LMFDB - Embedded newform 1225.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
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			$2.79129$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79129 q^{2} +5.79129 q^{4} +10.5826 q^{8} -3.00000 q^{9} +O(q^{10})$	$q+2.79129 q^{2} +5.79129 q^{4} +10.5826 q^{8} -3.00000 q^{9} +2.58258 q^{11} +17.9564 q^{16} -8.37386 q^{18} +7.20871 q^{22} -0.582576 q^{23} -10.1652 q^{29} +28.9564 q^{32} -17.3739 q^{36} +6.16515 q^{37} -10.5826 q^{43} +14.9564 q^{44} -1.62614 q^{46} +10.0000 q^{53} -28.3739 q^{58} +44.9129 q^{64} -11.7477 q^{67} -12.5826 q^{71} -31.7477 q^{72} +17.2087 q^{74} -17.7477 q^{79} +9.00000 q^{81} -29.5390 q^{86} +27.3303 q^{88} -3.37386 q^{92} -7.74773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9	$ 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9} - 4 q^{11} + 13 q^{16} - 3 q^{18} + 19 q^{22} + 8 q^{23} - 2 q^{29} + 35 q^{32} - 21 q^{36} - 6 q^{37} - 12 q^{43} + 7 q^{44} - 17 q^{46} + 20 q^{53} - 43 q^{58} + 44 q^{64} + 4 q^{67} - 16 q^{71} - 36 q^{72} + 39 q^{74} - 8 q^{79} + 18 q^{81} - 27 q^{86} + 18 q^{88} + 7 q^{92} + 12 q^{99}+O(q^{100}) $ 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9 - 4 * q^11 + 13 * q^16 - 3 * q^18 + 19 * q^22 + 8 * q^23 - 2 * q^29 + 35 * q^32 - 21 * q^36 - 6 * q^37 - 12 * q^43 + 7 * q^44 - 17 * q^46 + 20 * q^53 - 43 * q^58 + 44 * q^64 + 4 * q^67 - 16 * q^71 - 36 * q^72 + 39 * q^74 - 8 * q^79 + 18 * q^81 - 27 * q^86 + 18 * q^88 + 7 * q^92 + 12 * q^99

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 $

$2$	2.79129	1.97374	0.986869	−	0.161521i	$-0.0516399\pi$
$2$	2.79129	1.97374	0.986869	+	0.161521i	$0.0516399\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	5.79129	2.89564
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	10.5826	3.74151
$9$	−3.00000	−1.00000
$10$	0	0
$11$	2.58258	0.778676	0.389338	−	0.921095i	$-0.372704\pi$
$11$	2.58258	0.778676	0.389338	+	0.921095i	$0.372704\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	17.9564	4.48911
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−8.37386	−1.97374
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	7.20871	1.53690
$23$	−0.582576	−0.121475	−0.0607377	−	0.998154i	$-0.519345\pi$
$23$	−0.582576	−0.121475	−0.0607377	+	0.998154i	$0.519345\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.1652	−1.88762	−0.943811	−	0.330487i	$-0.892787\pi$
$29$	−10.1652	−1.88762	−0.943811	+	0.330487i	$0.892787\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	28.9564	5.11882
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−17.3739	−2.89564
$37$	6.16515	1.01354	0.506772	−	0.862080i	$-0.330838\pi$
$37$	6.16515	1.01354	0.506772	+	0.862080i	$0.330838\pi$
$38$	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
$43$	−10.5826	−1.61383	−0.806914	−	0.590669i	$-0.798864\pi$
$43$	−10.5826	−1.61383	−0.806914	+	0.590669i	$0.798864\pi$
$44$	14.9564	2.25477
$45$	0	0
$46$	−1.62614	−0.239761
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−28.3739	−3.72567
$59$	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
$64$	44.9129	5.61411
$65$	0	0
$66$	0	0
$67$	−11.7477	−1.43521	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	−11.7477	−1.43521	−0.717607	+	0.696449i	$0.754762\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.5826	−1.49328	−0.746639	−	0.665230i	$-0.768333\pi$
$71$	−12.5826	−1.49328	−0.746639	+	0.665230i	$0.768333\pi$
$72$	−31.7477	−3.74151
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	17.2087	2.00047
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−17.7477	−1.99678	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−17.7477	−1.99678	−0.998388	+	0.0567635i	$0.981922\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−29.5390	−3.18527
