LMFDB - Embedded newform 1089.3.c.h.604.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 11^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			604.3
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.h.604.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{4} +1.65445i q^{7} +O(q^{10})$	$q+4.00000 q^{4} +1.65445i q^{7} +5.75839i q^{13} +16.0000 q^{16} +33.4978i q^{19} -25.0000 q^{25} +6.61780i q^{28} +19.0526 q^{31} +57.1577 q^{37} +74.3441i q^{43} +46.2628 q^{49} +23.0336i q^{52} +96.5826i q^{61} +64.0000 q^{64} -133.368 q^{67} -121.937i q^{73} +133.991i q^{76} -103.674i q^{79} -9.52697 q^{91} +169.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4}+O(q^{10}) $ 4 * q + 16 * q^4	$ 4 q + 16 q^{4} + 64 q^{16} - 100 q^{25} - 196 q^{49} + 256 q^{64} - 724 q^{91} + 676 q^{97}+O(q^{100}) $ 4 * q + 16 * q^4 + 64 * q^16 - 100 * q^25 - 196 * q^49 + 256 * q^64 - 724 * q^91 + 676 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.65445i	0.236350i	0.992993	+	0.118175i	$0.0377044\pi$
$7$	1.65445i	0.236350i	−0.992993	+	0.118175i	$0.962296\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.75839i	0.442953i	0.975166	+	0.221477i	$0.0710876\pi$
$13$	5.75839i	0.442953i	−0.975166	+	0.221477i	$0.928912\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	33.4978i	1.76304i	0.472144	+	0.881521i	$0.343480\pi$
$19$	33.4978i	1.76304i	−0.472144	+	0.881521i	$0.656520\pi$
$20$	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
$25$	−25.0000	−1.00000
$26$	0	0
$27$	0	0
$28$	6.61780i	0.236350i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	19.0526	0.614599	0.307299	−	0.951613i	$-0.400575\pi$
$31$	19.0526	0.614599	0.307299	+	0.951613i	$0.400575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	57.1577	1.54480	0.772401	−	0.635135i	$-0.219056\pi$
$37$	57.1577	1.54480	0.772401	+	0.635135i	$0.219056\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	74.3441i	1.72893i	0.502691	+	0.864466i	$0.332343\pi$
$43$	74.3441i	1.72893i	−0.502691	+	0.864466i	$0.667657\pi$
$44$	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
$49$	46.2628	0.944139
$50$	0	0
$51$	0	0
$52$	23.0336i	0.442953i
$53$	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
$61$	96.5826i	1.58332i	0.610961	+	0.791661i	$0.290783\pi$
$61$	96.5826i	1.58332i	−0.610961	+	0.791661i	$0.709217\pi$
$62$	0	0
$63$	0	0
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	−133.368	−1.99057	−0.995283	−	0.0970149i	$-0.969071\pi$
$67$	−133.368	−1.99057	−0.995283	+	0.0970149i	$0.969071\pi$
$68$	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
$73$	− 121.937i	− 1.67037i	−0.549970	−	0.835184i	$-0.685361\pi$
$73$	− 121.937i	− 1.67037i	0.549970	−	0.835184i	$-0.314639\pi$
$74$	0	0
$75$	0	0
$76$	133.991i	1.76304i
$77$	0	0
$78$	0	0
$79$	− 103.674i	− 1.31232i	−0.754620	−	0.656162i	$-0.772179\pi$
$79$	− 103.674i	− 1.31232i	0.754620	−	0.656162i	$-0.227821\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	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
$91$	−9.52697	−0.104692
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	169.000	1.74227	0.871134	−	0.491045i	$-0.163385\pi$
$97$	169.000	1.74227	0.871134	+	0.491045i	$0.163385\pi$
$98$	0	0
$99$	0	0
$100$	−100.000	−1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	157.000	1.52427	0.762136	−	0.647417i	$-0.224151\pi$
$103$	157.000	1.52427	0.762136	+	0.647417i	$0.224151\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	217.467i	1.99511i	0.0698830	+	0.997555i	$0.477737\pi$
$109$	217.467i	1.99511i	−0.0698830	+	0.997555i	$0.522263\pi$
