LMFDB - Embedded newform 1225.2.b.c.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not 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:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.c.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} -1.00000 q^{16} +3.00000i q^{18} +4.00000i q^{22} -8.00000i q^{23} -2.00000 q^{29} +5.00000i q^{32} +3.00000 q^{36} -6.00000i q^{37} +12.0000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +10.0000i q^{53} -2.00000i q^{58} -7.00000 q^{64} +4.00000i q^{67} +16.0000 q^{71} +9.00000i q^{72} +6.00000 q^{74} -8.00000 q^{79} +9.00000 q^{81} -12.0000 q^{86} +12.0000i q^{88} -8.00000i q^{92} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 + 6 * q^9	$ 2 q + 2 q^{4} + 6 q^{9} + 8 q^{11} - 2 q^{16} - 4 q^{29} + 6 q^{36} + 8 q^{44} + 16 q^{46} - 14 q^{64} + 32 q^{71} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 24 q^{86} + 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^11 - 2 * q^16 - 4 * q^29 + 6 * q^36 + 8 * q^44 + 16 * q^46 - 14 * q^64 + 32 * q^71 + 12 * q^74 - 16 * q^79 + 18 * q^81 - 24 * q^86 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.00000i	1.06066i
$9$	3.00000	1.00000
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	3.00000i	0.707107i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	3.00000	0.500000
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	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
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	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
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	9.00000i	1.06066i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	12.0000i	1.27920i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	− 8.00000i	− 0.834058i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−10.0000	−0.971286
$107$	− 20.0000i	− 1.93347i	−0.255774	−	0.966736i	$-0.582330\pi$
$107$	− 20.0000i	− 1.93347i	0.255774	−	0.966736i	$-0.417670\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	3.00000i	0.265165i
$129$	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
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	16.0000i	1.34269i
$143$	0	0
$144$	−3.00000	−0.250000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000i	0.707107i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	12.0000i	0.914991i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	24.0000	1.76930
$185$	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
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	12.0000i	0.852803i
$199$	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
$207$	− 24.0000i	− 1.66812i
$208$	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	$0.635539\pi$
$212$	10.0000i	0.686803i
$213$	0	0
$214$	20.0000	1.36717
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 18.0000i	− 1.21911i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	5.00000i	0.321412i
$243$	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
$253$	− 32.0000i	− 2.01182i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 32.0000i	− 1.97320i	−0.163144	−	0.986602i	$-0.552164\pi$
$263$	− 32.0000i	− 1.97320i	0.163144	−	0.986602i	$-0.447836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	4.00000i	0.244339i
$269$	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
$274$	10.0000	0.604122
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	0	0
$288$	15.0000i	0.883883i
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	18.0000	1.04623
$297$	0	0
$298$	− 22.0000i	− 1.27443i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 24.0000i	− 1.38104i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	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	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 34.0000i	− 1.90963i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	− 34.0000i	− 1.90963i	0.297200	−	0.954815i	$-0.403947\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	9.00000	0.500000
$325$	0	0
$326$	−20.0000	−1.10770
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	36.0000	1.97874	0.989369	−	0.145424i	$-0.0464545\pi$
$331$	36.0000	1.97874	0.989369	+	0.145424i	$0.0464545\pi$
$332$	0	0
$333$	− 18.0000i	− 0.986394i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.0000i	1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$	30.0000i	1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	13.0000i	0.707107i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−36.0000	−1.94099
$345$	0	0
$346$	0	0
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	20.0000i	1.06600i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	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	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	8.00000i	0.417029i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	18.0000	0.916176
$387$	36.0000i	1.82998i
$388$	0	0
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	26.0000	1.30986
$395$	0	0
$396$	12.0000	0.603023
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 24.0000i	− 1.18964i
$408$	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
$414$	24.0000	1.17954
$415$	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
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	− 12.0000i	− 0.584151i
