LMFDB - Embedded newform 441.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{2} -25.0000 q^{4} -150.808 q^{8} +O(q^{10})$	$q+2.64575 q^{2} -25.0000 q^{4} -150.808 q^{8} -799.017 q^{11} +401.000 q^{16} -2114.00 q^{22} -1105.92 q^{23} -3125.00 q^{25} +5386.75 q^{29} +5886.80 q^{32} +8886.00 q^{37} -11748.0 q^{43} +19975.4 q^{44} -2926.00 q^{46} -8267.97 q^{50} -32712.1 q^{53} +14252.0 q^{58} +2743.00 q^{64} +69364.0 q^{67} +84923.3 q^{71} +23510.1 q^{74} +80168.0 q^{79} -31082.3 q^{86} +120498. q^{88} +27648.1 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{4}+O(q^{10}) $ 2 * q - 50 * q^4	$ 2 q - 50 q^{4} + 802 q^{16} - 4228 q^{22} - 6250 q^{25} + 17772 q^{37} - 23496 q^{43} - 5852 q^{46} + 28504 q^{58} + 5486 q^{64} + 138728 q^{67} + 160336 q^{79} + 240996 q^{88}+O(q^{100}) $ 2 * q - 50 * q^4 + 802 * q^16 - 4228 * q^22 - 6250 * q^25 + 17772 * q^37 - 23496 * q^43 - 5852 * q^46 + 28504 * q^58 + 5486 * q^64 + 138728 * q^67 + 160336 * q^79 + 240996 * q^88

Display 100 coefficients 10 coefficients

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 $

$2$	2.64575	0.467707	0.233854	−	0.972272i	$-0.424866\pi$
$2$	2.64575	0.467707	0.233854	+	0.972272i	$0.424866\pi$
$3$	0	0
$3$	0	0
$4$	−25.0000	−0.781250
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−150.808	−0.833103
$9$	0	0
$10$	0	0
$11$	−799.017	−1.99101	−0.995507	−	0.0946895i	$-0.969814\pi$
$11$	−799.017	−1.99101	−0.995507	+	0.0946895i	$0.969814\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	401.000	0.391602
$17$	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	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−2114.00	−0.931211
$23$	−1105.92	−0.435919	−0.217959	−	0.975958i	$-0.569940\pi$
$23$	−1105.92	−0.435919	−0.217959	+	0.975958i	$0.569940\pi$
$24$	0	0
$25$	−3125.00	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5386.75	1.18941	0.594705	−	0.803944i	$-0.297269\pi$
$29$	5386.75	1.18941	0.594705	+	0.803944i	$0.297269\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5886.80	1.01626
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8886.00	1.06709	0.533546	−	0.845771i	$-0.320859\pi$
$37$	8886.00	1.06709	0.533546	+	0.845771i	$0.320859\pi$
$38$	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
$43$	−11748.0	−0.968931	−0.484465	−	0.874810i	$-0.660986\pi$
$43$	−11748.0	−0.968931	−0.484465	+	0.874810i	$0.660986\pi$
$44$	19975.4	1.55548
$45$	0	0
$46$	−2926.00	−0.203882
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−8267.97	−0.467707
$51$	0	0
$52$	0	0
$53$	−32712.1	−1.59963	−0.799813	−	0.600250i	$-0.795068\pi$
$53$	−32712.1	−1.59963	−0.799813	+	0.600250i	$0.795068\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	14252.0	0.556296
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	2743.00	0.0837097
$65$	0	0
$66$	0	0
$67$	69364.0	1.88776	0.943881	−	0.330286i	$-0.107145\pi$
$67$	69364.0	1.88776	0.943881	+	0.330286i	$0.107145\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	84923.3	1.99931	0.999657	−	0.0261794i	$-0.00833410\pi$
$71$	84923.3	1.99931	0.999657	+	0.0261794i	$0.00833410\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	23510.1	0.499087
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	80168.0	1.44522	0.722609	−	0.691257i	$-0.242943\pi$
$79$	80168.0	1.44522	0.722609	+	0.691257i	$0.242943\pi$
$80$	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
$86$	−31082.3	−0.453176
$87$	0	0
$88$	120498.	1.65872
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	27648.1	0.340562
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	78125.0	0.781250
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−86548.0	−0.748156
$107$	227794.	1.92346	0.961729	−	0.274003i	$-0.0883478\pi$
$107$	227794.	1.92346	0.961729	+	0.274003i	$0.0883478\pi$
$108$	0	0
$109$	219582.	1.77023	0.885117	−	0.465369i	$-0.154078\pi$
$109$	219582.	1.77023	0.885117	+	0.465369i	$0.154078\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−241906.	−1.78218	−0.891089	−	0.453828i	$-0.850058\pi$
