LMFDB - Embedded newform 7569.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7569 = 3^{2} \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 4x^{2} + 4x + 1$ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.4
Root			$1.33826$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +4.63282 q^{7} -5.69033 q^{13} +4.00000 q^{16} +4.28135 q^{19} -5.00000 q^{25} -9.26564 q^{28} -8.64760 q^{31} +11.9534 q^{37} -13.0760 q^{43} +14.4630 q^{49} +11.3807 q^{52} -4.21917 q^{61} -8.00000 q^{64} -1.77267 q^{67} -16.2348 q^{73} -8.56271 q^{76} -3.39070 q^{79} -26.3623 q^{91} -17.5045 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 8 q^{4} + q^{7} - 2 q^{13} + 16 q^{16} - q^{19} - 20 q^{25} - 2 q^{28} - 4 q^{31} + 11 q^{37} + 5 q^{43} + 41 q^{49} + 4 q^{52} - 13 q^{61} - 32 q^{64} - 11 q^{67} - 10 q^{73} + 2 q^{76} - 13 q^{79}+ \cdots + 14 q^{97}+O(q^{100})$ 4 * q - 8 * q^4 + q^7 - 2 * q^13 + 16 * q^16 - q^19 - 20 * q^25 - 2 * q^28 - 4 * q^31 + 11 * q^37 + 5 * q^43 + 41 * q^49 + 4 * q^52 - 13 * q^61 - 32 * q^64 - 11 * q^67 - 10 * q^73 + 2 * q^76 - 13 * q^79 + 2 * q^91 + 14 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	4.63282	1.75104	0.875520	−	0.483181i	$-0.160519\pi$
$7$	4.63282	1.75104	0.875520	+	0.483181i	$0.160519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.69033	−1.57821	−0.789107	−	0.614256i	$-0.789456\pi$
$13$	−5.69033	−1.57821	−0.789107	+	0.614256i	$0.789456\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.28135	0.982210	0.491105	−	0.871100i	$-0.336593\pi$
$19$	4.28135	0.982210	0.491105	+	0.871100i	$0.336593\pi$
$20$	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
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−9.26564	−1.75104
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−8.64760	−1.55316	−0.776578	−	0.630022i	$-0.783046\pi$
$31$	−8.64760	−1.55316	−0.776578	+	0.630022i	$0.783046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.9534	1.96513	0.982564	−	0.185924i	$-0.0595278\pi$
$37$	11.9534	1.96513	0.982564	+	0.185924i	$0.0595278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−13.0760	−1.99408	−0.997038	−	0.0769089i	$-0.975495\pi$
$43$	−13.0760	−1.99408	−0.997038	+	0.0769089i	$0.975495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	14.4630	2.06614
$50$	0	0
$51$	0	0
$52$	11.3807	1.57821
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	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
$61$	−4.21917	−0.540209	−0.270105	−	0.962831i	$-0.587058\pi$
$61$	−4.21917	−0.540209	−0.270105	+	0.962831i	$0.587058\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−1.77267	−0.216566	−0.108283	−	0.994120i	$-0.534535\pi$
$67$	−1.77267	−0.216566	−0.108283	+	0.994120i	$0.534535\pi$
$68$	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
$73$	−16.2348	−1.90014	−0.950068	−	0.312044i	$-0.898986\pi$
$73$	−16.2348	−1.90014	−0.950068	+	0.312044i	$0.898986\pi$
$74$	0	0
$75$	0	0
$76$	−8.56271	−0.982210
$77$	0	0
$78$	0	0
$79$	−3.39070	−0.381484	−0.190742	−	0.981640i	$-0.561089\pi$
$79$	−3.39070	−0.381484	−0.190742	+	0.981640i	$0.561089\pi$
$80$	0	0
$81$	0	0
$82$	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		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−26.3623	−2.76352
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.5045	−1.77731	−0.888654	−	0.458577i	$-0.848359\pi$
$97$	−17.5045	−1.77731	−0.888654	+	0.458577i	$0.848359\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	2.88578	0.284345	0.142172	−	0.989842i	$-0.454591\pi$
$103$	2.88578	0.284345	0.142172	+	0.989842i	$0.454591\pi$
$104$	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
$109$	16.7842	1.60764	0.803819	−	0.594874i	$-0.202798\pi$
$109$	16.7842	1.60764	0.803819	+	0.594874i	$0.202798\pi$
$110$	0	0
$111$	0	0
$112$	18.5313	1.75104
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	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
$124$	17.2952	1.55316
$125$	0	0
$126$	0	0
