	0.844768i	$0.320262\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 17276.4i	− 1.74543i	−0.488231	−	0.872714i	$-0.662358\pi$
$461$	− 17276.4i	− 1.74543i	0.488231	−	0.872714i	$-0.337642\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15444.0	−1.48246
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	− 2390.74i	− 0.226629i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6851.23	−0.641440
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 5123.00i	− 0.468009i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	6116.42	0.538964
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20070.5i	1.74776i	0.486140	+	0.873881i	$0.338404\pi$
$509$	20070.5i	1.74776i	−0.486140	+	0.873881i	$0.661596\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10451.5i	0.878863i	0.898276	+	0.439432i	$0.144820\pi$
$521$	10451.5i	0.878863i	−0.898276	+	0.439432i	$0.855180\pi$
$522$	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
$529$	12167.0	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6465.81	0.525451
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−24460.0	−1.94384	−0.971920	−	0.235311i	$-0.924389\pi$
$541$	−24460.0	−1.94384	−0.971920	+	0.235311i	$0.924389\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10708.1i	0.841626i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	− 25539.7i	− 1.98544i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8626.00	0.656186	0.328093	−	0.944646i	$-0.393594\pi$
$557$	8626.00	0.656186	0.328093	+	0.944646i	$0.393594\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	6272.83i	0.467080i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15928.3	1.17355	0.586774	−	0.809751i	$-0.300398\pi$
$569$	15928.3	1.17355	0.586774	+	0.809751i	$0.300398\pi$
$570$	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
$577$	− 13716.4i	− 0.989638i	−0.868996	−	0.494819i	$-0.835234\pi$
$577$	− 13716.4i	− 0.989638i	0.868996	−	0.494819i	$-0.164766\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2369.22	−0.167445
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4996.75i	0.346024i	0.984920	+	0.173012i	$0.0553499\pi$
$593$	4996.75i	0.346024i	−0.984920	+	0.173012i	$0.944650\pi$
$594$	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
$601$	15700.4i	1.06561i	0.846238	+	0.532804i	$0.178862\pi$
$601$	15700.4i	1.06561i	−0.846238	+	0.532804i	$0.821138\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6287.00i	0.422484i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	30243.0	1.99266	0.996331	−	0.0855861i	$-0.0272763\pi$
$613$	30243.0	1.99266	0.996331	+	0.0855861i	$0.0272763\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29649.0	1.93456	0.967280	−	0.253712i	$-0.0816515\pi$
$617$	29649.0	1.93456	0.967280	+	0.253712i	$0.0816515\pi$
$618$	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
$625$	7755.95	0.496381
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 6054.46i	− 0.383795i
$630$	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
$641$	30916.1	1.90501	0.952507	−	0.304518i	$-0.0984954\pi$
$641$	30916.1	1.90501	0.952507	+	0.304518i	$0.0984954\pi$
$642$	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
$653$	24303.3	1.45645	0.728224	−	0.685340i	$-0.240346\pi$
$653$	24303.3	1.45645	0.728224	+	0.685340i	$0.240346\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	26112.3i	1.55059i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 30280.5i	− 1.78181i	−0.454190	−	0.890905i	$-0.650071\pi$
$661$	− 30280.5i	− 1.78181i	0.454190	−	0.890905i	$-0.349929\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−21333.4	−1.22191	−0.610953	−	0.791667i	$-0.709214\pi$
$673$	−21333.4	−1.22191	−0.610953	+	0.791667i	$0.709214\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33883.5i	1.92356i	0.273828	+	0.961779i	$0.411710\pi$
$677$	33883.5i	1.92356i	−0.273828	+	0.961779i	$0.588290\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	14636.0i	0.816370i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 10626.1i	− 0.587549i
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16374.4	0.889849
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36261.8	1.95377	0.976884	−	0.213771i	$-0.0685745\pi$
$701$	36261.8	1.95377	0.976884	+	0.213771i	$0.0685745\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20086.1	−1.06396	−0.531981	−	0.846757i	$-0.678552\pi$
$709$	−20086.1	−1.06396	−0.531981	+	0.846757i	$0.678552\pi$
$710$	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
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11182.2	0.572822
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−19683.0	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	22884.1i	1.15313i	0.817052	+	0.576563i	$0.195607\pi$
$733$	22884.1i	1.15313i	−0.817052	+	0.576563i	$0.804393\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	− 4440.11i	− 0.218353i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8860.05	−0.425395	−0.212697	−	0.977118i	$-0.568225\pi$
$757$	−8860.05	−0.425395	−0.212697	+	0.977118i	$0.568225\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39910.0i	1.90110i	0.310570	+	0.950550i	$0.399480\pi$
$761$	39910.0i	1.90110i	−0.310570	+	0.950550i	$0.600520\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5999.96	−0.283567