$113$	−241906.	−1.78218	−0.891089	+	0.453828i	$0.850058\pi$
$114$	0	0
$115$	0	0
$116$	−134669.	−0.929227
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	477377.	2.96414
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−262064.	−1.44178	−0.720888	−	0.693051i	$-0.756266\pi$
$127$	−262064.	−1.44178	−0.720888	+	0.693051i	$0.756266\pi$
$128$	−181120.	−0.977107
$129$	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
$134$	183520.	0.882920
$135$	0	0
$136$	0	0
$137$	260998.	1.18805	0.594027	−	0.804445i	$-0.297537\pi$
$137$	260998.	1.18805	0.594027	+	0.804445i	$0.297537\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	224686.	0.935094
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−222150.	−0.833666
$149$	−424474.	−1.56634	−0.783168	−	0.621810i	$-0.786398\pi$
$149$	−424474.	−1.56634	−0.783168	+	0.621810i	$0.786398\pi$
$150$	0	0
$151$	261624.	0.933760	0.466880	−	0.884321i	$-0.345378\pi$
$151$	261624.	0.933760	0.466880	+	0.884321i	$0.345378\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	212105.	0.675939
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	663100.	1.95483	0.977417	−	0.211318i	$-0.0677757\pi$
$163$	663100.	1.95483	0.977417	+	0.211318i	$0.0677757\pi$
$164$	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
$172$	293700.	0.756977
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−320406.	−0.779684
$177$	0	0
$178$	0	0
$179$	−627175.	−1.46304	−0.731520	−	0.681820i	$-0.761189\pi$
$179$	−627175.	−1.46304	−0.731520	+	0.681820i	$0.761189\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	166782.	0.363166
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	103317.	0.204921	0.102461	−	0.994737i	$-0.467328\pi$
$191$	103317.	0.204921	0.102461	+	0.994737i	$0.467328\pi$
$192$	0	0
$193$	385902.	0.745734	0.372867	−	0.927885i	$-0.378375\pi$
$193$	385902.	0.745734	0.372867	+	0.927885i	$0.378375\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.01882e6	−1.87038	−0.935190	−	0.354146i	$-0.884772\pi$
$197$	−1.01882e6	−1.87038	−0.935190	+	0.354146i	$0.884772\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	471274.	0.833103
$201$	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
$211$	1.09705e6	1.69637	0.848186	−	0.529699i	$-0.177695\pi$
$211$	1.09705e6	1.69637	0.848186	+	0.529699i	$0.177695\pi$
$212$	817802.	1.24971
$213$	0	0
$214$	602686.	0.899615
$215$	0	0
$216$	0	0
$217$	0	0
$218$	580959.	0.827951
$219$	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
$225$	0	0
$226$	−640024.	−0.833538
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	−812364.	−0.990902
$233$	−1.05345e6	−1.27123	−0.635617	−	0.772004i	$-0.719254\pi$
$233$	−1.05345e6	−1.27123	−0.635617	+	0.772004i	$0.719254\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.61113e6	1.82447	0.912233	−	0.409671i	$-0.134357\pi$
$239$	1.61113e6	1.82447	0.912233	+	0.409671i	$0.134357\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	1.26302e6	1.38635
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	883652.	0.867921
$254$	−693356.	−0.674329
$255$	0	0
$256$	−566975.	−0.540709
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.63936e6	−1.46145	−0.730725	−	0.682672i	$-0.760818\pi$
$263$	−1.63936e6	−1.46145	−0.730725	+	0.682672i	$0.760818\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−1.73410e6	−1.47481
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	690536.	0.555661
$275$	2.49693e6	1.99101
$276$	0	0
$277$	−2.55145e6	−1.99796	−0.998982	−	0.0451116i	$-0.985636\pi$
$277$	−2.55145e6	−1.99796	−0.998982	+	0.0451116i	$0.985636\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−718184.	−0.542588	−0.271294	−	0.962497i	$-0.587452\pi$
$281$	−718184.	−0.542588	−0.271294	+	0.962497i	$0.587452\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−2.12308e6	−1.56196
$285$	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
$295$	0	0
$296$	−1.34008e6	−0.888998
$297$	0	0
$298$	−1.12305e6	−0.732587
$299$	0	0