$127$	21.7236	1.92766	0.963830	−	0.266516i	$-0.0858726\pi$
$127$	21.7236	1.92766	0.963830	+	0.266516i	$0.0858726\pi$
$128$	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
$133$	19.8347	1.71989
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−21.4171	−1.81657	−0.908285	−	0.418351i	$-0.862608\pi$
$139$	−21.4171	−1.81657	−0.908285	+	0.418351i	$0.862608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−23.9068	−1.96513
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	24.5340	1.99655	0.998274	−	0.0587314i	$-0.0187055\pi$
$151$	24.5340	1.99655	0.998274	+	0.0587314i	$0.0187055\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.890649	−0.0710815	−0.0355408	−	0.999368i	$-0.511315\pi$
$157$	−0.890649	−0.0710815	−0.0355408	+	0.999368i	$0.511315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.8668	1.00780	0.503902	−	0.863761i	$-0.331897\pi$
$163$	12.8668	1.00780	0.503902	+	0.863761i	$0.331897\pi$
$164$	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
$169$	19.3799	1.49076
$170$	0	0
$171$	0	0
$172$	26.1521	1.99408
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−23.1641	−1.75104
$176$	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
$181$	−9.60799	−0.714156	−0.357078	−	0.934075i	$-0.616227\pi$
$181$	−9.60799	−0.714156	−0.357078	+	0.934075i	$0.616227\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−21.9329	−1.57876	−0.789382	−	0.613902i	$-0.789599\pi$
$193$	−21.9329	−1.57876	−0.789382	+	0.613902i	$0.789599\pi$
$194$	0	0
$195$	0	0
$196$	−28.9260	−2.06614
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	20.6163	1.46145	0.730726	−	0.682670i	$-0.239182\pi$
$199$	20.6163	1.46145	0.730726	+	0.682670i	$0.239182\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−22.7613	−1.57821
$209$	0	0
$210$	0	0
$211$	−29.0000	−1.99644	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−29.0000	−1.99644	−0.998221	+	0.0596196i	$0.981011\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−40.0628	−2.71964
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.0127	−0.737463	−0.368731	−	0.929536i	$-0.620208\pi$
$223$	−11.0127	−0.737463	−0.368731	+	0.929536i	$0.620208\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−29.0000	−1.91637	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−29.0000	−1.91637	−0.958187	+	0.286143i	$0.907627\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	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
$241$	30.6827	1.97645	0.988223	−	0.153021i	$-0.0489003\pi$
$241$	30.6827	1.97645	0.988223	+	0.153021i	$0.0489003\pi$
$242$	0	0
$243$	0	0
$244$	8.43834	0.540209
$245$	0	0
$246$	0	0
$247$	−24.3623	−1.55014
$248$	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
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	55.3780	3.44102
$260$	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
$268$	3.54534	0.216566
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.2983	0.919187	0.459593	−	0.888129i	$-0.347995\pi$
$277$	15.2983	0.919187	0.459593	+	0.888129i	$0.347995\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−31.9970	−1.90202	−0.951012	−	0.309155i	$-0.899954\pi$
$283$	−31.9970	−1.90202	−0.951012	+	0.309155i	$0.899954\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	32.4695	1.90014
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−60.5789	−3.49171
$302$	0	0
$303$	0	0
$304$	17.1254	0.982210
$305$	0	0
$306$	0	0
$307$	27.2975	1.55795	0.778976	−	0.627054i	$-0.215739\pi$
$307$	27.2975	1.55795	0.778976	+	0.627054i	$0.215739\pi$
$308$	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
$313$	−35.3155	−1.99615	−0.998075	−	0.0620147i	$-0.980247\pi$
$313$	−35.3155	−1.99615	−0.998075	+	0.0620147i	$0.980247\pi$
$314$	0	0
$315$	0	0
$316$	6.78141	0.381484
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	28.4516	1.57821
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−26.3613	−1.44895	−0.724475	−	0.689301i	$-0.757918\pi$