$423$	0	0
$424$	−30.0000	−1.45693
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 20.0000i	− 0.966736i
$429$	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	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−18.0000	−0.862044
$437$	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
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	0	0
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	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
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	22.0000	1.01913
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.0000i	2.20704i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.0000i	1.37361i
$478$	− 16.0000i	− 0.731823i
$479$	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
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	− 24.0000i	− 1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$	− 24.0000i	− 1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	44.0000	1.98569	0.992846	−	0.119401i	$-0.0380974\pi$
$491$	44.0000	1.98569	0.992846	+	0.119401i	$0.0380974\pi$
$492$	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	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	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
$506$	32.0000	1.42257
$507$	0	0
$508$	16.0000i	0.709885i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 11.0000i	− 0.486136i
$513$	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	$0.5\pi$
$522$	− 6.00000i	− 0.262613i
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	32.0000	1.39527
$527$	0	0
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	− 10.0000i	− 0.427179i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	0	0
$557$	46.0000i	1.94908i	0.224208	+	0.974541i	$0.428020\pi$
$557$	46.0000i	1.94908i	−0.224208	+	0.974541i	$0.571980\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	− 26.0000i	− 1.09674i
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	48.0000i	2.01404i
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−21.0000	−0.875000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	17.0000i	0.707107i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	40.0000i	1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	6.00000i	0.246598i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	−22.0000	−0.901155
$597$	0	0
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	−24.0000	−0.976546
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	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
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	− 24.0000i	− 0.954669i
$633$	0	0
$634$	34.0000	1.35031
$635$	0	0
$636$	0	0
$637$	0	0
$638$	− 8.00000i	− 0.316723i
$639$	48.0000	1.89885
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	27.0000i	1.06066i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	− 50.0000i	− 1.95665i	−0.207072	−	0.978326i	$-0.566394\pi$
$653$	− 50.0000i	− 1.95665i	0.207072	−	0.978326i	$-0.433606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	36.0000i	1.39918i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	18.0000	0.697486
$667$	16.0000i	0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 30.0000i	− 1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$	− 30.0000i	− 1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	−30.0000	−1.15556
$675$	0	0
$676$	13.0000	0.500000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	52.0000i	1.98972i	0.101237	+	0.994862i	$0.467720\pi$
$683$	52.0000i	1.98972i	−0.101237	+	0.994862i	$0.532280\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 12.0000i	− 0.457496i
$689$	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
$694$	−4.00000	−0.151838
$695$	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
$704$	−28.0000	−1.05529
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	− 8.00000i	− 0.298557i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	40.0000	1.47442
$737$	16.0000i	0.589368i
$738$	0	0
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.0000i	1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$	40.0000i	1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	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
$757$	− 54.0000i	− 1.96266i	−0.192323	−	0.981332i	$-0.561602\pi$
$757$	− 54.0000i	− 1.96266i	0.192323	−	0.981332i	$-0.438398\pi$
$758$	12.0000i	0.435860i
$759$	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
$764$	8.00000	0.289430
$765$	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
$772$	− 18.0000i	− 0.647834i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−36.0000	−1.29399
$775$	0	0
$776$	0	0
$777$	0	0
$778$	38.0000i	1.36237i
$779$	0	0
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 26.0000i	− 0.926212i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	36.0000i	1.27920i
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 34.0000i	− 1.20058i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	24.0000	0.841200
$815$	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
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	− 24.0000i	− 0.834058i
$829$	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
$841$	−25.0000	−0.862069
$842$	− 26.0000i	− 0.896019i
