LMFDB - Embedded newform 475.3.d.a.474.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(474,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("475.474");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.d (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:	$12.9428125571$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			474.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	475.474
Dual form			475.3.d.a.474.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{4} +5.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-4.00000 q^{4} +5.00000i q^{7} -9.00000 q^{9} +3.00000 q^{11} +16.0000 q^{16} -15.0000i q^{17} +19.0000 q^{19} -30.0000i q^{23} -20.0000i q^{28} +36.0000 q^{36} -85.0000i q^{43} -12.0000 q^{44} -75.0000i q^{47} +24.0000 q^{49} +103.000 q^{61} -45.0000i q^{63} -64.0000 q^{64} +60.0000i q^{68} -25.0000i q^{73} -76.0000 q^{76} +15.0000i q^{77} +81.0000 q^{81} +90.0000i q^{83} +120.000i q^{92} -27.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 18 q^{9}+O(q^{10}) $ 2 * q - 8 * q^4 - 18 * q^9	$ 2 q - 8 q^{4} - 18 q^{9} + 6 q^{11} + 32 q^{16} + 38 q^{19} + 72 q^{36} - 24 q^{44} + 48 q^{49} + 206 q^{61} - 128 q^{64} - 152 q^{76} + 162 q^{81} - 54 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 18 * q^9 + 6 * q^11 + 32 * q^16 + 38 * q^19 + 72 * q^36 - 24 * q^44 + 48 * q^49 + 206 * q^61 - 128 * q^64 - 152 * q^76 + 162 * q^81 - 54 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000i	0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$	5.00000i	0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	3.00000	0.272727	0.136364	−	0.990659i	$-0.456458\pi$
$11$	3.00000	0.272727	0.136364	+	0.990659i	$0.456458\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	− 15.0000i	− 0.882353i	−0.897420	−	0.441176i	$-0.854561\pi$
$17$	− 15.0000i	− 0.882353i	0.897420	−	0.441176i	$-0.145439\pi$
$18$	0	0
$19$	19.0000	1.00000
$19$	19.0000	1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 30.0000i	− 1.30435i	−0.758069	−	0.652174i	$-0.773857\pi$
$23$	− 30.0000i	− 1.30435i	0.758069	−	0.652174i	$-0.226143\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	− 20.0000i	− 0.714286i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	36.0000	1.00000
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 85.0000i	− 1.97674i	−0.152055	−	0.988372i	$-0.548589\pi$
$43$	− 85.0000i	− 1.97674i	0.152055	−	0.988372i	$-0.451411\pi$
$44$	−12.0000	−0.272727
$45$	0	0
$46$	0	0
$47$	− 75.0000i	− 1.59574i	−0.602826	−	0.797872i	$-0.705959\pi$
$47$	− 75.0000i	− 1.59574i	0.602826	−	0.797872i	$-0.294041\pi$
$48$	0	0
$49$	24.0000	0.489796
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	103.000	1.68852	0.844262	−	0.535930i	$-0.180039\pi$
$61$	103.000	1.68852	0.844262	+	0.535930i	$0.180039\pi$
$62$	0	0
$63$	− 45.0000i	− 0.714286i
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	60.0000i	0.882353i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 25.0000i	− 0.342466i	−0.985231	−	0.171233i	$-0.945225\pi$
$73$	− 25.0000i	− 0.342466i	0.985231	−	0.171233i	$-0.0547750\pi$
$74$	0	0
$75$	0	0
$76$	−76.0000	−1.00000
$77$	15.0000i	0.194805i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	90.0000i	1.08434i	0.840270	+	0.542169i	$0.182397\pi$
$83$	90.0000i	1.08434i	−0.840270	+	0.542169i	$0.817603\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	120.000i	1.30435i
$93$	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