$331$	−26.3613	−1.44895	−0.724475	+	0.689301i	$0.757918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	34.5747	1.86686
$344$	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
$349$	−35.9146	−1.92247	−0.961234	−	0.275736i	$-0.911079\pi$
$349$	−35.9146	−1.92247	−0.961234	+	0.275736i	$0.911079\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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
$361$	−0.670020	−0.0352642
$362$	0	0
$363$	0	0
$364$	52.7245	2.76352
$365$	0	0
$366$	0	0
$367$	4.00991	0.209316	0.104658	−	0.994508i	$-0.466625\pi$
$367$	4.00991	0.209316	0.104658	+	0.994508i	$0.466625\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.2783	1.04997	0.524985	−	0.851111i	$-0.324071\pi$
$373$	20.2783	1.04997	0.524985	+	0.851111i	$0.324071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	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
$388$	35.0089	1.77731
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.9357	1.45224	0.726119	−	0.687569i	$-0.241322\pi$
$397$	28.9357	1.45224	0.726119	+	0.687569i	$0.241322\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	49.2077	2.45121
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−29.0788	−1.43786	−0.718928	−	0.695085i	$-0.755367\pi$
$409$	−29.0788	−1.43786	−0.718928	+	0.695085i	$0.755367\pi$
$410$	0	0
$411$	0	0
$412$	−5.77157	−0.284345
$413$	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
$421$	−36.7509	−1.79113	−0.895564	−	0.444933i	$-0.853228\pi$
$421$	−36.7509	−1.79113	−0.895564	+	0.444933i	$0.853228\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19.5466	−0.945928
$428$	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
$433$	−18.7348	−0.900338	−0.450169	−	0.892943i	$-0.648636\pi$
$433$	−18.7348	−0.900338	−0.450169	+	0.892943i	$0.648636\pi$
$434$	0	0
$435$	0	0
$436$	−33.5685	−1.60764
$437$	0	0
$438$	0	0
$439$	−28.0793	−1.34015	−0.670077	−	0.742292i	$-0.733739\pi$
$439$	−28.0793	−1.34015	−0.670077	+	0.742292i	$0.733739\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−37.0625	−1.75104
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.13875	0.0532684	0.0266342	−	0.999645i	$-0.491521\pi$
$457$	1.13875	0.0532684	0.0266342	+	0.999645i	$0.491521\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	33.7697	1.56941	0.784705	−	0.619870i	$-0.212815\pi$
$463$	33.7697	1.56941	0.784705	+	0.619870i	$0.212815\pi$
$464$	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
$469$	−8.21245	−0.379216
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−21.4068	−0.982210
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−68.0188	−3.10139
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	41.6050	1.88530	0.942651	−	0.333781i	$-0.108325\pi$
$487$	41.6050	1.88530	0.942651	+	0.333781i	$0.108325\pi$
$488$	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
$496$	−34.5904	−1.55316
$497$	0	0
$498$	0	0
$499$	44.5811	1.99573	0.997863	−	0.0653408i	$-0.0208135\pi$
$499$	44.5811	1.99573	0.997863	+	0.0653408i	$0.0208135\pi$
$500$	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
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−43.4473	−1.92766
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−75.2127	−3.32721
$512$	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
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−11.0083	−0.481361	−0.240681	−	0.970604i	$-0.577371\pi$
$523$	−11.0083	−0.481361	−0.240681	+	0.970604i	$0.577371\pi$
$524$	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
$532$	−39.6695	−1.71989
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−26.6790	−1.14071	−0.570355	−	0.821398i	$-0.693195\pi$
$547$	−26.6790	−1.14071	−0.570355	+	0.821398i	$0.693195\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−15.7085	−0.667994
$554$	0	0
$555$	0	0
$556$	42.8341	1.81657
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	74.4069	3.14708
$560$	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