$110$	0	0
$111$	0	0
$112$	26.4712i	0.236350i
$113$	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
$121$	0	0
$122$	0	0
$123$	0	0
$124$	76.2102	0.614599
$125$	0	0
$126$	0	0
$127$	162.976i	1.28328i	0.767007	+	0.641639i	$0.221745\pi$
$127$	162.976i	1.28328i	−0.767007	+	0.641639i	$0.778255\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−55.4205	−0.416695
$134$	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
$139$	− 180.403i	− 1.29786i	−0.760847	−	0.648931i	$-0.775216\pi$
$139$	− 180.403i	− 1.29786i	0.760847	−	0.648931i	$-0.224784\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	228.631	1.54480
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	270.818i	1.79350i	0.442539	+	0.896749i	$0.354078\pi$
$151$	270.818i	1.79350i	−0.442539	+	0.896749i	$0.645922\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−247.683	−1.57760	−0.788800	−	0.614650i	$-0.789297\pi$
$157$	−247.683	−1.57760	−0.788800	+	0.614650i	$0.789297\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−37.0000	−0.226994	−0.113497	−	0.993538i	$-0.536205\pi$
$163$	−37.0000	−0.226994	−0.113497	+	0.993538i	$0.536205\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	135.841	0.803793
$170$	0	0
$171$	0	0
$172$	297.376i	1.72893i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 41.3613i	− 0.236350i
$176$	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
$181$	1.00000	0.00552486	0.00276243	−	0.999996i	$-0.499121\pi$
$181$	1.00000	0.00552486	0.00276243	+	0.999996i	$0.499121\pi$
$182$	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
$193$	− 152.406i	− 0.789668i	−0.918752	−	0.394834i	$-0.870802\pi$
$193$	− 152.406i	− 0.789668i	0.918752	−	0.394834i	$-0.129198\pi$
$194$	0	0
$195$	0	0
$196$	185.051	0.944139
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	285.788	1.43612	0.718061	−	0.695980i	$-0.245030\pi$
$199$	285.788	1.43612	0.718061	+	0.695980i	$0.245030\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	92.1342i	0.442953i
$209$	0	0
$210$	0	0
$211$	59.9272i	0.284015i	0.989866	+	0.142008i	$0.0453558\pi$
$211$	59.9272i	0.284015i	−0.989866	+	0.142008i	$0.954644\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	31.5215i	0.145260i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−83.0000	−0.372197	−0.186099	−	0.982531i	$-0.559584\pi$
$223$	−83.0000	−0.372197	−0.186099	+	0.982531i	$0.559584\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−457.261	−1.99677	−0.998387	−	0.0567686i	$-0.981920\pi$
$229$	−457.261	−1.99677	−0.998387	+	0.0567686i	$0.981920\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	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
$241$	− 476.575i	− 1.97749i	−0.149606	−	0.988746i	$-0.547801\pi$
$241$	− 476.575i	− 1.97749i	0.149606	−	0.988746i	$-0.452199\pi$
$242$	0	0
$243$	0	0
$244$	386.330i	1.58332i
$245$	0	0
$246$	0	0
$247$	−192.894	−0.780945
$248$	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
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	94.5645i	0.365114i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−533.472	−1.99057
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	− 171.809i	− 0.633981i	−0.948429	−	0.316990i	$-0.897328\pi$
$271$	− 171.809i	− 0.633981i	0.948429	−	0.316990i	$-0.102672\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 553.562i	− 1.99842i	−0.0397524	−	0.999210i	$-0.512657\pi$
$277$	− 553.562i	− 1.99842i	0.0397524	−	0.999210i	$-0.487343\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 528.465i	− 1.86737i	−0.358097	−	0.933684i	$-0.616574\pi$
