LMFDB - Embedded newform 399.1.h.c.398.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,1,Mod(398,399)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("399.398"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	399.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.199126940041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.53067.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.0.190563597.3

Embedding invariants

Embedding label			398.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	399.398

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -1.41421 q^{14} -1.41421 q^{15} -1.00000 q^{16} -1.41421 q^{17} +1.41421 q^{18} +1.00000 q^{19} +1.41421 q^{20} +1.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} -1.41421 q^{29} -2.00000 q^{30} -1.41421 q^{32} -2.00000 q^{34} -1.41421 q^{35} +1.00000 q^{36} +1.41421 q^{38} +1.41421 q^{42} +1.41421 q^{45} -1.41421 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.41421 q^{50} +1.41421 q^{51} +1.41421 q^{53} -1.41421 q^{54} -1.00000 q^{57} -2.00000 q^{58} -1.41421 q^{60} -1.00000 q^{63} -1.00000 q^{64} -1.41421 q^{68} -2.00000 q^{70} +1.41421 q^{71} -1.00000 q^{75} +1.00000 q^{76} -1.41421 q^{80} +1.00000 q^{81} +1.41421 q^{83} +1.00000 q^{84} -2.00000 q^{85} +1.41421 q^{87} +2.00000 q^{90} -2.00000 q^{94} +1.41421 q^{95} +1.41421 q^{96} +2.00000 q^{97} +1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{12} - 2 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 2 q^{28} - 4 q^{30} - 4 q^{34} + 2 q^{36} + 2 q^{48} + 2 q^{49} - 2 q^{57} - 4 q^{58}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^7 + 2 * q^9 + 4 * q^10 - 2 * q^12 - 2 * q^16 + 2 * q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 2 * q^28 - 4 * q^30 - 4 * q^34 + 2 * q^36 + 2 * q^48 + 2 * q^49 - 2 * q^57 - 4 * q^58 - 2 * q^63 - 2 * q^64 - 4 * q^70 - 2 * q^75 + 2 * q^76 + 2 * q^81 + 2 * q^84 - 4 * q^85 + 4 * q^90 - 4 * q^94 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−1.00000
$3$	−1.00000	−1.00000
$4$	1.00000	1.00000
$5$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	−1.41421	−1.41421
$7$	−1.00000	−1.00000
$7$	−1.00000	−1.00000
$8$	0	0
$9$	1.00000	1.00000
$10$	2.00000	2.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.00000	−1.00000
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−1.41421	−1.41421
$15$	−1.41421	−1.41421
$16$	−1.00000	−1.00000
$17$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$17$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$18$	1.41421	1.41421
$19$	1.00000	1.00000
$19$	1.00000	1.00000
$20$	1.41421	1.41421
$21$	1.00000	1.00000
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.00000	1.00000
$26$	0	0
$27$	−1.00000	−1.00000
$28$	−1.00000	−1.00000
$29$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$29$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$30$	−2.00000	−2.00000
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.41421	−1.41421
$33$	0	0
$34$	−2.00000	−2.00000
$35$	−1.41421	−1.41421
$36$	1.00000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.41421	1.41421
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	1.41421	1.41421
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.41421	1.41421
$46$	0	0
$47$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$47$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$48$	1.00000	1.00000
$49$	1.00000	1.00000
$50$	1.41421	1.41421
$51$	1.41421	1.41421
$52$	0	0
$53$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$53$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$54$	−1.41421	−1.41421
$55$	0	0
$56$	0	0
$57$	−1.00000	−1.00000
$58$	−2.00000	−2.00000
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1.41421	−1.41421
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−1.00000	−1.00000
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.41421	−1.41421
$69$	0	0
$70$	−2.00000	−2.00000
$71$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$71$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.00000	−1.00000
$76$	1.00000	1.00000
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.41421	−1.41421
$81$	1.00000	1.00000
$82$	0	0
$83$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$83$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$84$	1.00000	1.00000
$85$	−2.00000	−2.00000
$86$	0	0
$87$	1.41421	1.41421
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	2.00000	2.00000
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−2.00000	−2.00000
$95$	1.41421	1.41421
$96$	1.41421	1.41421
$97$	2.00000	2.00000	1.00000			$0$
$97$	2.00000	2.00000	1.00000			$0$
$98$	1.41421	1.41421
$99$	0	0
$100$	1.00000	1.00000
$101$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$101$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$102$	2.00000	2.00000
$103$	2.00000	2.00000	1.00000			$0$
$103$	2.00000	2.00000	1.00000			$0$
$104$	0	0
$105$	1.41421	1.41421
$106$	2.00000	2.00000
$107$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$107$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$108$	−1.00000	−1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	1.00000
