LMFDB - Embedded newform 5408.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5408 = 2^{5} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5408.a (trivial)

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:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.267949 q^{5} -3.00000 q^{9} +5.92820 q^{17} -4.92820 q^{25} -1.53590 q^{29} +4.26795 q^{37} +4.66025 q^{41} +0.803848 q^{45} -7.00000 q^{49} -3.53590 q^{53} +15.3923 q^{61} -13.1962 q^{73} +9.00000 q^{81} -1.58846 q^{85} +16.0000 q^{89} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{5} - 6 q^{9} - 2 q^{17} + 4 q^{25} - 10 q^{29} + 12 q^{37} - 8 q^{41} + 12 q^{45} - 14 q^{49} - 14 q^{53} + 10 q^{61} - 16 q^{73} + 18 q^{81} + 28 q^{85} + 32 q^{89} + 16 q^{97}+O(q^{100})$ 2 * q - 4 * q^5 - 6 * q^9 - 2 * q^17 + 4 * q^25 - 10 * q^29 + 12 * q^37 - 8 * q^41 + 12 * q^45 - 14 * q^49 - 14 * q^53 + 10 * q^61 - 16 * q^73 + 18 * q^81 + 28 * q^85 + 32 * q^89 + 16 * q^97

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
$5$	−0.267949	−0.119831	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	−0.267949	−0.119831	−0.0599153	+	0.998203i	$0.519083\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	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
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.92820	1.43780	0.718900	−	0.695113i	$-0.244646\pi$
$17$	5.92820	1.43780	0.718900	+	0.695113i	$0.244646\pi$
$18$	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		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.53590	−0.285209	−0.142605	−	0.989780i	$-0.545548\pi$
$29$	−1.53590	−0.285209	−0.142605	+	0.989780i	$0.545548\pi$
$30$	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
$37$	4.26795	0.701647	0.350823	−	0.936442i	$-0.385902\pi$
$37$	4.26795	0.701647	0.350823	+	0.936442i	$0.385902\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.66025	0.727809	0.363905	−	0.931436i	$-0.381443\pi$
$41$	4.66025	0.727809	0.363905	+	0.931436i	$0.381443\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0.803848	0.119831
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.53590	−0.485693	−0.242846	−	0.970065i	$-0.578081\pi$
$53$	−3.53590	−0.485693	−0.242846	+	0.970065i	$0.578081\pi$
$54$	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
$61$	15.3923	1.97078	0.985391	−	0.170305i	$-0.0544754\pi$
$61$	15.3923	1.97078	0.985391	+	0.170305i	$0.0544754\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−13.1962	−1.54449	−0.772246	−	0.635323i	$-0.780867\pi$
$73$	−13.1962	−1.54449	−0.772246	+	0.635323i	$0.780867\pi$
$74$	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
$85$	−1.58846	−0.172292
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.3205	−1.82296	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−18.3205	−1.82296	−0.911479	+	0.411346i	$0.865059\pi$
$102$	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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.85641	0.644996	0.322498	−	0.946570i	$-0.395477\pi$
$113$	6.85641	0.644996	0.322498	+	0.946570i	$0.395477\pi$
$114$	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
$125$	2.66025	0.237940
$126$	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
$131$	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
$137$	23.0526	1.96951	0.984757	−	0.173939i	$-0.0556494\pi$
$137$	23.0526	1.96951	0.984757	+	0.173939i	$0.0556494\pi$
$138$	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
$145$	0.411543	0.0341768
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.1244	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$149$	22.1244	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−17.7846	−1.43780
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.3923	1.70729	0.853646	−	0.520854i	$-0.174386\pi$
$157$	21.3923	1.70729	0.853646	+	0.520854i	$0.174386\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	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
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.0000	1.97674	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	26.0000	1.97674	0.988372	+	0.152057i	$0.0485898\pi$
$174$	0	0
$175$	0	0
$176$	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
$181$	−8.32051	−0.618458	−0.309229	−	0.950988i	$-0.600071\pi$
$181$	−8.32051	−0.618458	−0.309229	+	0.950988i	$0.600071\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.14359	−0.0840787
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	24.1244	1.73651	0.868255	−	0.496119i	$-0.165242\pi$