$283$	− 528.465i	− 1.86737i	0.358097	−	0.933684i	$-0.383426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	− 487.748i	− 1.67037i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−122.999	−0.408633
$302$	0	0
$303$	0	0
$304$	535.965i	1.76304i
$305$	0	0
$306$	0	0
$307$	− 574.491i	− 1.87130i	−0.352923	−	0.935652i	$-0.614812\pi$
$307$	− 574.491i	− 1.87130i	0.352923	−	0.935652i	$-0.385188\pi$
$308$	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
$313$	−609.682	−1.94787	−0.973933	−	0.226837i	$-0.927162\pi$
$313$	−609.682	−1.94787	−0.973933	+	0.226837i	$0.927162\pi$
$314$	0	0
$315$	0	0
$316$	− 414.694i	− 1.31232i
$317$	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
$325$	− 143.960i	− 0.442953i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	299.000	0.903323	0.451662	−	0.892189i	$-0.350831\pi$
$331$	299.000	0.903323	0.451662	+	0.892189i	$0.350831\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 330.314i	− 0.980161i	−0.871677	−	0.490080i	$-0.836967\pi$
$337$	− 330.314i	− 0.980161i	0.871677	−	0.490080i	$-0.163033\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	157.608i	0.459497i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	− 338.522i	− 0.969977i	−0.874521	−	0.484988i	$-0.838824\pi$
$349$	− 338.522i	− 0.969977i	0.874521	−	0.484988i	$-0.161176\pi$
$350$	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	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−761.104	−2.10832
$362$	0	0
$363$	0	0
$364$	−38.1079	−0.104692
$365$	0	0
$366$	0	0
$367$	−718.000	−1.95640	−0.978202	−	0.207657i	$-0.933416\pi$
$367$	−718.000	−1.95640	−0.978202	+	0.207657i	$0.933416\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 434.957i	− 1.16610i	−0.812435	−	0.583052i	$-0.801858\pi$
$373$	− 434.957i	− 1.16610i	0.812435	−	0.583052i	$-0.198142\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−694.000	−1.83113	−0.915567	−	0.402165i	$-0.868258\pi$
$379$	−694.000	−1.83113	−0.915567	+	0.402165i	$0.868258\pi$
$380$	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
$388$	676.000	1.74227
$389$	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
$397$	431.000	1.08564	0.542821	−	0.839848i	$-0.317356\pi$
$397$	431.000	1.08564	0.542821	+	0.839848i	$0.317356\pi$
$398$	0	0
$399$	0	0
$400$	−400.000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	109.712i	0.272238i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	468.390i	1.14521i	0.819832	+	0.572604i	$0.194067\pi$
$409$	468.390i	1.14521i	−0.819832	+	0.572604i	$0.805933\pi$
$410$	0	0
$411$	0	0
$412$	628.000	1.52427
$413$	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
$421$	−762.102	−1.81022	−0.905110	−	0.425178i	$-0.860211\pi$
$421$	−762.102	−1.81022	−0.905110	+	0.425178i	$0.860211\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−159.791	−0.374218
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	503.000	1.16166	0.580831	−	0.814024i	$-0.302728\pi$
$433$	503.000	1.16166	0.580831	+	0.814024i	$0.302728\pi$
$434$	0	0
$435$	0	0
$436$	869.868i	1.99511i
$437$	0	0
$438$	0	0
$439$	− 818.879i	− 1.86533i	−0.360745	−	0.932665i	$-0.617478\pi$
$439$	− 818.879i	− 1.86533i	0.360745	−	0.932665i	$-0.382522\pi$
$440$	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
$448$	105.885i	0.236350i
$449$	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
$457$	− 869.524i	− 1.90268i	−0.308147	−	0.951339i	$-0.599709\pi$
$457$	− 869.524i	− 1.90268i	0.308147	−	0.951339i	$-0.400291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	526.000	1.13607	0.568035	−	0.823005i	$-0.307704\pi$