$87$	0	0
$88$	27.3303	2.91342
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−3.37386	−0.351750
$93$	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
$99$	−7.74773	−0.778676
$100$	0	0
$101$	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
$106$	27.9129	2.71114
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	0	0
$109$	0.165151	0.0158186	0.00790932	−	0.999969i	$-0.497482\pi$
$109$	0.165151	0.0158186	0.00790932	+	0.999969i	$0.497482\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.3303	−1.63030	−0.815149	−	0.579252i	$-0.803345\pi$
$113$	−17.3303	−1.63030	−0.815149	+	0.579252i	$0.803345\pi$
$114$	0	0
$115$	0	0
$116$	−58.8693	−5.46588
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.33030	−0.393664
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	21.7477	1.92980	0.964899	−	0.262620i	$-0.0845865\pi$
$127$	21.7477	1.92980	0.964899	+	0.262620i	$0.0845865\pi$
$128$	67.4519	5.96196
$129$	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
$134$	−32.7913	−2.83274
$135$	0	0
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	−35.1216	−2.94734
$143$	0	0
$144$	−53.8693	−4.48911
$145$	0	0
$146$	0	0
$147$	0	0
$148$	35.7042	2.93486
$149$	−20.1652	−1.65199	−0.825997	−	0.563675i	$-0.809387\pi$
$149$	−20.1652	−1.65199	−0.825997	+	0.563675i	$0.809387\pi$
$150$	0	0
$151$	7.41742	0.603621	0.301811	−	0.953368i	$-0.402409\pi$
$151$	7.41742	0.603621	0.301811	+	0.953368i	$0.402409\pi$
$152$	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	$0.5\pi$
$158$	−49.5390	−3.94111
$159$	0	0
$160$	0	0
$161$	0	0
$162$	25.1216	1.97374
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	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
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−61.2867	−4.67307
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	46.3739	3.49556
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−6.16515	−0.454501
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	27.3303	1.96728	0.983639	−	0.180150i	$-0.0576584\pi$
$193$	27.3303	1.96728	0.983639	+	0.180150i	$0.0576584\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.83485	−0.273222	−0.136611	−	0.990625i	$-0.543621\pi$
$197$	−3.83485	−0.273222	−0.136611	+	0.990625i	$0.543621\pi$
$198$	−21.6261	−1.53690
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.74773	0.121475
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	57.9129	3.97747
$213$	0	0
$214$	55.8258	3.81617
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0.460985	0.0312218
$219$	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
$226$	−48.3739	−3.21778
$227$	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
$232$	−107.573	−7.06255
$233$	−7.33030	−0.480224	−0.240112	−	0.970745i	$-0.577184\pi$
$233$	−7.33030	−0.480224	−0.240112	+	0.970745i	$0.577184\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−12.0871	−0.776990
$243$	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
$253$	−1.50455	−0.0945900
$254$	60.7042	3.80892
$255$	0	0
$256$	98.4519	6.15324
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	30.4955	1.88762
$262$	0	0
$263$	20.5826	1.26918	0.634588	−	0.772851i	$-0.281170\pi$
$263$	20.5826	1.26918	0.634588	+	0.772851i	$0.281170\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−68.0345	−4.15587
$269$	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
$274$	27.9129	1.68628
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.33030	−0.317979	−0.158990	−	0.987280i	$-0.550824\pi$
$281$	−5.33030	−0.317979	−0.158990	+	0.987280i	$0.550824\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−72.8693	−4.32400