$193$	24.1244	1.73651	0.868255	+	0.496119i	$0.165242\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	28.0000	1.99492	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	28.0000	1.99492	0.997459	+	0.0712470i	$0.0226979\pi$
$198$	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
$205$	−1.24871	−0.0872138
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	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
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	14.7846	0.985641
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	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
$241$	21.9808	1.41591	0.707953	−	0.706260i	$-0.249619\pi$
$241$	21.9808	1.41591	0.707953	+	0.706260i	$0.249619\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.87564	0.119831
$246$	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
$257$	−26.7128	−1.66630	−0.833150	−	0.553047i	$-0.813465\pi$
$257$	−26.7128	−1.66630	−0.833150	+	0.553047i	$0.813465\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.60770	0.285209
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0.947441	0.0582008
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	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
$277$	15.2487	0.916206	0.458103	−	0.888899i	$-0.348529\pi$
$277$	15.2487	0.916206	0.458103	+	0.888899i	$0.348529\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.6603	−1.47111	−0.735554	−	0.677466i	$-0.763078\pi$
$281$	−24.6603	−1.47111	−0.735554	+	0.677466i	$0.763078\pi$
$282$	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
$289$	18.1436	1.06727
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−31.4449	−1.83703	−0.918514	−	0.395388i	$-0.870610\pi$
$293$	−31.4449	−1.83703	−0.918514	+	0.395388i	$0.870610\pi$
$294$	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
$305$	−4.12436	−0.236160
$306$	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
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.05256	0.283780	0.141890	−	0.989882i	$-0.454682\pi$
$317$	5.05256	0.283780	0.141890	+	0.989882i	$0.454682\pi$
$318$	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
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−12.8038	−0.701647
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−36.7128	−1.99987	−0.999937	−	0.0112091i	$-0.996432\pi$
$337$	−36.7128	−1.99987	−0.999937	+	0.0112091i	$0.996432\pi$
$338$	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		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	36.0000	1.92704	0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000	1.92704	0.963518	+	0.267644i	$0.0862451\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.4449	−1.14139	−0.570697	−	0.821160i	$-0.693327\pi$
$353$	−21.4449	−1.14139	−0.570697	+	0.821160i	$0.693327\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.53590	0.185077
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−13.9808	−0.727809
$370$	0	0
$371$	0	0
$372$	0	0
$373$	38.1769	1.97673	0.988363	−	0.152115i	$-0.0486083\pi$
$373$	38.1769	1.97673	0.988363	+	0.152115i	$0.0486083\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	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
$389$	34.3205	1.74012	0.870059	−	0.492947i	$-0.164080\pi$
$389$	34.3205	1.74012	0.870059	+	0.492947i	$0.164080\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.0000	−0.602263	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000	−0.602263	−0.301131	+	0.953583i	$0.597364\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.7321	1.08525	0.542623	−	0.839976i	$-0.317431\pi$
$401$	21.7321	1.08525	0.542623	+	0.839976i	$0.317431\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.41154	−0.119831
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.1962	1.24587	0.622935	−	0.782274i	$-0.285940\pi$
$409$	25.1962	1.24587	0.622935	+	0.782274i	$0.285940\pi$
$410$	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
$421$	39.9808	1.94854	0.974272	−	0.225377i	$-0.0723615\pi$
$421$	39.9808	1.94854	0.974272	+	0.225377i	$0.0723615\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−29.2154	−1.41715
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−3.78461	−0.181877	−0.0909384	−	0.995857i	$-0.528987\pi$
$433$	−3.78461	−0.181877	−0.0909384	+	0.995857i	$0.528987\pi$
$434$	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
$441$	21.0000	1.00000
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−4.28719	−0.203232
$446$	0	0
$447$	0	0
$448$	0	0
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	40.3731	1.88857	0.944286	−	0.329125i	$-0.106754\pi$