$843$	0	0
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	60.0000	2.05076
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	32.0000i	1.08992i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	0	0
$872$	− 54.0000i	− 1.82867i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	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
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	0	0
$885$	0	0
$886$	−20.0000	−0.671913
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	36.0000	1.20605
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 2.00000i	− 0.0667409i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	0	0
$907$	60.0000i	1.99227i	0.0878507	+	0.996134i	$0.472000\pi$
$907$	60.0000i	1.99227i	−0.0878507	+	0.996134i	$0.528000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	0	0
$914$	6.00000	0.198462
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−40.0000	−1.31448
$927$	0	0
$928$	− 10.0000i	− 0.328266i
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	− 22.0000i	− 0.720634i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	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
$946$	−48.0000	−1.56061
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 58.0000i	− 1.87880i	−0.342817	−	0.939402i	$-0.611381\pi$
$953$	− 58.0000i	− 1.87880i	0.342817	−	0.939402i	$-0.388619\pi$
$954$	−30.0000	−0.971286
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 60.0000i	− 1.93347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	15.0000i	0.482118i
$969$	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
$974$	24.0000	0.769010
$975$	0	0
$976$	0	0
$977$	46.0000i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$	46.0000i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−54.0000	−1.72409
$982$	44.0000i	1.40410i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	96.0000	3.05262
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 36.0000i	− 1.13956i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.c.99.2		2
5.2	odd	4		1225.2.a.c.1.1		1
5.3	odd	4		49.2.a.a.1.1	✓	1
5.4	even	2	inner	1225.2.b.c.99.1		2
7.6	odd	2	CM	1225.2.b.c.99.2		2
15.8	even	4		441.2.a.c.1.1		1
20.3	even	4		784.2.a.f.1.1		1
35.3	even	12		49.2.c.a.30.1		2
35.13	even	4		49.2.a.a.1.1	✓	1
35.18	odd	12		49.2.c.a.30.1		2
35.23	odd	12		49.2.c.a.18.1		2
35.27	even	4		1225.2.a.c.1.1		1
35.33	even	12		49.2.c.a.18.1		2
35.34	odd	2	inner	1225.2.b.c.99.1		2
40.3	even	4		3136.2.a.o.1.1		1
40.13	odd	4		3136.2.a.n.1.1		1
55.43	even	4		5929.2.a.c.1.1		1
60.23	odd	4		7056.2.a.bg.1.1		1
65.38	odd	4		8281.2.a.d.1.1		1
105.23	even	12		441.2.e.d.361.1		2
105.38	odd	12		441.2.e.d.226.1		2
105.53	even	12		441.2.e.d.226.1		2
105.68	odd	12		441.2.e.d.361.1		2
105.83	odd	4		441.2.a.c.1.1		1
140.3	odd	12		784.2.i.f.177.1		2
140.23	even	12		784.2.i.f.753.1		2
140.83	odd	4		784.2.a.f.1.1		1
140.103	odd	12		784.2.i.f.753.1		2
140.123	even	12		784.2.i.f.177.1		2
280.13	even	4		3136.2.a.n.1.1		1
280.83	odd	4		3136.2.a.o.1.1		1
385.153	odd	4		5929.2.a.c.1.1		1
420.83	even	4		7056.2.a.bg.1.1		1
455.363	even	4		8281.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.2.a.a.1.1	✓	1	5.3	odd	4
49.2.a.a.1.1	✓	1	35.13	even	4
49.2.c.a.18.1		2	35.23	odd	12
49.2.c.a.18.1		2	35.33	even	12
49.2.c.a.30.1		2	35.3	even	12
49.2.c.a.30.1		2	35.18	odd	12
441.2.a.c.1.1		1	15.8	even	4
441.2.a.c.1.1		1	105.83	odd	4
441.2.e.d.226.1		2	105.38	odd	12
441.2.e.d.226.1		2	105.53	even	12
441.2.e.d.361.1		2	105.23	even	12
441.2.e.d.361.1		2	105.68	odd	12
784.2.a.f.1.1		1	20.3	even	4
784.2.a.f.1.1		1	140.83	odd	4
784.2.i.f.177.1		2	140.3	odd	12
784.2.i.f.177.1		2	140.123	even	12
784.2.i.f.753.1		2	140.23	even	12
784.2.i.f.753.1		2	140.103	odd	12
1225.2.a.c.1.1		1	5.2	odd	4
1225.2.a.c.1.1		1	35.27	even	4
1225.2.b.c.99.1		2	5.4	even	2	inner
1225.2.b.c.99.1		2	35.34	odd	2	inner
1225.2.b.c.99.2		2	1.1	even	1	trivial
1225.2.b.c.99.2		2	7.6	odd	2	CM
3136.2.a.n.1.1		1	40.13	odd	4
3136.2.a.n.1.1		1	280.13	even	4
3136.2.a.o.1.1		1	40.3	even	4
3136.2.a.o.1.1		1	280.83	odd	4
5929.2.a.c.1.1		1	55.43	even	4
5929.2.a.c.1.1		1	385.153	odd	4
7056.2.a.bg.1.1		1	60.23	odd	4
7056.2.a.bg.1.1		1	420.83	even	4
8281.2.a.d.1.1		1	65.38	odd	4
8281.2.a.d.1.1		1	455.363	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} -1.00000 q^{16} +3.00000i q^{18} +4.00000i q^{22} -8.00000i q^{23} -2.00000 q^{29} +5.00000i q^{32} +3.00000 q^{36} -6.00000i q^{37} +12.0000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +10.0000i q^{53} -2.00000i q^{58} -7.00000 q^{64} +4.00000i q^{67} +16.0000 q^{71} +9.00000i q^{72} +6.00000 q^{74} -8.00000 q^{79} +9.00000 q^{81} -12.0000 q^{86} +12.0000i q^{88} -8.00000i q^{92} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 6 * q^9	\( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{11} - 2 q^{16} - 4 q^{29} + 6 q^{36} + 8 q^{44} + 16 q^{46} - 14 q^{64} + 32 q^{71} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 24 q^{86} + 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^11 - 2 * q^16 - 4 * q^29 + 6 * q^36 + 8 * q^44 + 16 * q^46 - 14 * q^64 + 32 * q^71 + 12 * q^74 - 16 * q^79 + 18 * q^81 - 24 * q^86 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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