$99$	−27.0000	−0.272727
$100$	0	0
$101$	−102.000	−1.00990	−0.504950	−	0.863148i	$-0.668489\pi$
$101$	−102.000	−1.00990	−0.504950	+	0.863148i	$0.668489\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	80.0000i	0.714286i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	75.0000	0.630252
$120$	0	0
$121$	−112.000	−0.925620
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−213.000	−1.62595	−0.812977	−	0.582296i	$-0.802155\pi$
$131$	−213.000	−1.62595	−0.812977	+	0.582296i	$0.802155\pi$
$132$	0	0
$133$	95.0000i	0.714286i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 255.000i	− 1.86131i	−0.365893	−	0.930657i	$-0.619236\pi$
$137$	− 255.000i	− 1.86131i	0.365893	−	0.930657i	$-0.380764\pi$
$138$	0	0
$139$	197.000	1.41727	0.708633	−	0.705577i	$-0.249312\pi$
$139$	197.000	1.41727	0.708633	+	0.705577i	$0.249312\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−144.000	−1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	177.000	1.18792	0.593960	−	0.804495i	$-0.297564\pi$
$149$	177.000	1.18792	0.593960	+	0.804495i	$0.297564\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	135.000i	0.882353i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.0636943i	−0.999493	−	0.0318471i	$-0.989861\pi$
$157$	− 10.0000i	− 0.0636943i	0.999493	−	0.0318471i	$-0.0101390\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	150.000	0.931677
$162$	0	0
$163$	250.000i	1.53374i	0.641801	+	0.766871i	$0.278187\pi$
$163$	250.000i	1.53374i	−0.641801	+	0.766871i	$0.721813\pi$
$164$	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
$169$	−169.000	−1.00000
$170$	0	0
$171$	−171.000	−1.00000
$172$	340.000i	1.97674i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	48.0000	0.272727
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 45.0000i	− 0.240642i
$188$	300.000i	1.59574i
$189$	0	0
$190$	0	0
$191$	−93.0000	−0.486911	−0.243455	−	0.969912i	$-0.578281\pi$
$191$	−93.0000	−0.486911	−0.243455	+	0.969912i	$0.578281\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	−96.0000	−0.489796
$197$	− 90.0000i	− 0.456853i	−0.973561	−	0.228426i	$-0.926642\pi$
$197$	− 90.0000i	− 0.456853i	0.973561	−	0.228426i	$-0.0733580\pi$
$198$	0	0
$199$	−227.000	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$199$	−227.000	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	270.000i	1.30435i
$208$	0	0
$209$	57.0000	0.272727
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	17.0000	0.0742358	0.0371179	−	0.999311i	$-0.488182\pi$
$229$	17.0000	0.0742358	0.0371179	+	0.999311i	$0.488182\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 465.000i	− 1.99571i	−0.0654770	−	0.997854i	$-0.520857\pi$
$233$	− 465.000i	− 1.99571i	0.0654770	−	0.997854i	$-0.479143\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	453.000	1.89540	0.947699	−	0.319166i	$-0.103403\pi$
$239$	453.000	1.89540	0.947699	+	0.319166i	$0.103403\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	−412.000	−1.68852
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.0000	0.107570	0.0537849	−	0.998553i	$-0.482871\pi$
$251$	27.0000	0.107570	0.0537849	+	0.998553i	$0.482871\pi$
$252$	180.000i	0.714286i
$253$	− 90.0000i	− 0.355731i
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 405.000i	− 1.53992i	−0.638090	−	0.769962i	$-0.720275\pi$
$263$	− 405.000i	− 1.53992i	0.638090	−	0.769962i	$-0.279725\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−142.000	−0.523985	−0.261993	−	0.965070i	$-0.584380\pi$
