LMFDB - Embedded newform 1089.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$1089 = 3^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	1089.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:	$174.657979776$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 121)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-32.0000 q^{4} -57.0000 q^{5} +1024.00 q^{16} +1824.00 q^{20} -981.000 q^{23} +124.000 q^{25} -7775.00 q^{31} +1267.00 q^{37} -24708.0 q^{47} -16807.0 q^{49} -34806.0 q^{53} +24825.0 q^{59} -32768.0 q^{64} -72917.0 q^{67} +66273.0 q^{71} -58368.0 q^{80} +91089.0 q^{89} +31392.0 q^{92} -163183. q^{97} +O(q^{100})

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
$4$	−32.0000	−1.00000
$5$	−57.0000	−1.01965	−0.509823	−	0.860279i	$-0.670289\pi$
$5$	−57.0000	−1.01965	−0.509823	+	0.860279i	$0.670289\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	1024.00	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1824.00	1.01965
$21$	0	0
$22$	0	0
$23$	−981.000	−0.386678	−0.193339	−	0.981132i	$-0.561932\pi$
$23$	−981.000	−0.386678	−0.193339	+	0.981132i	$0.561932\pi$
$24$	0	0
$25$	124.000	0.0396800
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−7775.00	−1.45310	−0.726551	−	0.687112i	$-0.758878\pi$
$31$	−7775.00	−1.45310	−0.726551	+	0.687112i	$0.758878\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1267.00	0.152150	0.0760751	−	0.997102i	$-0.475761\pi$
$37$	1267.00	0.152150	0.0760751	+	0.997102i	$0.475761\pi$
$38$	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
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−24708.0	−1.63152	−0.815761	−	0.578389i	$-0.803682\pi$
$47$	−24708.0	−1.63152	−0.815761	+	0.578389i	$0.803682\pi$
$48$	0	0
$49$	−16807.0	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−34806.0	−1.70202	−0.851010	−	0.525150i	$-0.824009\pi$
$53$	−34806.0	−1.70202	−0.851010	+	0.525150i	$0.824009\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	24825.0	0.928452	0.464226	−	0.885717i	$-0.346333\pi$
$59$	24825.0	0.928452	0.464226	+	0.885717i	$0.346333\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	−32768.0	−1.00000
$65$	0	0
$66$	0	0
$67$	−72917.0	−1.98446	−0.992229	−	0.124427i	$-0.960291\pi$
$67$	−72917.0	−1.98446	−0.992229	+	0.124427i	$0.960291\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	66273.0	1.56024	0.780119	−	0.625631i	$-0.215159\pi$
$71$	66273.0	1.56024	0.780119	+	0.625631i	$0.215159\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	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	$0.5\pi$
$80$	−58368.0	−1.01965
$81$	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
$89$	91089.0	1.21896	0.609482	−	0.792800i	$-0.291377\pi$
$89$	91089.0	1.21896	0.609482	+	0.792800i	$0.291377\pi$
$90$	0	0
$91$	0	0
$92$	31392.0	0.386678
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−163183.	−1.76094	−0.880472	−	0.474098i	$-0.842774\pi$
$97$	−163183.	−1.76094	−0.880472	+	0.474098i	$0.842774\pi$
$98$	0	0
$99$	0	0
$100$	−3968.00	−0.0396800
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−180244.	−1.67405	−0.837024	−	0.547167i	$-0.815706\pi$
$103$	−180244.	−1.67405	−0.837024	+	0.547167i	$0.815706\pi$
$104$	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
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−192381.	−1.41731	−0.708657	−	0.705553i	$-0.750699\pi$
$113$	−192381.	−1.41731	−0.708657	+	0.705553i	$0.750699\pi$
$114$	0	0
$115$	55917.0	0.394275
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	248800.	1.45310
$125$	171057.	0.979187
$126$	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
$137$	263283.	1.19845	0.599227	−	0.800579i	$-0.295475\pi$
$137$	263283.	1.19845	0.599227	+	0.800579i	$0.295475\pi$
$138$	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
$148$	−40544.0	−0.152150
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	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
$155$	443175.	1.48165
$156$	0	0
$157$	280117.	0.906965	0.453482	−	0.891265i	$-0.350182\pi$
$157$	280117.	0.906965	0.453482	+	0.891265i	$0.350182\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	164144.	0.483900	0.241950	−	0.970289i	$-0.422213\pi$
$163$	164144.	0.483900	0.241950	+	0.970289i	$0.422213\pi$