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−86.8693	−5.11882
$289$	−17.0000	−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
$296$	65.2432	3.79218
$297$	0	0
$298$	−56.2867	−3.26060
$299$	0	0
$300$	0	0
$301$	0	0
$302$	20.7042	1.19139
$303$	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
$316$	−102.782	−5.78195
$317$	−26.1652	−1.46958	−0.734791	−	0.678294i	$-0.762720\pi$
$317$	−26.1652	−1.46958	−0.734791	+	0.678294i	$0.762720\pi$
$318$	0	0
$319$	−26.2523	−1.46985
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	52.1216	2.89564
$325$	0	0
$326$	55.8258	3.09190
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.5826	−1.24125	−0.620625	−	0.784107i	$-0.713121\pi$
$331$	−22.5826	−1.24125	−0.620625	+	0.784107i	$0.713121\pi$
$332$	0	0
$333$	−18.4955	−1.01354
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	−36.2867	−1.97374
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−111.991	−6.03815
$345$	0	0
$346$	0	0
$347$	34.0780	1.82940	0.914702	−	0.404128i	$-0.132425\pi$
$347$	34.0780	1.82940	0.914702	+	0.404128i	$0.132425\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	74.7822	3.98590
$353$	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
$358$	11.1652	0.590097
$359$	28.0780	1.48190	0.740951	−	0.671559i	$-0.234375\pi$
$359$	28.0780	1.48190	0.740951	+	0.671559i	$0.234375\pi$
$360$	0	0
$361$	−19.0000	−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	$0.5\pi$
$368$	−10.4610	−0.545317
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	38.4955	1.99322	0.996610	−	0.0822766i	$-0.0262191\pi$
$373$	38.4955	1.99322	0.996610	+	0.0822766i	$0.0262191\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	38.0780	1.95594	0.977969	−	0.208752i	$-0.0669403\pi$
$379$	38.0780	1.95594	0.977969	+	0.208752i	$0.0669403\pi$
$380$	0	0
$381$	0	0
$382$	22.3303	1.14252
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	76.2867	3.88289
$387$	31.7477	1.61383
$388$	0	0
$389$	9.83485	0.498647	0.249323	−	0.968420i	$-0.419792\pi$
$389$	9.83485	0.498647	0.249323	+	0.968420i	$0.419792\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−10.7042	−0.539268
$395$	0	0
$396$	−44.8693	−2.25477
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.3303	1.76431	0.882156	−	0.470958i	$-0.156092\pi$
$401$	35.3303	1.76431	0.882156	+	0.470958i	$0.156092\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.9220	0.789223
$408$	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
$414$	4.87841	0.239761
$415$	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
$421$	40.4955	1.97363	0.986814	−	0.161859i	$-0.0517491\pi$
$421$	40.4955	1.97363	0.986814	+	0.161859i	$0.0517491\pi$
$422$	−33.4955	−1.63053
$423$	0	0
$424$	105.826	5.13935
$425$	0	0
$426$	0	0
$427$	0	0
$428$	115.826	5.59865
$429$	0	0
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0.956439	0.0458051
$437$	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
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	35.6606	1.68293	0.841464	−	0.540313i	$-0.181694\pi$
$449$	35.6606	1.68293	0.841464	+	0.540313i	$0.181694\pi$
$450$	0	0
$451$	0	0
$452$	−100.365	−4.72076
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.6606	−1.85524	−0.927622	−	0.373519i	$-0.878151\pi$
$457$	−39.6606	−1.85524	−0.927622	+	0.373519i	$0.878151\pi$
$458$	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
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	−182.530	−8.47374
$465$	0	0
$466$	−20.4610	−0.947837
$467$	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
$473$	−27.3303	−1.25665
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.0000	−1.37361
$478$	44.6606	2.04273
$479$	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
$484$	−25.0780	−1.13991
$485$	0	0