$271$	−142.000	−0.523985	−0.261993	+	0.965070i	$0.584380\pi$
$272$	− 240.000i	− 0.882353i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 535.000i	− 1.93141i	−0.259646	−	0.965704i	$-0.583606\pi$
$277$	− 535.000i	− 1.93141i	0.259646	−	0.965704i	$-0.416394\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	395.000i	1.39576i	0.716215	+	0.697880i	$0.245873\pi$
$283$	395.000i	1.39576i	−0.716215	+	0.697880i	$0.754127\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	64.0000	0.221453
$290$	0	0
$291$	0	0
$292$	100.000i	0.342466i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	425.000	1.41196
$302$	0	0
$303$	0	0
$304$	304.000	1.00000
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	− 60.0000i	− 0.194805i
$309$	0	0
$310$	0	0
$311$	603.000	1.93891	0.969453	−	0.245276i	$-0.0788785\pi$
$311$	603.000	1.93891	0.969453	+	0.245276i	$0.0788785\pi$
$312$	0	0
$313$	− 590.000i	− 1.88498i	−0.334229	−	0.942492i	$-0.608476\pi$
$313$	− 590.000i	− 1.88498i	0.334229	−	0.942492i	$-0.391524\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 285.000i	− 0.882353i
$324$	−324.000	−1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	375.000	1.13982
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 360.000i	− 1.08434i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	365.000i	1.06414i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 675.000i	− 1.94524i	−0.232391	−	0.972622i	$-0.574655\pi$
$347$	− 675.000i	− 1.94524i	0.232391	−	0.972622i	$-0.425345\pi$
$348$	0	0
$349$	−527.000	−1.51003	−0.755014	−	0.655708i	$-0.772370\pi$
$349$	−527.000	−1.51003	−0.755014	+	0.655708i	$0.772370\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 510.000i	− 1.44476i	−0.691497	−	0.722380i	$-0.743048\pi$
$353$	− 510.000i	− 1.44476i	0.691497	−	0.722380i	$-0.256952\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−243.000	−0.676880	−0.338440	−	0.940988i	$-0.609899\pi$
$359$	−243.000	−0.676880	−0.338440	+	0.940988i	$0.609899\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 50.0000i	− 0.136240i	−0.997677	−	0.0681199i	$-0.978300\pi$
$367$	− 50.0000i	− 0.136240i	0.997677	−	0.0681199i	$-0.0217000\pi$
$368$	− 480.000i	− 1.30435i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	765.000i	1.97674i
$388$	0	0
$389$	153.000	0.393316	0.196658	−	0.980472i	$-0.436991\pi$
$389$	153.000	0.393316	0.196658	+	0.980472i	$0.436991\pi$
$390$	0	0
$391$	−450.000	−1.15090
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	108.000	0.272727
$397$	745.000i	1.87657i	0.345857	+	0.938287i	$0.387588\pi$
$397$	745.000i	1.87657i	−0.345857	+	0.938287i	$0.612412\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	408.000	1.00990
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	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
$419$	−762.000	−1.81862	−0.909308	−	0.416124i	$-0.863388\pi$
$419$	−762.000	−1.81862	−0.909308	+	0.416124i	$0.863388\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	675.000i	1.59574i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	515.000i	1.20609i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 570.000i	− 1.30435i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−216.000	−0.489796
$442$	0	0
$443$	− 45.0000i	− 0.101580i	−0.998709	−	0.0507901i	$-0.983826\pi$
$443$	− 45.0000i	− 0.101580i	0.998709	−	0.0507901i	$-0.0161739\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 320.000i	− 0.714286i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	625.000i	1.36761i	0.729663	+	0.683807i	$0.239677\pi$