$300$	0	0
$301$	0	0
$302$	692192.	0.436726
$303$	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
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−2.00420e6	−1.12908
$317$	3.57144e6	1.99616	0.998079	−	0.0619605i	$-0.0197353\pi$
$317$	3.57144e6	1.99616	0.998079	+	0.0619605i	$0.0197353\pi$
$318$	0	0
$319$	−4.30410e6	−2.36813
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	1.75440e6	0.914290
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.97148e6	−1.49074	−0.745371	−	0.666650i	$-0.767727\pi$
$331$	−2.97148e6	−1.49074	−0.745371	+	0.666650i	$0.767727\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.15965e6	1.99518	0.997590	−	0.0693859i	$-0.0221040\pi$
$337$	4.15965e6	1.99518	0.997590	+	0.0693859i	$0.0221040\pi$
$338$	−982349.	−0.467707
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	1.77169e6	0.807220
$345$	0	0
$346$	0	0
$347$	3.85255e6	1.71761	0.858804	−	0.512304i	$-0.171208\pi$
$347$	3.85255e6	1.71761	0.858804	+	0.512304i	$0.171208\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−4.70365e6	−2.02338
$353$	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
$358$	−1.65935e6	−0.684275
$359$	−2.37241e6	−0.971523	−0.485762	−	0.874091i	$-0.661458\pi$
$359$	−2.37241e6	−0.971523	−0.485762	+	0.874091i	$0.661458\pi$
$360$	0	0
$361$	−2.47610e6	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−443476.	−0.170707
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	599302.	0.223035	0.111518	−	0.993762i	$-0.464429\pi$
$373$	599302.	0.223035	0.111518	+	0.993762i	$0.464429\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	5.59273e6	1.99998	0.999991	−	0.00429827i	$-0.00136819\pi$
$379$	5.59273e6	1.99998	0.999991	+	0.00429827i	$0.00136819\pi$
$380$	0	0
$381$	0	0
$382$	273350.	0.0958431
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	1.02100e6	0.348785
$387$	0	0
$388$	0	0
$389$	−5.83451e6	−1.95492	−0.977462	−	0.211109i	$-0.932292\pi$
$389$	−5.83451e6	−1.95492	−0.977462	+	0.211109i	$0.932292\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−2.69553e6	−0.874790
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−1.25312e6	−0.391602
$401$	−2.25352e6	−0.699844	−0.349922	−	0.936779i	$-0.613792\pi$
$401$	−2.25352e6	−0.699844	−0.349922	+	0.936779i	$0.613792\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.10006e6	−2.12460
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	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
$421$	2.07477e6	0.570513	0.285257	−	0.958451i	$-0.407921\pi$
$421$	2.07477e6	0.570513	0.285257	+	0.958451i	$0.407921\pi$
$422$	2.90253e6	0.793405
$423$	0	0
$424$	4.93324e6	1.33265
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−5.69485e6	−1.50270
$429$	0	0
$430$	0	0
$431$	−2.37311e6	−0.615353	−0.307676	−	0.951491i	$-0.599551\pi$
$431$	−2.37311e6	−0.615353	−0.307676	+	0.951491i	$0.599551\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−5.48955e6	−1.38299
$437$	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
$443$	−6.49509e6	−1.57245	−0.786223	−	0.617942i	$-0.787967\pi$
$443$	−6.49509e6	−1.57245	−0.786223	+	0.617942i	$0.787967\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.30677e6	1.94454	0.972269	−	0.233866i	$-0.0751378\pi$
$449$	8.30677e6	1.94454	0.972269	+	0.233866i	$0.0751378\pi$
$450$	0	0
$451$	0	0
$452$	6.04766e6	1.39233
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.77969e6	1.29453	0.647267	−	0.762263i	$-0.275912\pi$
$457$	5.77969e6	1.29453	0.647267	+	0.762263i	$0.275912\pi$
$458$	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
$463$	2.88620e6	0.625711	0.312856	−	0.949801i	$-0.398714\pi$
$463$	2.88620e6	0.625711	0.312856	+	0.949801i	$0.398714\pi$
$464$	2.16009e6	0.465775
$465$	0	0
$466$	−2.78718e6	−0.594565
$467$	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
$473$	9.38685e6	1.92915
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	4.26265e6	0.853316
$479$	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
$484$	−1.19344e7	−2.31573