$463$	526.000	1.13607	0.568035	+	0.823005i	$0.307704\pi$
$464$	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
$469$	− 220.651i	− 0.470470i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 837.445i	− 1.76304i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	329.136i	0.684275i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−152.420	−0.312978	−0.156489	−	0.987680i	$-0.550018\pi$
$487$	−152.420	−0.312978	−0.156489	+	0.987680i	$0.550018\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	304.841	0.614599
$497$	0	0
$498$	0	0
$499$	476.314	0.954537	0.477269	−	0.878758i	$-0.341627\pi$
$499$	476.314	0.954537	0.477269	+	0.878758i	$0.341627\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	651.905i	1.28328i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	201.739	0.394792
$512$	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
$523$	1041.78i	1.99194i	0.0897041	+	0.995968i	$0.471408\pi$
$523$	1041.78i	1.99194i	−0.0897041	+	0.995968i	$0.528592\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	−221.682	−0.416695
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	956.501i	1.76802i	0.467463	+	0.884012i	$0.345168\pi$
$541$	956.501i	1.76802i	−0.467463	+	0.884012i	$0.654832\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1043.65i	1.90796i	0.299873	+	0.953979i	$0.403056\pi$
$547$	1043.65i	1.90796i	−0.299873	+	0.953979i	$0.596944\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	171.523	0.310168
$554$	0	0
$555$	0	0
$556$	− 721.611i	− 1.29786i
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−428.102	−0.765836
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	466.668i	0.817282i	0.912695	+	0.408641i	$0.133997\pi$
$571$	466.668i	0.817282i	−0.912695	+	0.408641i	$0.866003\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	514.419	0.891541	0.445770	−	0.895147i	$-0.352930\pi$
$577$	514.419	0.891541	0.445770	+	0.895147i	$0.352930\pi$
$578$	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
$589$	638.219i	1.08356i
$590$	0	0
$591$	0	0
$592$	914.523	1.54480
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	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
$601$	− 787.809i	− 1.31083i	−0.755269	−	0.655415i	$-0.772494\pi$
$601$	− 787.809i	− 1.31083i	0.755269	−	0.655415i	$-0.227506\pi$
$602$	0	0
$603$	0	0
$604$	1083.27i	1.79350i
$605$	0	0
$606$	0	0
$607$	− 1212.45i	− 1.99745i	−0.0504797	−	0.998725i	$-0.516075\pi$
$607$	− 1212.45i	− 1.99745i	0.0504797	−	0.998725i	$-0.483925\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 962.112i	− 1.56951i	−0.619804	−	0.784757i	$-0.712788\pi$
$613$	− 962.112i	− 1.56951i	0.619804	−	0.784757i	$-0.287212\pi$
$614$	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
$619$	−214.000	−0.345719	−0.172859	−	0.984947i	$-0.555301\pi$
$619$	−214.000	−0.345719	−0.172859	+	0.984947i	$0.555301\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	−990.733	−1.57760
$629$	0	0
$630$	0	0
$631$	1066.94	1.69088	0.845438	−	0.534073i	$-0.179339\pi$
$631$	1066.94	1.69088	0.845438	+	0.534073i	$0.179339\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	266.399i	0.418209i
$638$	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
$643$	−923.000	−1.43546	−0.717729	−	0.696322i	$-0.754819\pi$
$643$	−923.000	−1.43546	−0.717729	+	0.696322i	$0.754819\pi$
$644$	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
$652$	−148.000	−0.226994
$653$	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	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−552.524	−0.835891	−0.417946	−	0.908472i	$-0.637250\pi$