$457$	625.000i	1.36761i	−0.729663	+	0.683807i	$0.760323\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	447.000	0.969631	0.484816	−	0.874616i	$-0.338887\pi$
$461$	447.000	0.969631	0.484816	+	0.874616i	$0.338887\pi$
$462$	0	0
$463$	755.000i	1.63067i	0.578990	+	0.815335i	$0.303447\pi$
$463$	755.000i	1.63067i	−0.578990	+	0.815335i	$0.696553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 915.000i	− 1.95931i	−0.200677	−	0.979657i	$-0.564314\pi$
$467$	− 915.000i	− 1.95931i	0.200677	−	0.979657i	$-0.435686\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 255.000i	− 0.539112i
$474$	0	0
$475$	0	0
$476$	−300.000	−0.630252
$477$	0	0
$478$	0	0
$479$	942.000	1.96660	0.983299	−	0.182000i	$-0.0582571\pi$
$479$	942.000	1.96660	0.983299	+	0.182000i	$0.0582571\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	448.000	0.925620
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−918.000	−1.86965	−0.934827	−	0.355104i	$-0.884446\pi$
$491$	−918.000	−1.86965	−0.934827	+	0.355104i	$0.884446\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−523.000	−1.04810	−0.524048	−	0.851689i	$-0.675579\pi$
$499$	−523.000	−1.04810	−0.524048	+	0.851689i	$0.675579\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	930.000i	1.84891i	0.381295	+	0.924453i	$0.375478\pi$
$503$	930.000i	1.84891i	−0.381295	+	0.924453i	$0.624522\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	125.000	0.244618
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 225.000i	− 0.435203i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	852.000	1.62595
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−371.000	−0.701323
$530$	0	0
$531$	0	0
$532$	− 380.000i	− 0.714286i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	72.0000	0.133581
$540$	0	0
$541$	−457.000	−0.844732	−0.422366	−	0.906425i	$-0.638800\pi$
$541$	−457.000	−0.844732	−0.422366	+	0.906425i	$0.638800\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	1020.00i	1.86131i
$549$	−927.000	−1.68852
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−788.000	−1.41727
$557$	− 1095.00i	− 1.96589i	−0.183903	−	0.982944i	$-0.558873\pi$
$557$	− 1095.00i	− 1.96589i	0.183903	−	0.982944i	$-0.441127\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	405.000i	0.714286i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	458.000	0.802102	0.401051	−	0.916056i	$-0.368645\pi$
$571$	458.000	0.802102	0.401051	+	0.916056i	$0.368645\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	576.000	1.00000
$577$	1145.00i	1.98440i	0.124648	+	0.992201i	$0.460220\pi$
$577$	1145.00i	1.98440i	−0.124648	+	0.992201i	$0.539780\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−450.000	−0.774527
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1125.00i	1.91652i	0.285890	+	0.958262i	$0.407711\pi$
$587$	1125.00i	1.91652i	−0.285890	+	0.958262i	$0.592289\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 30.0000i	− 0.0505902i	−0.999680	−	0.0252951i	$-0.991947\pi$
$593$	− 30.0000i	− 0.0505902i	0.999680	−	0.0252951i	$-0.00805254\pi$
$594$	0	0
$595$	0	0
$596$	−708.000	−1.18792
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 540.000i	− 0.882353i
$613$	295.000i	0.481240i	0.970619	+	0.240620i	$0.0773507\pi$
$613$	295.000i	0.481240i	−0.970619	+	0.240620i	$0.922649\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1065.00i	1.72609i	0.505124	+	0.863047i	$0.331447\pi$