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−13.5257	−0.566031	−0.283015	−	0.959115i	$-0.591335\pi$
$571$	−13.5257	−0.566031	−0.283015	+	0.959115i	$0.591335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−47.8757	−1.99309	−0.996546	−	0.0830461i	$-0.973535\pi$
$577$	−47.8757	−1.99309	−0.996546	+	0.0830461i	$0.973535\pi$
$578$	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
$589$	−37.0234	−1.52552
$590$	0	0
$591$	0	0
$592$	47.8136	1.96513
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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
$601$	13.4946	0.550454	0.275227	−	0.961379i	$-0.411247\pi$
$601$	13.4946	0.550454	0.275227	+	0.961379i	$0.411247\pi$
$602$	0	0
$603$	0	0
$604$	−49.0680	−1.99655
$605$	0	0
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−49.2140	−1.98773	−0.993867	−	0.110586i	$-0.964727\pi$
$613$	−49.2140	−1.98773	−0.993867	+	0.110586i	$0.964727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−13.7347	−0.552044	−0.276022	−	0.961151i	$-0.589016\pi$
$619$	−13.7347	−0.552044	−0.276022	+	0.961151i	$0.589016\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	1.78130	0.0710815
$629$	0	0
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−82.2992	−3.26081
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	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
$652$	−25.7335	−1.00780
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	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
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	0	0
$676$	−38.7597	−1.49076
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−81.0950	−3.11214
$680$	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
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−52.3041	−1.99408
$689$	0	0
$690$	0	0
$691$	50.8406	1.93407	0.967035	−	0.254645i	$-0.0819585\pi$
$691$	50.8406	1.93407	0.967035	+	0.254645i	$0.0819585\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	46.3282	1.75104
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	51.1768	1.93017
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−0.372321	−0.0139828	−0.00699141	−	0.999976i	$-0.502225\pi$
$709$	−0.372321	−0.0139828	−0.00699141	+	0.999976i	$0.502225\pi$
$710$	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
$721$	13.3693	0.497899
$722$	0	0
$723$	0	0
$724$	19.2160	0.714156
$725$	0	0
$726$	0	0
$727$	−26.4069	−0.979377	−0.489688	−	0.871898i	$-0.662889\pi$
$727$	−26.4069	−0.979377	−0.489688	+	0.871898i	$0.662889\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−35.2182	−1.30081	−0.650407	−	0.759586i	$-0.725402\pi$
$733$	−35.2182	−1.30081	−0.650407	+	0.759586i	$0.725402\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.84695	−0.178298	−0.0891490	−	0.996018i	$-0.528415\pi$
$739$	−4.84695	−0.178298	−0.0891490	+	0.996018i	$0.528415\pi$
$740$	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
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	47.6664	1.73937	0.869687	−	0.493604i	$-0.164321\pi$
$751$	47.6664	1.73937	0.869687	+	0.493604i	$0.164321\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.0000	−1.05402	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000	−1.05402	−0.527011	+	0.849858i	$0.676688\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	77.7583	2.81504
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	52.0949	1.87859	0.939295	−	0.343112i	$-0.111481\pi$
$769$	52.0949	1.87859	0.939295	+	0.343112i	$0.111481\pi$
$770$	0	0
$771$	0	0
$772$	43.8658	1.57876
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	43.2380	1.55316
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	57.8520	2.06614
$785$	0	0
$786$	0	0
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.0085	0.852566
$794$	0	0
$795$	0	0
$796$	−41.2327	−1.46145
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	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		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	27.1886	0.954722	0.477361	−	0.878707i	$-0.341593\pi$