$766$	0	0
$767$	0	0
$768$	0	0
$769$	24813.6i	1.16359i	0.813335	+	0.581796i	$0.197650\pi$
$769$	24813.6i	1.16359i	−0.813335	+	0.581796i	$0.802350\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	437.150i	0.0203405i	0.999948	+	0.0101702i	$0.00323734\pi$
$773$	437.150i	0.0203405i	−0.999948	+	0.0101702i	$0.996763\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17641.2	0.802089
$786$	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
$791$	0	0
$792$	0	0
$793$	17572.3	0.786900
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44989.0i	− 1.99949i	−0.0226228	−	0.999744i	$-0.507202\pi$
$797$	− 44989.0i	− 1.99949i	0.0226228	−	0.999744i	$-0.492798\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	− 39839.2i	− 1.75737i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23270.0	1.01129	0.505643	−	0.862743i	$-0.331255\pi$
$809$	23270.0	1.01129	0.505643	+	0.862743i	$0.331255\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1850.00	0.0786424	0.0393212	−	0.999227i	$-0.487480\pi$
$821$	1850.00	0.0786424	0.0393212	+	0.999227i	$0.487480\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	− 5429.39i	− 0.227467i	−0.993511	−	0.113734i	$-0.963719\pi$
$829$	− 5429.39i	− 0.227467i	0.993511	−	0.113734i	$-0.0362811\pi$
$830$	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
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−12531.0	−0.513797
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8747.45i	0.356120i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	49805.3i	1.99918i	0.0286392	+	0.999590i	$0.490883\pi$
$853$	49805.3i	1.99918i	−0.0286392	+	0.999590i	$0.509117\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34817.8i	1.38781i	0.720066	+	0.693905i	$0.244111\pi$
$857$	34817.8i	1.38781i	−0.720066	+	0.693905i	$0.755889\pi$
$858$	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	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−16159.0	−0.635169
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 39162.2i	− 1.51826i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8959.04	0.344955	0.172477	−	0.985013i	$-0.444823\pi$
$877$	8959.04	0.344955	0.172477	+	0.985013i	$0.444823\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 26431.8i	− 1.01079i	−0.862887	−	0.505397i	$-0.831346\pi$
$881$	− 26431.8i	− 1.01079i	0.862887	−	0.505397i	$-0.168654\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	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
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 26910.1i	− 0.995011i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5355.32	0.196704
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	34961.9i	1.27570i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	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
$925$	13215.3	0.469748
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 10346.7i	− 0.365407i	−0.983168	−	0.182703i	$-0.941515\pi$
$929$	− 10346.7i	− 0.365407i	0.983168	−	0.182703i	$-0.0584848\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42362.4i	− 1.47697i	−0.674271	−	0.738484i	$-0.735542\pi$
$937$	− 42362.4i	− 1.47697i	0.674271	−	0.738484i	$-0.264458\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47534.3i	1.64673i	0.567511	+	0.823366i	$0.307907\pi$
$941$	47534.3i	1.64673i	−0.567511	+	0.823366i	$0.692093\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−17966.3	−0.614553
$950$	0	0
$951$	0	0
$952$	0	0
$953$	56758.0	1.92925	0.964623	−	0.263632i	$-0.0849205\pi$
$953$	56758.0	1.92925	0.964623	+	0.263632i	$0.0849205\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 340.093i	− 0.0113451i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23808.3	−0.779626	−0.389813	−	0.920894i	$-0.627460\pi$
$977$	−23808.3	−0.779626	−0.389813	+	0.920894i	$0.627460\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−61208.6	−1.99209
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	− 5545.41i	− 0.179382i
$986$	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
$997$	− 11324.8i	− 0.359740i	−0.983690	−	0.179870i	$-0.942432\pi$
$997$	− 11324.8i	− 0.359740i	0.983690	−	0.179870i	$-0.0575677\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.e.783.3	yes	4
4.3	odd	2	CM	784.4.f.e.783.3	yes	4
7.6	odd	2	inner	784.4.f.e.783.2	✓	4
28.27	even	2	inner	784.4.f.e.783.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.e.783.2	✓	4	7.6	odd	2	inner
784.4.f.e.783.2	✓	4	28.27	even	2	inner
784.4.f.e.783.3	yes	4	1.1	even	1	trivial
784.4.f.e.783.3	yes	4	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.72352i q^{5} -27.0000 q^{9} +O(q^{10})\)	\(q+4.72352i q^{5} -27.0000 q^{9} -18.5770i q^{13} -47.0456i q^{17} +102.688 q^{25} +108.894 q^{29} +128.693 q^{37} +348.054i q^{41} -127.535i q^{45} +572.000 q^{53} +945.916i q^{61} +87.7490 q^{65} -967.123i q^{73} +729.000 q^{81} +222.221 q^{85} +1475.53i q^{89} +1450.45i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} - 500 q^{25} + 2288 q^{53} + 4192 q^{65} + 2916 q^{81} - 5328 q^{85}+O(q^{100}) \) 4 * q - 108 * q^9 - 500 * q^25 + 2288 * q^53 + 4192 * q^65 + 2916 * q^81 - 5328 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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