$617$	1065.00i	1.72609i	−0.505124	+	0.863047i	$0.668553\pi$
$618$	0	0
$619$	662.000	1.06947	0.534733	−	0.845021i	$-0.320412\pi$
$619$	662.000	1.06947	0.534733	+	0.845021i	$0.320412\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	40.0000i	0.0636943i
$629$	0	0
$630$	0	0
$631$	−1037.00	−1.64342	−0.821712	−	0.569904i	$-0.806981\pi$
$631$	−1037.00	−1.64342	−0.821712	+	0.569904i	$0.806981\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1115.00i	1.73406i	0.498257	+	0.867030i	$0.333974\pi$
$643$	1115.00i	1.73406i	−0.498257	+	0.867030i	$0.666026\pi$
$644$	−600.000	−0.931677
$645$	0	0
$646$	0	0
$647$	1005.00i	1.55332i	0.629918	+	0.776662i	$0.283088\pi$
$647$	1005.00i	1.55332i	−0.629918	+	0.776662i	$0.716912\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 1000.00i	− 1.53374i
$653$	375.000i	0.574273i	0.957890	+	0.287136i	$0.0927033\pi$
$653$	375.000i	0.574273i	−0.957890	+	0.287136i	$0.907297\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	225.000i	0.342466i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	309.000	0.460507
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	676.000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	684.000	1.00000
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 1360.00i	− 1.97674i
$689$	0	0
$690$	0	0
$691$	−157.000	−0.227207	−0.113603	−	0.993526i	$-0.536239\pi$
$691$	−157.000	−0.227207	−0.113603	+	0.993526i	$0.536239\pi$
$692$	0	0
$693$	− 135.000i	− 0.194805i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1098.00	1.56633	0.783167	−	0.621812i	$-0.213603\pi$
$701$	1098.00	1.56633	0.783167	+	0.621812i	$0.213603\pi$
$702$	0	0
$703$	0	0
$704$	−192.000	−0.272727
$705$	0	0
$706$	0	0
$707$	− 510.000i	− 0.721358i
$708$	0	0
$709$	1318.00	1.85896	0.929478	−	0.368877i	$-0.120258\pi$
$709$	1318.00	1.85896	0.929478	+	0.368877i	$0.120258\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−963.000	−1.33936	−0.669680	−	0.742650i	$-0.733569\pi$
$719$	−963.000	−1.33936	−0.669680	+	0.742650i	$0.733569\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	85.0000i	0.116919i	0.998290	+	0.0584594i	$0.0186188\pi$
$727$	85.0000i	0.116919i	−0.998290	+	0.0584594i	$0.981381\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	−1275.00	−1.74419
$732$	0	0
$733$	− 1270.00i	− 1.73261i	−0.499519	−	0.866303i	$-0.666490\pi$
$733$	− 1270.00i	− 1.73261i	0.499519	−	0.866303i	$-0.333510\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−547.000	−0.740189	−0.370095	−	0.928994i	$-0.620675\pi$
$739$	−547.000	−0.740189	−0.370095	+	0.928994i	$0.620675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 810.000i	− 1.08434i
$748$	180.000i	0.240642i
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 1200.00i	− 1.59574i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	785.000i	1.03699i	0.855081	+	0.518494i	$0.173507\pi$
$757$	785.000i	1.03699i	−0.855081	+	0.518494i	$0.826493\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1503.00	1.97503	0.987516	−	0.157516i	$-0.0503486\pi$
$761$	1503.00	1.97503	0.987516	+	0.157516i	$0.0503486\pi$
$762$	0	0
$763$	0	0
$764$	372.000	0.486911
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1063.00	−1.38231	−0.691157	−	0.722704i	$-0.742899\pi$
$769$	−1063.00	−1.38231	−0.691157	+	0.722704i	$0.742899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	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