$164$	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
$169$	−371293.	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	840189.	1.95995	0.979974	−	0.199127i	$-0.0638106\pi$
$179$	840189.	1.95995	0.979974	+	0.199127i	$0.0638106\pi$
$180$	0	0
$181$	279875.	0.634991	0.317496	−	0.948260i	$-0.397158\pi$
$181$	279875.	0.634991	0.317496	+	0.948260i	$0.397158\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−72219.0	−0.155139
$186$	0	0
$187$	0	0
$188$	790656.	1.63152
$189$	0	0
$190$	0	0
$191$	272325.	0.540137	0.270069	−	0.962841i	$-0.412954\pi$
$191$	272325.	0.540137	0.270069	+	0.962841i	$0.412954\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	537824.	1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	799900.	1.43187	0.715934	−	0.698168i	$-0.246001\pi$
$199$	799900.	1.43187	0.715934	+	0.698168i	$0.246001\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	1.11379e6	1.70202
$213$	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
$223$	−247531.	−0.333325	−0.166662	−	0.986014i	$-0.553299\pi$
$223$	−247531.	−0.333325	−0.166662	+	0.986014i	$0.553299\pi$
$224$	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
$229$	−1.17102e6	−1.47563	−0.737815	−	0.675003i	$-0.764142\pi$
$229$	−1.17102e6	−1.47563	−0.737815	+	0.675003i	$0.764142\pi$
$230$	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
$235$	1.40836e6	1.66358
$236$	−794400.	−0.928452
$237$	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		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	957999.	1.01965
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.84812e6	−1.85160	−0.925799	−	0.378017i	$-0.876606\pi$
$251$	−1.84812e6	−1.85160	−0.925799	+	0.378017i	$0.876606\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.04858e6	1.00000
$257$	−339858.	−0.320970	−0.160485	−	0.987038i	$-0.551306\pi$
$257$	−339858.	−0.320970	−0.160485	+	0.987038i	$0.551306\pi$
$258$	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
$265$	1.98394e6	1.73546
$266$	0	0
$267$	0	0
$268$	2.33334e6	1.98446
$269$	1.16085e6	0.978127	0.489064	−	0.872248i	$-0.337339\pi$
$269$	1.16085e6	0.978127	0.489064	+	0.872248i	$0.337339\pi$
$270$	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		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	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
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−2.12074e6	−1.56024
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41986e6	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−1.41502e6	−0.946693
$296$	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
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.36505e6	1.97284	0.986418	−	0.164257i	$-0.0525226\pi$
$311$	3.36505e6	1.97284	0.986418	+	0.164257i	$0.0525226\pi$
$312$	0	0
$313$	1.04882e6	0.605117	0.302559	−	0.953131i	$-0.402159\pi$
$313$	1.04882e6	0.605117	0.302559	+	0.953131i	$0.402159\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.28263e6	−1.83474	−0.917369	−	0.398037i	$-0.869691\pi$
$317$	−3.28263e6	−1.83474	−0.917369	+	0.398037i	$0.869691\pi$
$318$	0	0
$319$	0	0
$320$	1.86778e6	1.01965
$321$	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
$331$	736925.	0.369703	0.184852	−	0.982766i	$-0.440820\pi$
$331$	736925.	0.369703	0.184852	+	0.982766i	$0.440820\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.15627e6	2.02345
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	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
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.37977e6	1.87074	0.935372	−	0.353665i	$-0.115065\pi$
$353$	4.37977e6	1.87074	0.935372	+	0.353665i	$0.115065\pi$
$354$	0	0
$355$	−3.77756e6	−1.59089
$356$	−2.91485e6	−1.21896
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−2.47610e6	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.31317e6	−0.508926	−0.254463	−	0.967082i	$-0.581899\pi$
$367$	−1.31317e6	−0.508926	−0.254463	+	0.967082i	$0.581899\pi$
$368$	−1.00454e6	−0.386678
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.88862e6	0.675379	0.337690	−	0.941258i	$-0.390355\pi$
$379$	1.88862e6	0.675379	0.337690	+	0.941258i	$0.390355\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.23267e6	−1.82275	−0.911373	−	0.411581i	$-0.864977\pi$