$661$	−552.524	−0.835891	−0.417946	+	0.908472i	$0.637250\pi$
$662$	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
$673$	− 1293.99i	− 1.92272i	−0.275287	−	0.961362i	$-0.588773\pi$
$673$	− 1293.99i	− 1.92272i	0.275287	−	0.961362i	$-0.411227\pi$
$674$	0	0
$675$	0	0
$676$	543.364	0.803793
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	279.602i	0.411785i
$680$	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
$688$	1189.51i	1.72893i
$689$	0	0
$690$	0	0
$691$	1019.00	1.47467	0.737337	−	0.675525i	$-0.236083\pi$
$691$	1019.00	1.47467	0.737337	+	0.675525i	$0.236083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	− 165.445i	− 0.236350i
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	1914.66i	2.72355i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1066.94	1.50486	0.752428	−	0.658674i	$-0.228882\pi$
$709$	1066.94	1.50486	0.752428	+	0.658674i	$0.228882\pi$
$710$	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
$721$	259.749i	0.360262i
$722$	0	0
$723$	0	0
$724$	4.00000	0.00552486
$725$	0	0
$726$	0	0
$727$	−1371.78	−1.88691	−0.943455	−	0.331499i	$-0.892446\pi$
$727$	−1371.78	−1.88691	−0.943455	+	0.331499i	$0.892446\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1466.00i	− 1.99999i	−0.00252286	−	0.999997i	$-0.500803\pi$
$733$	− 1466.00i	− 1.99999i	0.00252286	−	0.999997i	$-0.499197\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1395.54i	1.88842i	0.329350	+	0.944208i	$0.393171\pi$
$739$	1395.54i	1.88842i	−0.329350	+	0.944208i	$0.606829\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	590.629	0.786457	0.393229	−	0.919441i	$-0.371358\pi$
$751$	590.629	0.786457	0.393229	+	0.919441i	$0.371358\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−95.2628	−0.125843	−0.0629213	−	0.998018i	$-0.520042\pi$
$757$	−95.2628	−0.125843	−0.0629213	+	0.998018i	$0.520042\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−359.788	−0.471544
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1453.04i	1.88952i	0.327765	+	0.944759i	$0.393705\pi$
$769$	1453.04i	1.88952i	−0.327765	+	0.944759i	$0.606295\pi$
$770$	0	0
$771$	0	0
$772$	− 609.624i	− 0.789668i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−476.314	−0.614599
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	740.205	0.944139
$785$	0	0
$786$	0	0
$787$	1241.67i	1.57773i	0.614568	+	0.788864i	$0.289331\pi$
$787$	1241.67i	1.57773i	−0.614568	+	0.788864i	$0.710669\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−556.160	−0.701337
$794$	0	0
$795$	0	0
$796$	1143.15	1.43612
$797$	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	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	1311.79i	1.61749i	0.588157	+	0.808747i	$0.299854\pi$
$811$	1311.79i	1.61749i	−0.588157	+	0.808747i	$0.700146\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2490.37	−3.04818
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1621.00	1.96962	0.984812	−	0.173626i	$-0.0555484\pi$
$823$	1621.00	1.96962	0.984812	+	0.173626i	$0.0555484\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1609.00	1.94089	0.970446	−	0.241317i	$-0.0775794\pi$
$829$	1609.00	1.94089	0.970446	+	0.241317i	$0.0775794\pi$
$830$	0	0
$831$	0	0
$832$	368.537i	0.442953i
$833$	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
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	239.709i	0.284015i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 1705.51i	− 1.99942i	−0.0240202	−	0.999711i	$-0.507647\pi$