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	384.000	0.489796
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	360.000i	0.456853i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	908.000	1.14070
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−1125.00	−1.40801
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 75.0000i	− 0.0933998i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1593.00	1.96910	0.984549	−	0.175110i	$-0.0560282\pi$
$809$	1593.00	1.96910	0.984549	+	0.175110i	$0.0560282\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1615.00i	− 1.97674i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1167.00	1.42144	0.710719	−	0.703476i	$-0.248370\pi$
$821$	1167.00	1.42144	0.710719	+	0.703476i	$0.248370\pi$
$822$	0	0
$823$	− 1565.00i	− 1.90158i	−0.309837	−	0.950790i	$-0.600274\pi$
$823$	− 1565.00i	− 1.90158i	0.309837	−	0.950790i	$-0.399726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	− 1080.00i	− 1.30435i
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 360.000i	− 0.432173i
$834$	0	0
$835$	0	0
$836$	−228.000	−0.272727
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 560.000i	− 0.661157i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 1030.00i	− 1.20750i	−0.797173	−	0.603751i	$-0.793672\pi$
$853$	− 1030.00i	− 1.20750i	0.797173	−	0.603751i	$-0.206328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	1493.00	1.73807	0.869034	−	0.494753i	$-0.164741\pi$
$859$	1493.00	1.73807	0.869034	+	0.494753i	$0.164741\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−537.000	−0.609535	−0.304767	−	0.952427i	$-0.598579\pi$
$881$	−537.000	−0.609535	−0.304767	+	0.952427i	$0.598579\pi$
$882$	0	0
$883$	835.000i	0.945640i	0.881159	+	0.472820i	$0.156764\pi$
$883$	835.000i	0.945640i	−0.881159	+	0.472820i	$0.843236\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	243.000	0.272727
$892$	0	0
$893$	− 1425.00i	− 1.59574i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	918.000	1.00990
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	270.000i	0.295728i
$914$	0	0
$915$	0	0
$916$	−68.0000	−0.0742358
$917$	− 1065.00i	− 1.16140i
$918$	0	0
$919$	−1762.00	−1.91730	−0.958651	−	0.284585i	$-0.908144\pi$
$919$	−1762.00	−1.91730	−0.958651	+	0.284585i	$0.908144\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−642.000	−0.691066	−0.345533	−	0.938407i	$-0.612302\pi$
$929$	−642.000	−0.691066	−0.345533	+	0.938407i	$0.612302\pi$
$930$	0	0
$931$	456.000	0.489796
$932$	1860.00i	1.99571i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 335.000i	− 0.357524i	−0.983892	−	0.178762i	$-0.942791\pi$
$937$	− 335.000i	− 0.357524i	0.983892	−	0.178762i	$-0.0572092\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1830.00i	1.93242i	0.257760	+	0.966209i	$0.417016\pi$
$947$	1830.00i	1.93242i	−0.257760	+	0.966209i	$0.582984\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	−1812.00	−1.89540
$957$	0	0
$958$	0	0
$959$	1275.00	1.32951
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1790.00i	1.85109i	0.378643	+	0.925543i	$0.376391\pi$
$967$	1790.00i	1.85109i	−0.378643	+	0.925543i	$0.623609\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	985.000i	1.01233i
$974$	0	0
$975$	0	0
$976$	1648.00	1.68852
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2550.00	−2.57836
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1975.00i	− 1.98094i	−0.137718	−	0.990471i	$-0.543977\pi$