$113$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$114$	−1.41421	−1.41421
$115$	0	0
$116$	−1.41421	−1.41421
$117$	0	0
$118$	0	0
$119$	1.41421	1.41421
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−1.41421	−1.41421
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$131$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$132$	0	0
$133$	−1.00000	−1.00000
$134$	0	0
$135$	−1.41421	−1.41421
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−1.41421	−1.41421
$141$	1.41421	1.41421
$142$	2.00000	2.00000
$143$	0	0
$144$	−1.00000	−1.00000
$145$	−2.00000	−2.00000
$146$	0	0
$147$	−1.00000	−1.00000
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.41421	−1.41421
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−1.41421	−1.41421
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−1.41421	−1.41421
$160$	−2.00000	−2.00000
$161$	0	0
$162$	1.41421	1.41421
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	2.00000	2.00000
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−1.00000
$170$	−2.82843	−2.82843
$171$	1.00000	1.00000
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	2.00000	2.00000
$175$	−1.00000	−1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$179$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$180$	1.41421	1.41421
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1.41421	−1.41421
$189$	1.00000	1.00000
$190$	2.00000	2.00000
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.00000	1.00000
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	2.82843	2.82843
$195$	0	0
$196$	1.00000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	−2.00000	−2.00000
$203$	1.41421	1.41421
$204$	1.41421	1.41421
$205$	0	0
$206$	2.82843	2.82843
$207$	0	0
$208$	0	0
$209$	0	0
$210$	2.00000	2.00000
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.41421	1.41421
$213$	−1.41421	−1.41421
$214$	−2.00000	−2.00000
$215$	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	$0.5\pi$
$224$	1.41421	1.41421
$225$	1.00000	1.00000
$226$	−2.00000	−2.00000
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−1.00000	−1.00000
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	−2.00000	−2.00000
$236$	0	0
$237$	0	0
$238$	2.00000	2.00000
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.41421	1.41421
$241$	−2.00000	−2.00000	−1.00000			$\pi$
$241$	−2.00000	−2.00000	−1.00000			$\pi$
$242$	1.41421	1.41421
$243$	−1.00000	−1.00000
$244$	0	0
$245$	1.41421	1.41421
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.41421	−1.41421
$250$	0	0
$251$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$251$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$252$	−1.00000	−1.00000
$253$	0	0
$254$	0	0
$255$	2.00000	2.00000
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.41421	−1.41421
$262$	2.00000	2.00000
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	2.00000	2.00000
$266$	−1.41421	−1.41421
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−2.00000	−2.00000
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	1.41421	1.41421
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$281$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$282$	2.00000	2.00000
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.41421	1.41421
$285$	−1.41421	−1.41421
$286$	0	0
$287$	0	0
$288$	−1.41421	−1.41421
$289$	1.00000	1.00000
$290$	−2.82843	−2.82843
$291$	−2.00000	−2.00000
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.41421	−1.41421
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−1.00000	−1.00000
$301$	0	0
$302$	0	0
$303$	1.41421	1.41421
$304$	−1.00000	−1.00000
$305$	0	0
$306$	−2.00000	−2.00000
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−2.00000	−2.00000
$310$	0	0
$311$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$311$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−1.41421	−1.41421
$316$	0	0
$317$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$317$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$318$	−2.00000	−2.00000
$319$	0	0
$320$	−1.41421	−1.41421
$321$	1.41421	1.41421
$322$	0	0
$323$	−1.41421	−1.41421
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.41421	1.41421
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.41421	1.41421
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−1.00000	−1.00000
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.41421	−1.41421
$339$	1.41421	1.41421
$340$	−2.00000	−2.00000
$341$	0	0
$342$	1.41421	1.41421
$343$	−1.00000	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	1.41421	1.41421
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.41421	−1.41421
$351$	0	0
$352$	0	0
$353$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$353$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$354$	0	0
$355$	2.00000	2.00000
$356$	0	0
$357$	−1.41421	−1.41421
$358$	−2.00000	−2.00000
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	−2.82843	−2.82843
$363$	−1.00000	−1.00000