$486$	0	0
$487$	−44.0780	−1.99737	−0.998683	−	0.0513038i	$-0.983662\pi$
$487$	−44.0780	−1.99737	−0.998683	+	0.0513038i	$0.983662\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.4174	−0.786037	−0.393019	−	0.919530i	$-0.628569\pi$
$491$	−17.4174	−0.786037	−0.393019	+	0.919530i	$0.628569\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	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
$506$	−4.19962	−0.186696
$507$	0	0
$508$	125.947	5.58801
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	139.904	6.18293
$513$	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	$0.5\pi$
$522$	85.1216	3.72567
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	57.4519	2.50502
$527$	0	0
$528$	0	0
$529$	−22.6606	−0.985244
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−124.321	−5.36986
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.4955	−0.451235	−0.225617	−	0.974216i	$-0.572440\pi$
$541$	−10.4955	−0.451235	−0.225617	+	0.974216i	$0.572440\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.25227	0.352842	0.176421	−	0.984315i	$-0.443548\pi$
$547$	8.25227	0.352842	0.176421	+	0.984315i	$0.443548\pi$
$548$	57.9129	2.47392
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	27.9129	1.18590
$555$	0	0
$556$	0	0
$557$	13.8348	0.586201	0.293101	−	0.956082i	$-0.405313\pi$
$557$	13.8348	0.586201	0.293101	+	0.956082i	$0.405313\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−14.8784	−0.627608
$563$	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
$568$	−133.156	−5.58710
$569$	25.6606	1.07575	0.537874	−	0.843025i	$-0.319228\pi$
$569$	25.6606	1.07575	0.537874	+	0.843025i	$0.319228\pi$
$570$	0	0
$571$	−43.2432	−1.80967	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−43.2432	−1.80967	−0.904835	+	0.425762i	$0.860006\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−134.739	−5.61411
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−47.4519	−1.97374
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	25.8258	1.06959
$584$	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
$592$	110.704	4.54991
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−116.782	−4.78359
$597$	0	0
$598$	0	0
$599$	−48.0780	−1.96442	−0.982208	−	0.187799i	$-0.939865\pi$
$599$	−48.0780	−1.96442	−0.982208	+	0.187799i	$0.939865\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	35.2432	1.43521
$604$	42.9564	1.74787
$605$	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
$613$	8.49545	0.343128	0.171564	−	0.985173i	$-0.445118\pi$
$613$	8.49545	0.343128	0.171564	+	0.985173i	$0.445118\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−49.6606	−1.99926	−0.999630	−	0.0271876i	$-0.991345\pi$
$617$	−49.6606	−1.99926	−0.999630	+	0.0271876i	$0.991345\pi$
$618$	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
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	33.2432	1.32339	0.661695	−	0.749773i	$-0.269837\pi$
$631$	33.2432	1.32339	0.661695	+	0.749773i	$0.269837\pi$
$632$	−187.817	−7.47095
$633$	0	0
$634$	−73.0345	−2.90057
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−73.2777	−2.90109
$639$	37.7477	1.49328
$640$	0	0
$641$	−4.66970	−0.184442	−0.0922210	−	0.995739i	$-0.529397\pi$
$641$	−4.66970	−0.184442	−0.0922210	+	0.995739i	$0.529397\pi$
$642$	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	$0.5\pi$
$648$	95.2432	3.74151
$649$	0	0
$650$	0	0
$651$	0	0
$652$	115.826	4.53609
$653$	−50.0000	−1.95665	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−50.0000	−1.95665	−0.978326	+	0.207072i	$0.933606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−63.0345	−2.44990
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−51.6261	−2.00047
$667$	5.92197	0.229300