$485$	0	0
$486$	0	0
$487$	2.76146e6	0.527615	0.263807	−	0.964575i	$-0.415022\pi$
$487$	2.76146e6	0.527615	0.263807	+	0.964575i	$0.415022\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.01881e6	−1.12670	−0.563349	−	0.826219i	$-0.690487\pi$
$491$	−6.01881e6	−1.12670	−0.563349	+	0.826219i	$0.690487\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.11204e7	1.99925	0.999626	−	0.0273386i	$-0.00870324\pi$
$499$	1.11204e7	1.99925	0.999626	+	0.0273386i	$0.00870324\pi$
$500$	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
$506$	2.33792e6	0.405933
$507$	0	0
$508$	6.55160e6	1.12639
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	4.29577e6	0.724213
$513$	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
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−4.33733e6	−0.683530
$527$	0	0
$528$	0	0
$529$	−5.21328e6	−0.809975
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−1.04606e7	−1.57270
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.11261e7	1.63437	0.817186	−	0.576374i	$-0.195533\pi$
$541$	1.11261e7	1.63437	0.817186	+	0.576374i	$0.195533\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.23604e6	−0.319529	−0.159765	−	0.987155i	$-0.551074\pi$
$547$	−2.23604e6	−0.319529	−0.159765	+	0.987155i	$0.551074\pi$
$548$	−6.52495e6	−0.928167
$549$	0	0
$550$	6.60625e6	0.931211
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−6.75050e6	−0.934462
$555$	0	0
$556$	0	0
$557$	−1.32485e7	−1.80938	−0.904690	−	0.426070i	$-0.859898\pi$
$557$	−1.32485e7	−1.80938	−0.904690	+	0.426070i	$0.859898\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−1.90014e6	−0.253772
$563$	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
$568$	−1.28071e7	−1.66564
$569$	1.13522e7	1.46994	0.734971	−	0.678099i	$-0.237196\pi$
$569$	1.13522e7	1.46994	0.734971	+	0.678099i	$0.237196\pi$
$570$	0	0
$571$	6.33912e6	0.813653	0.406826	−	0.913506i	$-0.366635\pi$
$571$	6.33912e6	0.813653	0.406826	+	0.913506i	$0.366635\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.45601e6	0.435919
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−3.75659e6	−0.467707
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.61375e7	3.18488
$584$	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
$592$	3.56329e6	0.417875
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	1.06118e7	1.22370
$597$	0	0
$598$	0	0
$599$	1.60293e7	1.82536	0.912679	−	0.408677i	$-0.134010\pi$
$599$	1.60293e7	1.82536	0.912679	+	0.408677i	$0.134010\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	−6.54060e6	−0.729500
$605$	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
$613$	1.75514e7	1.88652	0.943258	−	0.332060i	$-0.107744\pi$
$613$	1.75514e7	1.88652	0.943258	+	0.332060i	$0.107744\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.75090e7	1.85160	0.925802	−	0.378008i	$-0.123391\pi$
$617$	1.75090e7	1.85160	0.925802	+	0.378008i	$0.123391\pi$
$618$	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
$625$	9.76562e6	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.99786e7	1.99752	0.998760	−	0.0497844i	$-0.0158534\pi$
$631$	1.99786e7	1.99752	0.998760	+	0.0497844i	$0.0158534\pi$
$632$	−1.20900e7	−1.20402
$633$	0	0
$634$	9.44913e6	0.933617
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−1.13876e7	−1.10759
$639$	0	0
$640$	0	0
$641$	1.73407e7	1.66694	0.833471	−	0.552563i	$-0.186350\pi$
$641$	1.73407e7	1.66694	0.833471	+	0.552563i	$0.186350\pi$
$642$	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
$652$	−1.65775e7	−1.52721
$653$	1.88262e7	1.72775	0.863873	−	0.503710i	$-0.168032\pi$
$653$	1.88262e7	1.72775	0.863873	+	0.503710i	$0.168032\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.38990e6	0.842263	0.421131	−	0.907000i	$-0.361633\pi$
$659$	9.38990e6	0.842263	0.421131	+	0.907000i	$0.361633\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−7.86179e6	−0.697230
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.95734e6	−0.518487