$383$	−5.23267e6	−1.82275	−0.911373	+	0.411581i	$0.864977\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	5.22186e6	1.76094
$389$	5.54408e6	1.85761	0.928806	−	0.370566i	$-0.120836\pi$
$389$	5.54408e6	1.85761	0.928806	+	0.370566i	$0.120836\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.56024e6	0.496839	0.248420	−	0.968653i	$-0.420089\pi$
$397$	1.56024e6	0.496839	0.248420	+	0.968653i	$0.420089\pi$
$398$	0	0
$399$	0	0
$400$	126976.	0.0396800
$401$	5.71485e6	1.77478	0.887389	−	0.461022i	$-0.152517\pi$
$401$	5.71485e6	1.77478	0.887389	+	0.461022i	$0.152517\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	5.76781e6	1.67405
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−68664.0	−0.0191071	−0.00955353	−	0.999954i	$-0.503041\pi$
$419$	−68664.0	−0.0191071	−0.00955353	+	0.999954i	$0.503041\pi$
$420$	0	0
$421$	6.85705e6	1.88552	0.942762	−	0.333466i	$-0.108218\pi$
$421$	6.85705e6	1.88552	0.942762	+	0.333466i	$0.108218\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	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
$433$	−5.10513e6	−1.30854	−0.654270	−	0.756261i	$-0.727024\pi$
$433$	−5.10513e6	−1.30854	−0.654270	+	0.756261i	$0.727024\pi$
$434$	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
$443$	−4.17713e6	−1.01127	−0.505637	−	0.862746i	$-0.668742\pi$
$443$	−4.17713e6	−1.01127	−0.505637	+	0.862746i	$0.668742\pi$
$444$	0	0
$445$	−5.19207e6	−1.24291
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.63476e6	0.850864	0.425432	−	0.904990i	$-0.360122\pi$
$449$	3.63476e6	0.850864	0.425432	+	0.904990i	$0.360122\pi$
$450$	0	0
$451$	0	0
$452$	6.15619e6	1.41731
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	−1.78934e6	−0.394275
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	7.10982e6	1.54137	0.770684	−	0.637218i	$-0.219915\pi$
$463$	7.10982e6	1.54137	0.770684	+	0.637218i	$0.219915\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.20853e6	0.680792	0.340396	−	0.940282i	$-0.389439\pi$
$467$	3.20853e6	0.680792	0.340396	+	0.940282i	$0.389439\pi$
$468$	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
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.30143e6	1.79554
$486$	0	0
$487$	4.40023e6	0.840724	0.420362	−	0.907357i	$-0.361903\pi$
$487$	4.40023e6	0.840724	0.420362	+	0.907357i	$0.361903\pi$
$488$	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
$496$	−7.96160e6	−1.45310
$497$	0	0
$498$	0	0
$499$	−7.47980e6	−1.34474	−0.672370	−	0.740215i	$-0.734724\pi$
$499$	−7.47980e6	−1.34474	−0.672370	+	0.740215i	$0.734724\pi$
$500$	−5.47382e6	−0.979187
$501$	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
$508$	0	0
$509$	1.09082e7	1.86621	0.933103	−	0.359609i	$-0.117090\pi$
$509$	1.09082e7	1.86621	0.933103	+	0.359609i	$0.117090\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.02739e7	1.70694
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.23256e7	1.98936	0.994679	−	0.103024i	$-0.0328519\pi$
$521$	1.23256e7	1.98936	0.994679	+	0.103024i	$0.0328519\pi$
$522$	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
$529$	−5.47398e6	−0.850480
$530$	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
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−8.42506e6	−1.19845
$549$	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		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	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
$565$	1.09657e7	1.44516
$566$	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
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−121644.	−0.0153434
$576$	0	0
$577$	−8.05397e6	−1.00709	−0.503547	−	0.863968i	$-0.667972\pi$
$577$	−8.05397e6	−1.00709	−0.503547	+	0.863968i	$0.667972\pi$
$578$	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
$587$	1.29130e7	1.54679	0.773396	−	0.633923i	$-0.218556\pi$
$587$	1.29130e7	1.54679	0.773396	+	0.633923i	$0.218556\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	1.29741e6	0.152150
$593$	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
$599$	−1.46844e7	−1.67220	−0.836100	−	0.548577i	$-0.815170\pi$
$599$	−1.46844e7	−1.67220	−0.836100	+	0.548577i	$0.815170\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	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