$853$	− 1705.51i	− 1.99942i	0.0240202	−	0.999711i	$-0.492353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1549.00	1.80326	0.901630	−	0.432509i	$-0.142371\pi$
$859$	1549.00	1.80326	0.901630	+	0.432509i	$0.142371\pi$
$860$	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
$868$	126.086i	0.145260i
$869$	0	0
$870$	0	0
$871$	− 767.984i	− 0.881727i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1004.39i	1.14526i	0.819814	+	0.572630i	$0.194077\pi$
$877$	1004.39i	1.14526i	−0.819814	+	0.572630i	$0.805923\pi$
$878$	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
$883$	1238.42	1.40251	0.701255	−	0.712911i	$-0.252623\pi$
$883$	1238.42	1.40251	0.701255	+	0.712911i	$0.252623\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−269.636	−0.303303
$890$	0	0
$891$	0	0
$892$	−332.000	−0.372197
$893$	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
$907$	1453.00	1.60198	0.800992	−	0.598675i	$-0.204306\pi$
$907$	1453.00	1.60198	0.800992	+	0.598675i	$0.204306\pi$
$908$	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
$916$	−1829.05	−1.99677
$917$	0	0
$918$	0	0
$919$	1256.09i	1.36680i	0.730044	+	0.683400i	$0.239499\pi$
$919$	1256.09i	1.36680i	−0.730044	+	0.683400i	$0.760501\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1428.94	−1.54480
$926$	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
$931$	1549.70i	1.66456i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1702.03i	1.81647i	0.418465	+	0.908233i	$0.362568\pi$
$937$	1702.03i	1.81647i	−0.418465	+	0.908233i	$0.637432\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	702.160	0.739895
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−598.000	−0.622268
$962$	0	0
$963$	0	0
$964$	− 1906.30i	− 1.97749i
$965$	0	0
$966$	0	0
$967$	− 1176.89i	− 1.21706i	−0.793532	−	0.608529i	$-0.791760\pi$
$967$	− 1176.89i	− 1.21706i	0.793532	−	0.608529i	$-0.208240\pi$
$968$	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
$973$	298.468	0.306750
$974$	0	0
$975$	0	0
$976$	1545.32i	1.58332i
$977$	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
$988$	−771.574	−0.780945
$989$	0	0
$990$	0	0
$991$	1981.47	1.99946	0.999731	−	0.0232089i	$-0.00738828\pi$
$991$	1981.47	1.99946	0.999731	+	0.0232089i	$0.00738828\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1092.50i	1.09578i	0.836550	+	0.547891i	$0.184569\pi$
$997$	1092.50i	1.09578i	−0.836550	+	0.547891i	$0.815431\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.h.604.3	yes	4
3.2	odd	2	CM	1089.3.c.h.604.3	yes	4
11.10	odd	2	inner	1089.3.c.h.604.2	✓	4
33.32	even	2	inner	1089.3.c.h.604.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.c.h.604.2	✓	4	11.10	odd	2	inner
1089.3.c.h.604.2	✓	4	33.32	even	2	inner
1089.3.c.h.604.3	yes	4	1.1	even	1	trivial
1089.3.c.h.604.3	yes	4	3.2	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000 q^{4} +1.65445i q^{7} +O(q^{10})\)	\(q+4.00000 q^{4} +1.65445i q^{7} +5.75839i q^{13} +16.0000 q^{16} +33.4978i q^{19} -25.0000 q^{25} +6.61780i q^{28} +19.0526 q^{31} +57.1577 q^{37} +74.3441i q^{43} +46.2628 q^{49} +23.0336i q^{52} +96.5826i q^{61} +64.0000 q^{64} -133.368 q^{67} -121.937i q^{73} +133.991i q^{76} -103.674i q^{79} -9.52697 q^{91} +169.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4}+O(q^{10}) \) 4 * q + 16 * q^4	\( 4 q + 16 q^{4} + 64 q^{16} - 100 q^{25} - 196 q^{49} + 256 q^{64} - 724 q^{91} + 676 q^{97}+O(q^{100}) \) 4 * q + 16 * q^4 + 64 * q^16 - 100 * q^25 - 196 * q^49 + 256 * q^64 - 724 * q^91 + 676 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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