$457$	40.3731	1.88857	0.944286	+	0.329125i	$0.106754\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.9090	1.06698	0.533488	−	0.845807i	$-0.320881\pi$
$461$	22.9090	1.06698	0.533488	+	0.845807i	$0.320881\pi$
$462$	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
$467$	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
$477$	10.6077	0.485693
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.14359	−0.0973356
$486$	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
$493$	−9.10512	−0.410074
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	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
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	4.90897	0.218446
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−30.6603	−1.35899	−0.679496	−	0.733679i	$-0.737801\pi$
$509$	−30.6603	−1.35899	−0.679496	+	0.733679i	$0.737801\pi$
$510$	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
$521$	23.6410	1.03573	0.517866	−	0.855462i	$-0.326727\pi$
$521$	23.6410	1.03573	0.517866	+	0.855462i	$0.326727\pi$
$522$	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
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	46.3731	1.99373	0.996867	−	0.0790969i	$-0.0252036\pi$
$541$	46.3731	1.99373	0.996867	+	0.0790969i	$0.0252036\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.35898	−0.229554
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−46.1769	−1.97078
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−46.9090	−1.98760	−0.993798	−	0.111198i	$-0.964531\pi$
$557$	−46.9090	−1.98760	−0.993798	+	0.111198i	$0.964531\pi$
$558$	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
$565$	−1.83717	−0.0772903
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	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
$577$	−25.7321	−1.07124	−0.535620	−	0.844459i	$-0.679922\pi$
$577$	−25.7321	−1.07124	−0.535620	+	0.844459i	$0.679922\pi$
$578$	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
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−31.8372	−1.30740	−0.653698	−	0.756756i	$-0.726783\pi$
$593$	−31.8372	−1.30740	−0.653698	+	0.756756i	$0.726783\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	46.5692	1.89960	0.949799	−	0.312861i	$-0.101287\pi$
$601$	46.5692	1.89960	0.949799	+	0.312861i	$0.101287\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.94744	0.119831
$606$	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
$613$	47.4449	1.91628	0.958140	−	0.286300i	$-0.0924254\pi$
$613$	47.4449	1.91628	0.958140	+	0.286300i	$0.0924254\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.9090	−1.96900	−0.984500	−	0.175382i	$-0.943884\pi$
$617$	−48.9090	−1.96900	−0.984500	+	0.175382i	$0.943884\pi$
$618$	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
$625$	23.9282	0.957128
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.3013	1.00883
$630$	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
$639$	0	0
$640$	0	0
$641$	−31.9282	−1.26109	−0.630544	−	0.776153i	$-0.717168\pi$
$641$	−31.9282	−1.26109	−0.630544	+	0.776153i	$0.717168\pi$
$642$	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
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	39.5885	1.54449
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−49.3013	−1.91760	−0.958799	−	0.284087i	$-0.908310\pi$
$661$	−49.3013	−1.91760	−0.958799	+	0.284087i	$0.908310\pi$
$662$	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
$673$	43.7846	1.68777	0.843886	−	0.536522i	$-0.180262\pi$
$673$	43.7846	1.68777	0.843886	+	0.536522i	$0.180262\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−6.17691	−0.236008
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	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
$697$	27.6269	1.04644
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−47.9808	−1.80196	−0.900978	−	0.433865i	$-0.857149\pi$
$709$	−47.9808	−1.80196	−0.900978	+	0.433865i	$0.857149\pi$
$710$	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
$725$	7.56922	0.281114
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−44.7654	−1.65345	−0.826723	−	0.562609i	$-0.809798\pi$
$733$	−44.7654	−1.65345	−0.826723	+	0.562609i	$0.809798\pi$
$734$	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
$745$	−5.92820	−0.217193
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−40.0000	−1.45000	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000	−1.45000	−0.724999	+	0.688749i	$0.758160\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.76537	0.172292