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.79256e7	−1.89566	−0.947829	−	0.318781i	$-0.896727\pi$
$617$	−1.79256e7	−1.89566	−0.947829	+	0.318781i	$0.896727\pi$
$618$	0	0
$619$	−1.91271e6	−0.200642	−0.100321	−	0.994955i	$-0.531987\pi$
$619$	−1.91271e6	−0.200642	−0.100321	+	0.994955i	$0.531987\pi$
$620$	−1.41816e7	−1.48165
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.01377e7	−1.03811
$626$	0	0
$627$	0	0
$628$	−8.96374e6	−0.906965
$629$	0	0
$630$	0	0
$631$	1.28703e7	1.28681	0.643405	−	0.765526i	$-0.277521\pi$
$631$	1.28703e7	1.28681	0.643405	+	0.765526i	$0.277521\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.50793e7	−1.44956	−0.724779	−	0.688982i	$-0.758058\pi$
$641$	−1.50793e7	−1.44956	−0.724779	+	0.688982i	$0.758058\pi$
$642$	0	0
$643$	−2.09678e7	−1.99998	−0.999988	−	0.00489569i	$-0.998442\pi$
$643$	−2.09678e7	−1.99998	−0.999988	+	0.00489569i	$0.998442\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.18662e6	0.674938	0.337469	−	0.941337i	$-0.390429\pi$
$647$	7.18662e6	0.674938	0.337469	+	0.941337i	$0.390429\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−5.25261e6	−0.483900
$653$	−2.06540e7	−1.89549	−0.947746	−	0.319026i	$-0.896644\pi$
$653$	−2.06540e7	−1.89549	−0.947746	+	0.319026i	$0.896644\pi$
$654$	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
$661$	2.15101e7	1.91487	0.957433	−	0.288657i	$-0.0932086\pi$
$661$	2.15101e7	1.91487	0.957433	+	0.288657i	$0.0932086\pi$
$662$	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
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	1.18814e7	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.67323e7	−1.37248	−0.686238	−	0.727377i	$-0.740739\pi$
$683$	−1.67323e7	−1.37248	−0.686238	+	0.727377i	$0.740739\pi$
$684$	0	0
$685$	−1.50071e7	−1.22200
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.50313e7	1.99429	0.997146	−	0.0754971i	$-0.0240544\pi$
$691$	2.50313e7	1.99429	0.997146	+	0.0754971i	$0.0240544\pi$
$692$	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
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.59156e7	1.93618	0.968091	−	0.250598i	$-0.0806271\pi$
$709$	2.59156e7	1.93618	0.968091	+	0.250598i	$0.0806271\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.62728e6	0.561883
$714$	0	0
$715$	0	0
$716$	−2.68860e7	−1.95995
$717$	0	0
$718$	0	0
$719$	30039.0	0.00216702	0.00108351	−	0.999999i	$-0.499655\pi$
$719$	30039.0	0.00216702	0.00108351	+	0.999999i	$0.499655\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	−8.95600e6	−0.634991
$725$	0	0
$726$	0	0
$727$	−1.70878e7	−1.19908	−0.599542	−	0.800343i	$-0.704651\pi$
$727$	−1.70878e7	−1.19908	−0.599542	+	0.800343i	$0.704651\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	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	$0.5\pi$
$740$	2.31101e6	0.155139
$741$	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
$751$	−2.56094e7	−1.65691	−0.828455	−	0.560055i	$-0.810780\pi$
$751$	−2.56094e7	−1.65691	−0.828455	+	0.560055i	$0.810780\pi$
$752$	−2.53010e7	−1.63152
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.95760e7	1.24161	0.620805	−	0.783965i	$-0.286806\pi$
$757$	1.95760e7	1.24161	0.620805	+	0.783965i	$0.286806\pi$
$758$	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
$763$	0	0
$764$	−8.71440e6	−0.540137
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.18995e7	−0.716275	−0.358137	−	0.933669i	$-0.616588\pi$
$773$	−1.18995e7	−0.716275	−0.358137	+	0.933669i	$0.616588\pi$
$774$	0	0
$775$	−964100.	−0.0576591
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.72104e7	−1.00000
$785$	−1.59667e7	−0.924784
$786$	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
$796$	−2.55968e7	−1.43187
$797$	9.42078e6	0.525341	0.262670	−	0.964886i	$-0.415397\pi$
$797$	9.42078e6	0.525341	0.262670	+	0.964886i	$0.415397\pi$
$798$	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
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.35621e6	−0.493408
$816$	0	0
$817$	0	0
$818$	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
$823$	3.57048e7	1.83750	0.918749	−	0.394843i	$-0.129201\pi$
$823$	3.57048e7	1.83750	0.918749	+	0.394843i	$0.129201\pi$