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.38435e6	−0.117817	−0.0589085	−	0.998263i	$-0.518762\pi$
$673$	−1.38435e6	−0.117817	−0.0589085	+	0.998263i	$0.518762\pi$
$674$	1.10054e7	0.933160
$675$	0	0
$676$	9.28232e6	0.781250
$677$	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
$683$	−1.18403e7	−0.971206	−0.485603	−	0.874179i	$-0.661400\pi$
$683$	−1.18403e7	−0.971206	−0.485603	+	0.874179i	$0.661400\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−4.71095e6	−0.379435
$689$	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
$694$	1.01929e7	0.803338
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.55582e7	−1.96443	−0.982213	−	0.187771i	$-0.939874\pi$
$701$	−2.55582e7	−1.96443	−0.982213	+	0.187771i	$0.939874\pi$
$702$	0	0
$703$	0	0
$704$	−2.19170e6	−0.166667
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.43225e7	1.07005	0.535023	−	0.844837i	$-0.320303\pi$
$709$	1.43225e7	1.07005	0.535023	+	0.844837i	$0.320303\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.56794e7	1.14300
$717$	0	0
$718$	−6.27680e6	−0.454388
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−6.55114e6	−0.467707
$723$	0	0
$724$	0	0
$725$	−1.68336e7	−1.18941
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	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
$736$	−6.51035e6	−0.443006
$737$	−5.54230e7	−3.75856
$738$	0	0
$739$	−2.64893e6	−0.178427	−0.0892133	−	0.996013i	$-0.528435\pi$
$739$	−2.64893e6	−0.178427	−0.0892133	+	0.996013i	$0.528435\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.68294e7	1.11840	0.559199	−	0.829033i	$-0.311109\pi$
$743$	1.68294e7	1.11840	0.559199	+	0.829033i	$0.311109\pi$
$744$	0	0
$745$	0	0
$746$	1.58560e6	0.104315
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.51088e7	−1.62452	−0.812260	−	0.583295i	$-0.801763\pi$
$751$	−2.51088e7	−1.62452	−0.812260	+	0.583295i	$0.801763\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.78870e7	1.13448	0.567242	−	0.823551i	$-0.308011\pi$
$757$	1.78870e7	1.13448	0.567242	+	0.823551i	$0.308011\pi$
$758$	1.47970e7	0.935406
$759$	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
$764$	−2.58291e6	−0.160095
$765$	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
$772$	−9.64755e6	−0.582604
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−1.54367e7	−0.914332
$779$	0	0
$780$	0	0
$781$	−6.78552e7	−3.98066
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	2.54704e7	1.46123
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−1.83962e7	−1.01626
$801$	0	0
$802$	−5.96226e6	−0.327322
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.23828e7	1.73957	0.869787	−	0.493428i	$-0.164256\pi$
$809$	3.23828e7	1.73957	0.869787	+	0.493428i	$0.164256\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	−1.87850e7	−0.993689
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.50170e7	−0.777546	−0.388773	−	0.921334i	$-0.627101\pi$
$821$	−1.50170e7	−0.777546	−0.388773	+	0.921334i	$0.627101\pi$
$822$	0	0
$823$	7.08675e6	0.364710	0.182355	−	0.983233i	$-0.441628\pi$
$823$	7.08675e6	0.364710	0.182355	+	0.983233i	$0.441628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.37491e7	0.699054	0.349527	−	0.936926i	$-0.386342\pi$
$827$	1.37491e7	0.699054	0.349527	+	0.936926i	$0.386342\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	8.50592e6	0.414698
$842$	5.48934e6	0.266833
$843$	0	0
$844$	−2.74263e7	−1.32529
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−1.31175e7	−0.626416
$849$	0	0
$850$	0	0
$851$	−9.82724e6	−0.465166
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−3.43531e7	−1.60244
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	−6.27865e6	−0.287805
$863$	3.39442e7	1.55145	0.775727	−	0.631068i	$-0.217383\pi$
$863$	3.39442e7	1.55145	0.775727	+	0.631068i	$0.217383\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.40556e7	−2.87745
$870$	0	0
$871$	0	0
$872$	−3.31147e7	−1.47479
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.33428e7	−1.90291	−0.951453	−	0.307793i	$-0.900410\pi$