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−44.0000	−1.58257	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−44.0000	−1.58257	−0.791285	+	0.611448i	$0.790588\pi$
$774$	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
$785$	−5.73205	−0.204586
$786$	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
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−48.0000	−1.69600
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−43.4974	−1.52929	−0.764644	−	0.644453i	$-0.777085\pi$
$809$	−43.4974	−1.52929	−0.764644	+	0.644453i	$0.777085\pi$
$810$	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
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	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
$829$	44.3205	1.53931	0.769657	−	0.638457i	$-0.220427\pi$
$829$	44.3205	1.53931	0.769657	+	0.638457i	$0.220427\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−41.4974	−1.43780
$834$	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
$841$	−26.6410	−0.918656
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	57.8372	1.98031	0.990153	−	0.139986i	$-0.0447058\pi$
$853$	57.8372	1.98031	0.990153	+	0.139986i	$0.0447058\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.9282	1.22728	0.613642	−	0.789584i	$-0.289704\pi$
$857$	35.9282	1.22728	0.613642	+	0.789584i	$0.289704\pi$
$858$	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
$865$	−6.96668	−0.236874
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−24.0000	−0.812277
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.2295	1.49352	0.746762	−	0.665092i	$-0.231608\pi$
$877$	44.2295	1.49352	0.746762	+	0.665092i	$0.231608\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	52.7128	1.77594	0.887970	−	0.459902i	$-0.152115\pi$
$881$	52.7128	1.77594	0.887970	+	0.459902i	$0.152115\pi$
$882$	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
$887$	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
$901$	−20.9615	−0.698330
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.22947	0.0741102
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	54.9615	1.82296
$910$	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
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−21.0333	−0.691571
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.8372	0.650836	0.325418	−	0.945570i	$-0.394495\pi$
$929$	19.8372	0.650836	0.325418	+	0.945570i	$0.394495\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.5692	−0.737304	−0.368652	−	0.929567i	$-0.620181\pi$
$937$	−22.5692	−0.737304	−0.368652	+	0.929567i	$0.620181\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.0000	−0.651981	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000	−0.651981	−0.325991	+	0.945373i	$0.605698\pi$
$942$	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
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.46410	−0.208087
$966$	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
$971$	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
$977$	−49.6936	−1.58984	−0.794919	−	0.606715i	$-0.792487\pi$
$977$	−49.6936	−1.58984	−0.794919	+	0.606715i	$0.792487\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−60.0000	−1.91565
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−7.50258	−0.239052
$986$	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
$997$	−41.3923	−1.31091	−0.655454	−	0.755235i	$-0.727523\pi$
$997$	−41.3923	−1.31091	−0.655454	+	0.755235i	$0.727523\pi$
$998$	0	0
$999$	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	5408.2.a.r.1.2		2
4.3	odd	2	CM	5408.2.a.r.1.2		2
13.2	odd	12		416.2.w.b.225.2	✓	4
13.7	odd	12		416.2.w.b.257.1	yes	4
13.12	even	2		5408.2.a.bc.1.1		2
52.7	even	12		416.2.w.b.257.1	yes	4
52.15	even	12		416.2.w.b.225.2	✓	4
52.51	odd	2		5408.2.a.bc.1.1		2
104.59	even	12		832.2.w.f.257.2		4
104.67	even	12		832.2.w.f.641.1		4
104.85	odd	12		832.2.w.f.257.2		4
104.93	odd	12		832.2.w.f.641.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.w.b.225.2	✓	4	13.2	odd	12
416.2.w.b.225.2	✓	4	52.15	even	12
416.2.w.b.257.1	yes	4	13.7	odd	12
416.2.w.b.257.1	yes	4	52.7	even	12
832.2.w.f.257.2		4	104.59	even	12
832.2.w.f.257.2		4	104.85	odd	12
832.2.w.f.641.1		4	104.67	even	12
832.2.w.f.641.1		4	104.93	odd	12
5408.2.a.r.1.2		2	1.1	even	1	trivial
5408.2.a.r.1.2		2	4.3	odd	2	CM
5408.2.a.bc.1.1		2	13.12	even	2
5408.2.a.bc.1.1		2	52.51	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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	5408.2.a.r.1.2

Level	$5408$
Weight	$2$
Character	5408.1
Self dual	yes
Analytic conductor	$43.183$
Analytic rank	$0$
Dimension	$2$
CM discriminant	-4
Inner twists	$2$