$824$	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
$829$	1.90687e7	0.963684	0.481842	−	0.876258i	$-0.339968\pi$
$829$	1.90687e7	0.963684	0.481842	+	0.876258i	$0.339968\pi$
$830$	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
$839$	−3.93590e7	−1.93036	−0.965182	−	0.261577i	$-0.915757\pi$
$839$	−3.93590e7	−1.93036	−0.965182	+	0.261577i	$0.915757\pi$
$840$	0	0
$841$	−2.05111e7	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.11637e7	1.01965
$846$	0	0
$847$	0	0
$848$	−3.56413e7	−1.70202
$849$	0	0
$850$	0	0
$851$	−1.24293e6	−0.0588331
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	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
$859$	−1.50484e7	−0.695835	−0.347918	−	0.937525i	$-0.613111\pi$
$859$	−1.50484e7	−0.695835	−0.347918	+	0.937525i	$0.613111\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.79604e6	−0.310620	−0.155310	−	0.987866i	$-0.549638\pi$
$863$	−6.79604e6	−0.310620	−0.155310	+	0.987866i	$0.549638\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.12278e6	−0.309179	−0.154589	−	0.987979i	$-0.549405\pi$
$881$	−7.12278e6	−0.309179	−0.154589	+	0.987979i	$0.549405\pi$
$882$	0	0
$883$	6.29994e6	0.271916	0.135958	−	0.990715i	$-0.456589\pi$
$883$	6.29994e6	0.271916	0.135958	+	0.990715i	$0.456589\pi$
$884$	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
$892$	7.92099e6	0.333325
$893$	0	0
$894$	0	0
$895$	−4.78908e7	−1.99845
$896$	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
$905$	−1.59529e7	−0.647467
$906$	0	0
$907$	3.06168e7	1.23578	0.617891	−	0.786264i	$-0.287987\pi$
$907$	3.06168e7	1.23578	0.617891	+	0.786264i	$0.287987\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.26963e7	1.70449	0.852245	−	0.523143i	$-0.175241\pi$
$911$	4.26963e7	1.70449	0.852245	+	0.523143i	$0.175241\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	3.74728e7	1.47563
$917$	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
$925$	157108.	0.00603732
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.83412e7	−1.07740	−0.538702	−	0.842497i	$-0.681085\pi$
$929$	−2.83412e7	−1.07740	−0.538702	+	0.842497i	$0.681085\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	−4.50674e7	−1.66358
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	2.54208e7	0.928452
$945$	0	0
$946$	0	0
$947$	1.96062e7	0.710427	0.355213	−	0.934785i	$-0.384408\pi$
$947$	1.96062e7	0.710427	0.355213	+	0.934785i	$0.384408\pi$
$948$	0	0
$949$	0	0
$950$	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
$955$	−1.55225e7	−0.550749
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.18215e7	1.11151
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	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
$971$	−4.57445e7	−1.55701	−0.778504	−	0.627639i	$-0.784021\pi$
$971$	−4.57445e7	−1.55701	−0.778504	+	0.627639i	$0.784021\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.70589e7	1.57727	0.788633	−	0.614864i	$-0.210789\pi$
$977$	4.70589e7	1.57727	0.788633	+	0.614864i	$0.210789\pi$
$978$	0	0
$979$	0	0
$980$	−3.06560e7	−1.01965
$981$	0	0
$982$	0	0
$983$	6.05507e7	1.99864	0.999322	−	0.0368136i	$-0.0117208\pi$
$983$	6.05507e7	1.99864	0.999322	+	0.0368136i	$0.0117208\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−6.17681e7	−1.99793	−0.998965	−	0.0454936i	$-0.985514\pi$
$991$	−6.17681e7	−1.99793	−0.998965	+	0.0454936i	$0.985514\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.55943e7	−1.46000
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	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	1089.6.a.e.1.1		1
3.2	odd	2		121.6.a.a.1.1	✓	1
11.10	odd	2	CM	1089.6.a.e.1.1		1
33.32	even	2		121.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.6.a.a.1.1	✓	1	3.2	odd	2
121.6.a.a.1.1	✓	1	33.32	even	2
1089.6.a.e.1.1		1	1.1	even	1	trivial
1089.6.a.e.1.1		1	11.10	odd	2	CM

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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 $\F_{q}$

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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	1089.6.a.e.1.1

Level	$1089$
Weight	$6$
Character	1089.1
Self dual	yes
Analytic conductor	$174.658$
Analytic rank	$0$
Dimension	$1$
CM discriminant	-11
Inner twists	$2$