$997$	− 1975.00i	− 1.98094i	0.137718	−	0.990471i	$-0.456023\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.a.474.2		2
5.2	odd	4		19.3.b.a.18.1	✓	1
5.3	odd	4		475.3.c.a.151.1		1
5.4	even	2	inner	475.3.d.a.474.1		2
15.2	even	4		171.3.c.a.37.1		1
19.18	odd	2	CM	475.3.d.a.474.2		2
20.7	even	4		304.3.e.a.113.1		1
40.27	even	4		1216.3.e.b.1025.1		1
40.37	odd	4		1216.3.e.a.1025.1		1
60.47	odd	4		2736.3.o.a.721.1		1
95.2	even	36		361.3.f.a.262.1		6
95.7	odd	12		361.3.d.a.293.1		2
95.12	even	12		361.3.d.a.293.1		2
95.17	odd	36		361.3.f.a.262.1		6
95.18	even	4		475.3.c.a.151.1		1
95.22	even	36		361.3.f.a.333.1		6
95.27	even	12		361.3.d.a.69.1		2
95.32	even	36		361.3.f.a.116.1		6
95.37	even	4		19.3.b.a.18.1	✓	1
95.42	odd	36		361.3.f.a.307.1		6
95.47	odd	36		361.3.f.a.299.1		6
95.52	even	36		361.3.f.a.127.1		6
95.62	odd	36		361.3.f.a.127.1		6
95.67	even	36		361.3.f.a.299.1		6
95.72	even	36		361.3.f.a.307.1		6
95.82	odd	36		361.3.f.a.116.1		6
95.87	odd	12		361.3.d.a.69.1		2
95.92	odd	36		361.3.f.a.333.1		6
95.94	odd	2	inner	475.3.d.a.474.1		2
285.227	odd	4		171.3.c.a.37.1		1
380.227	odd	4		304.3.e.a.113.1		1
760.37	even	4		1216.3.e.a.1025.1		1
760.227	odd	4		1216.3.e.b.1025.1		1
1140.227	even	4		2736.3.o.a.721.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.a.18.1	✓	1	5.2	odd	4
19.3.b.a.18.1	✓	1	95.37	even	4
171.3.c.a.37.1		1	15.2	even	4
171.3.c.a.37.1		1	285.227	odd	4
304.3.e.a.113.1		1	20.7	even	4
304.3.e.a.113.1		1	380.227	odd	4
361.3.d.a.69.1		2	95.27	even	12
361.3.d.a.69.1		2	95.87	odd	12
361.3.d.a.293.1		2	95.7	odd	12
361.3.d.a.293.1		2	95.12	even	12
361.3.f.a.116.1		6	95.32	even	36
361.3.f.a.116.1		6	95.82	odd	36
361.3.f.a.127.1		6	95.52	even	36
361.3.f.a.127.1		6	95.62	odd	36
361.3.f.a.262.1		6	95.2	even	36
361.3.f.a.262.1		6	95.17	odd	36
361.3.f.a.299.1		6	95.47	odd	36
361.3.f.a.299.1		6	95.67	even	36
361.3.f.a.307.1		6	95.42	odd	36
361.3.f.a.307.1		6	95.72	even	36
361.3.f.a.333.1		6	95.22	even	36
361.3.f.a.333.1		6	95.92	odd	36
475.3.c.a.151.1		1	5.3	odd	4
475.3.c.a.151.1		1	95.18	even	4
475.3.d.a.474.1		2	5.4	even	2	inner
475.3.d.a.474.1		2	95.94	odd	2	inner
475.3.d.a.474.2		2	1.1	even	1	trivial
475.3.d.a.474.2		2	19.18	odd	2	CM
1216.3.e.a.1025.1		1	40.37	odd	4
1216.3.e.a.1025.1		1	760.37	even	4
1216.3.e.b.1025.1		1	40.27	even	4
1216.3.e.b.1025.1		1	760.227	odd	4
2736.3.o.a.721.1		1	60.47	odd	4
2736.3.o.a.721.1		1	1140.227	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000 q^{4} +5.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-4.00000 q^{4} +5.00000i q^{7} -9.00000 q^{9} +3.00000 q^{11} +16.0000 q^{16} -15.0000i q^{17} +19.0000 q^{19} -30.0000i q^{23} -20.0000i q^{28} +36.0000 q^{36} -85.0000i q^{43} -12.0000 q^{44} -75.0000i q^{47} +24.0000 q^{49} +103.000 q^{61} -45.0000i q^{63} -64.0000 q^{64} +60.0000i q^{68} -25.0000i q^{73} -76.0000 q^{76} +15.0000i q^{77} +81.0000 q^{81} +90.0000i q^{83} +120.000i q^{92} -27.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 18 q^{9}+O(q^{10}) \) 2 * q - 8 * q^4 - 18 * q^9	\( 2 q - 8 q^{4} - 18 q^{9} + 6 q^{11} + 32 q^{16} + 38 q^{19} + 72 q^{36} - 24 q^{44} + 48 q^{49} + 206 q^{61} - 128 q^{64} - 152 q^{76} + 162 q^{81} - 54 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 18 * q^9 + 6 * q^11 + 32 * q^16 + 38 * q^19 + 72 * q^36 - 24 * q^44 + 48 * q^49 + 206 * q^61 - 128 * q^64 - 152 * q^76 + 162 * q^81 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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