$877$	−4.33428e7	−1.90291	−0.951453	+	0.307793i	$0.900410\pi$
$878$	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
$883$	3.94011e7	1.70062	0.850308	−	0.526286i	$-0.176416\pi$
$883$	3.94011e7	1.70062	0.850308	+	0.526286i	$0.176416\pi$
$884$	0	0
$885$	0	0
$886$	−1.71844e7	−0.735445
$887$	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
$898$	2.19776e7	0.909474
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	3.64814e7	1.48474
$905$	0	0
$906$	0	0
$907$	−4.48347e7	−1.80966	−0.904828	−	0.425777i	$-0.860001\pi$
$907$	−4.48347e7	−1.80966	−0.904828	+	0.425777i	$0.860001\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.13974e7	−0.454997	−0.227498	−	0.973778i	$-0.573055\pi$
$911$	−1.13974e7	−0.454997	−0.227498	+	0.973778i	$0.573055\pi$
$912$	0	0
$913$	0	0
$914$	1.52916e7	0.605463
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.06716e7	−1.97914	−0.989569	−	0.144059i	$-0.953984\pi$
$919$	−5.06716e7	−1.97914	−0.989569	+	0.144059i	$0.953984\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.77688e7	−1.06709
$926$	7.63617e6	0.292650
$927$	0	0
$928$	3.17107e7	1.20875
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	2.63363e7	0.993152
$933$	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
$940$	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
$946$	2.48353e7	0.902279
$947$	4.63272e6	0.167865	0.0839326	−	0.996471i	$-0.473252\pi$
$947$	4.63272e6	0.167865	0.0839326	+	0.996471i	$0.473252\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.51804e7	1.96812	0.984062	−	0.177824i	$-0.0569057\pi$
$953$	5.51804e7	1.96812	0.984062	+	0.177824i	$0.0569057\pi$
$954$	0	0
$955$	0	0
$956$	−4.02783e7	−1.42536
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.86292e7	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.00222e7	0.688566	0.344283	−	0.938866i	$-0.388122\pi$
$967$	2.00222e7	0.688566	0.344283	+	0.938866i	$0.388122\pi$
$968$	−7.19922e7	−2.46943
$969$	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
$974$	7.30615e6	0.246769
$975$	0	0
$976$	0	0
$977$	3.25961e7	1.09252	0.546260	−	0.837616i	$-0.316051\pi$
$977$	3.25961e7	1.09252	0.546260	+	0.837616i	$0.316051\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−1.59243e7	−0.526964
$983$	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
$988$	0	0
$989$	1.29924e7	0.422375
$990$	0	0
$991$	5.73144e7	1.85387	0.926936	−	0.375219i	$-0.122433\pi$
$991$	5.73144e7	1.85387	0.926936	+	0.375219i	$0.122433\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	2.94217e7	0.935065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.o.1.2	yes	2
3.2	odd	2	inner	441.6.a.o.1.1	✓	2
7.6	odd	2	CM	441.6.a.o.1.2	yes	2
21.20	even	2	inner	441.6.a.o.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.6.a.o.1.1	✓	2	3.2	odd	2	inner
441.6.a.o.1.1	✓	2	21.20	even	2	inner
441.6.a.o.1.2	yes	2	1.1	even	1	trivial
441.6.a.o.1.2	yes	2	7.6	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} -25.0000 q^{4} -150.808 q^{8} +O(q^{10})\)	\(q+2.64575 q^{2} -25.0000 q^{4} -150.808 q^{8} -799.017 q^{11} +401.000 q^{16} -2114.00 q^{22} -1105.92 q^{23} -3125.00 q^{25} +5386.75 q^{29} +5886.80 q^{32} +8886.00 q^{37} -11748.0 q^{43} +19975.4 q^{44} -2926.00 q^{46} -8267.97 q^{50} -32712.1 q^{53} +14252.0 q^{58} +2743.00 q^{64} +69364.0 q^{67} +84923.3 q^{71} +23510.1 q^{74} +80168.0 q^{79} -31082.3 q^{86} +120498. q^{88} +27648.1 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{4}+O(q^{10}) \) 2 * q - 50 * q^4	\( 2 q - 50 q^{4} + 802 q^{16} - 4228 q^{22} - 6250 q^{25} + 17772 q^{37} - 23496 q^{43} - 5852 q^{46} + 28504 q^{58} + 5486 q^{64} + 138728 q^{67} + 160336 q^{79} + 240996 q^{88}+O(q^{100}) \) 2 * q - 50 * q^4 + 802 * q^16 - 4228 * q^22 - 6250 * q^25 + 17772 * q^37 - 23496 * q^43 - 5852 * q^46 + 28504 * q^58 + 5486 * q^64 + 138728 * q^67 + 160336 * q^79 + 240996 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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