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−83.7386	−3.22549
$675$	0	0
$676$	−75.2867	−2.89564
$677$	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
$683$	−30.5826	−1.17021	−0.585105	−	0.810958i	$-0.698947\pi$
$683$	−30.5826	−1.17021	−0.585105	+	0.810958i	$0.698947\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−190.025	−7.24465
$689$	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
$694$	95.1216	3.61077
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	115.991	4.37157
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	53.2432	1.99678
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	23.1652	0.865722
$717$	0	0
$718$	78.3739	2.92489
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−53.0345	−1.97374
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	−16.8693	−0.621811
$737$	−30.3394	−1.11757
$738$	0	0
$739$	12.2523	0.450707	0.225354	−	0.974277i	$-0.427646\pi$
$739$	12.2523	0.450707	0.225354	+	0.974277i	$0.427646\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	0	0
$746$	107.452	3.93409
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.1652	−1.31444	−0.657222	−	0.753697i	$-0.728269\pi$
$757$	−36.1652	−1.31444	−0.657222	+	0.753697i	$0.728269\pi$
$758$	106.287	3.86051
$759$	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
$764$	46.3303	1.67617
$765$	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
$772$	158.278	5.69654
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	88.6170	3.18527
$775$	0	0
$776$	0	0
$777$	0	0
$778$	27.4519	0.984198
$779$	0	0
$780$	0	0
$781$	−32.4955	−1.16278
$782$	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	$0.5\pi$
$788$	−22.2087	−0.791153
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−81.9909	−2.91342
$793$	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
$800$	0	0
$801$	0	0
$802$	98.6170	3.48229
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	55.6606	1.95692	0.978461	−	0.206430i	$-0.0661846\pi$
$809$	55.6606	1.95692	0.978461	+	0.206430i	$0.0661846\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	44.4428	1.55772
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	−25.2432	−0.879922	−0.439961	−	0.898017i	$-0.645008\pi$
$823$	−25.2432	−0.879922	−0.439961	+	0.898017i	$0.645008\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.0780	1.88048	0.940239	−	0.340516i	$-0.110602\pi$
$827$	54.0780	1.88048	0.940239	+	0.340516i	$0.110602\pi$
$828$	10.1216	0.351750
$829$	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
$841$	74.3303	2.56311
$842$	113.034	3.89543
$843$	0	0
$844$	−69.4955	−2.39213
$845$	0	0
$846$	0	0
$847$	0	0
$848$	179.564	6.16627
$849$	0	0
$850$	0	0
$851$	−3.59167	−0.123121
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	211.652	7.23410
$857$	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
$862$	89.3212	3.04229
$863$	−46.4083	−1.57976	−0.789879	−	0.613263i	$-0.789857\pi$
$863$	−46.4083	−1.57976	−0.789879	+	0.613263i	$0.789857\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−45.8348	−1.55484
$870$	0	0
$871$	0	0
$872$	1.74773	0.0591855
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	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
$883$	−56.4083	−1.89829	−0.949146	−	0.314837i	$-0.898050\pi$
$883$	−56.4083	−1.89829	−0.949146	+	0.314837i	$0.898050\pi$
$884$	0	0
$885$	0	0
$886$	55.8258	1.87550
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	23.2432	0.778676
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	99.5390	3.32166
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−183.399	−6.09977
$905$	0	0
$906$	0	0
$907$	−60.0000	−1.99227	−0.996134	−	0.0878507i	$-0.972000\pi$