$811$	27.1886	0.954722	0.477361	+	0.878707i	$0.341593\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−55.9831	−1.95860
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	37.6415	1.31210	0.656051	−	0.754717i	$-0.272226\pi$
$823$	37.6415	1.31210	0.656051	+	0.754717i	$0.272226\pi$
$824$	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
$829$	56.5233	1.96314	0.981568	−	0.191115i	$-0.0612103\pi$
$829$	56.5233	1.96314	0.981568	+	0.191115i	$0.0612103\pi$
$830$	0	0
$831$	0	0
$832$	45.5226	1.57821
$833$	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
$841$	0	0
$842$	0	0
$843$	0	0
$844$	58.0000	1.99644
$845$	0	0
$846$	0	0
$847$	−50.9610	−1.75104
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	58.0000	1.98588	0.992941	−	0.118609i	$-0.0378434\pi$
$853$	58.0000	1.98588	0.992941	+	0.118609i	$0.0378434\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	55.4857	1.89315	0.946574	−	0.322488i	$-0.104519\pi$
$859$	55.4857	1.89315	0.946574	+	0.322488i	$0.104519\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	0	0
$868$	80.1255	2.71964
$869$	0	0
$870$	0	0
$871$	10.0871	0.341787
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.2902	0.448777	0.224389	−	0.974500i	$-0.427962\pi$
$877$	13.2902	0.448777	0.224389	+	0.974500i	$0.427962\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	58.4796	1.96800	0.983998	−	0.178182i	$-0.0570216\pi$
$883$	58.4796	1.96800	0.983998	+	0.178182i	$0.0570216\pi$
$884$	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
$889$	100.642	3.37541
$890$	0	0
$891$	0	0
$892$	22.0253	0.737463
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.89076	−0.195599	−0.0977997	−	0.995206i	$-0.531180\pi$
$907$	−5.89076	−0.195599	−0.0977997	+	0.995206i	$0.531180\pi$
$908$	0	0
$909$	0	0
$910$	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
$916$	58.0000	1.91637
$917$	0	0
$918$	0	0
$919$	−45.7199	−1.50816	−0.754080	−	0.656783i	$-0.771917\pi$
$919$	−45.7199	−1.50816	−0.754080	+	0.656783i	$0.771917\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−59.7670	−1.96513
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	61.9212	2.02939
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	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
$949$	92.3811	2.99882
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	43.7810	1.41229
$962$	0	0
$963$	0	0
$964$	−61.3654	−1.97645
$965$	0	0
$966$	0	0
$967$	60.6577	1.95062	0.975310	−	0.220840i	$-0.0708800\pi$
$967$	60.6577	1.95062	0.975310	+	0.220840i	$0.0708800\pi$
$968$	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
$973$	−99.2213	−3.18089
$974$	0	0
$975$	0	0
$976$	−16.8767	−0.540209
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	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
$988$	48.7246	1.55014
$989$	0	0
$990$	0	0
$991$	−4.02453	−0.127843	−0.0639217	−	0.997955i	$-0.520361\pi$
$991$	−4.02453	−0.127843	−0.0639217	+	0.997955i	$0.520361\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.6466	−1.25562	−0.627811	−	0.778366i	$-0.716049\pi$
$997$	−39.6466	−1.25562	−0.627811	+	0.778366i	$0.716049\pi$
$998$	0	0
$999$	0	0

Display

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.u.1.4	✓	4
3.2	odd	2	CM	7569.2.a.u.1.4	✓	4
29.28	even	2		7569.2.a.v.1.4	yes	4
87.86	odd	2		7569.2.a.v.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.u.1.4	✓	4	1.1	even	1	trivial
7569.2.a.u.1.4	✓	4	3.2	odd	2	CM
7569.2.a.v.1.4	yes	4	29.28	even	2
7569.2.a.v.1.4	yes	4	87.86	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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Modular forms

Varieties

Fields

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Groups

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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	7569.2.a.u.1.4

Level	$7569$
Weight	$2$
Character	7569.1
Self dual	yes
Analytic conductor	$60.439$
Analytic rank	$1$
Dimension	$4$
CM discriminant	-3
Inner twists	$2$