$907$	−60.0000	−1.99227	−0.996134	+	0.0878507i	$0.972000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−58.4083	−1.93515	−0.967577	−	0.252575i	$-0.918722\pi$
$911$	−58.4083	−1.93515	−0.967577	+	0.252575i	$0.918722\pi$
$912$	0	0
$913$	0	0
$914$	−110.704	−3.66177
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.07803	−0.266470	−0.133235	−	0.991084i	$-0.542536\pi$
$919$	−8.07803	−0.266470	−0.133235	+	0.991084i	$0.542536\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	111.652	3.66910
$927$	0	0
$928$	−294.347	−9.66240
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	−42.4519	−1.39056
$933$	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
$946$	−76.2867	−2.48030
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	47.3303	1.53318	0.766589	−	0.642138i	$-0.221952\pi$
$953$	47.3303	1.53318	0.766589	+	0.642138i	$0.221952\pi$
$954$	−83.7386	−2.71114
$955$	0	0
$956$	92.6606	2.99686
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−60.0000	−1.93347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	−45.8258	−1.47290
$969$	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
$974$	−123.034	−3.94228
$975$	0	0
$976$	0	0
$977$	59.6606	1.90871	0.954356	−	0.298672i	$-0.0965435\pi$
$977$	59.6606	1.90871	0.954356	+	0.298672i	$0.0965435\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−0.495454	−0.0158186
$982$	−48.6170	−1.55143
$983$	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
$989$	6.16515	0.196040
$990$	0	0
$991$	−38.4083	−1.22008	−0.610040	−	0.792370i	$-0.708847\pi$
$991$	−38.4083	−1.22008	−0.610040	+	0.792370i	$0.708847\pi$
$992$	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	$0.5\pi$
$998$	100.486	3.18084
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.t.1.2	yes	2
5.2	odd	4		1225.2.b.e.99.4		4
5.3	odd	4		1225.2.b.e.99.1		4
5.4	even	2		1225.2.a.o.1.1	✓	2
7.6	odd	2	CM	1225.2.a.t.1.2	yes	2
35.13	even	4		1225.2.b.e.99.1		4
35.27	even	4		1225.2.b.e.99.4		4
35.34	odd	2		1225.2.a.o.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.2.a.o.1.1	✓	2	5.4	even	2
1225.2.a.o.1.1	✓	2	35.34	odd	2
1225.2.a.t.1.2	yes	2	1.1	even	1	trivial
1225.2.a.t.1.2	yes	2	7.6	odd	2	CM
1225.2.b.e.99.1		4	5.3	odd	4
1225.2.b.e.99.1		4	35.13	even	4
1225.2.b.e.99.4		4	5.2	odd	4
1225.2.b.e.99.4		4	35.27	even	4

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79129 q^{2} +5.79129 q^{4} +10.5826 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+2.79129 q^{2} +5.79129 q^{4} +10.5826 q^{8} -3.00000 q^{9} +2.58258 q^{11} +17.9564 q^{16} -8.37386 q^{18} +7.20871 q^{22} -0.582576 q^{23} -10.1652 q^{29} +28.9564 q^{32} -17.3739 q^{36} +6.16515 q^{37} -10.5826 q^{43} +14.9564 q^{44} -1.62614 q^{46} +10.0000 q^{53} -28.3739 q^{58} +44.9129 q^{64} -11.7477 q^{67} -12.5826 q^{71} -31.7477 q^{72} +17.2087 q^{74} -17.7477 q^{79} +9.00000 q^{81} -29.5390 q^{86} +27.3303 q^{88} -3.37386 q^{92} -7.74773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9	\( 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9} - 4 q^{11} + 13 q^{16} - 3 q^{18} + 19 q^{22} + 8 q^{23} - 2 q^{29} + 35 q^{32} - 21 q^{36} - 6 q^{37} - 12 q^{43} + 7 q^{44} - 17 q^{46} + 20 q^{53} - 43 q^{58} + 44 q^{64} + 4 q^{67} - 16 q^{71} - 36 q^{72} + 39 q^{74} - 8 q^{79} + 18 q^{81} - 27 q^{86} + 18 q^{88} + 7 q^{92} + 12 q^{99}+O(q^{100}) \) 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9 - 4 * q^11 + 13 * q^16 - 3 * q^18 + 19 * q^22 + 8 * q^23 - 2 * q^29 + 35 * q^32 - 21 * q^36 - 6 * q^37 - 12 * q^43 + 7 * q^44 - 17 * q^46 + 20 * q^53 - 43 * q^58 + 44 * q^64 + 4 * q^67 - 16 * q^71 - 36 * q^72 + 39 * q^74 - 8 * q^79 + 18 * q^81 - 27 * q^86 + 18 * q^88 + 7 * q^92 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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