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.41421	−1.41421
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	1.41421	1.41421
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	1.41421	1.41421
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	2.00000	2.00000
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.41421	−1.41421
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	1.00000	1.00000
$400$	−1.00000	−1.00000
$401$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$401$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$402$	0	0
$403$	0	0
$404$	−1.41421	−1.41421
$405$	1.41421	1.41421
$406$	2.00000	2.00000
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	2.00000	2.00000
$413$	0	0
$414$	0	0
$415$	2.00000	2.00000
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$419$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$420$	1.41421	1.41421
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	−1.41421	−1.41421
$424$	0	0
$425$	−1.41421	−1.41421
$426$	−2.00000	−2.00000
$427$	0	0
$428$	−1.41421	−1.41421
$429$	0	0
$430$	0	0
$431$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$431$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$432$	1.00000	1.00000
$433$	−2.00000	−2.00000	−1.00000			$\pi$
$433$	−2.00000	−2.00000	−1.00000			$\pi$
$434$	0	0
$435$	2.00000	2.00000
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	1.00000	1.00000
$449$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$449$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$450$	1.41421	1.41421
$451$	0	0
$452$	−1.41421	−1.41421
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	1.41421	1.41421
$460$	0	0
$461$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$461$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$462$	0	0
$463$	2.00000	2.00000	1.00000			$0$
$463$	2.00000	2.00000	1.00000			$0$
$464$	1.41421	1.41421
$465$	0	0
$466$	0	0
$467$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$467$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$468$	0	0
$469$	0	0
$470$	−2.82843	−2.82843
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.00000	1.00000
$476$	1.41421	1.41421
$477$	1.41421	1.41421
$478$	0	0
$479$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$479$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$480$	2.00000	2.00000
$481$	0	0
$482$	−2.82843	−2.82843
$483$	0	0
$484$	1.00000	1.00000
$485$	2.82843	2.82843
$486$	−1.41421	−1.41421
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	2.00000	2.00000
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	2.00000	2.00000
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.41421	−1.41421
$498$	−2.00000	−2.00000
$499$	2.00000	2.00000	1.00000			$0$
$499$	2.00000	2.00000	1.00000			$0$
$500$	0	0
$501$	0	0
$502$	−2.00000	−2.00000
$503$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$503$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$504$	0	0
$505$	−2.00000	−2.00000
$506$	0	0
$507$	1.00000	1.00000
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	2.82843	2.82843
$511$	0	0
$512$	1.41421	1.41421
$513$	−1.00000	−1.00000
$514$	0	0
$515$	2.82843	2.82843
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−2.00000	−2.00000
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	1.41421	1.41421
$525$	1.00000	1.00000
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	2.82843	2.82843
$531$	0	0
$532$	−1.00000	−1.00000
$533$	0	0
$534$	0	0
$535$	−2.00000	−2.00000
$536$	0	0
$537$	1.41421	1.41421
$538$	0	0
$539$	0	0
$540$	−1.41421	−1.41421
$541$	−2.00000	−2.00000	−1.00000			$\pi$
$541$	−2.00000	−2.00000	−1.00000			$\pi$
$542$	0	0
$543$	2.00000	2.00000
$544$	2.00000	2.00000
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.41421	−1.41421
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	1.41421	1.41421
$561$	0	0
$562$	2.00000	2.00000
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.41421	1.41421
$565$	−2.00000	−2.00000
$566$	0	0
$567$	−1.00000	−1.00000
$568$	0	0
$569$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$569$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$570$	−2.00000	−2.00000
$571$	−2.00000	−2.00000	−1.00000			$\pi$
$571$	−2.00000	−2.00000	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−1.00000	−1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.41421	1.41421
$579$	0	0
$580$	−2.00000	−2.00000
$581$	−1.41421	−1.41421
$582$	−2.82843	−2.82843
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$587$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$588$	−1.00000	−1.00000
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$593$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$594$	0	0
$595$	2.00000	2.00000
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$599$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$600$	0	0
$601$	2.00000	2.00000	1.00000			$0$
$601$	2.00000	2.00000	1.00000			$0$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.41421	1.41421
$606$	2.00000	2.00000
$607$	−2.00000	−2.00000	−1.00000			$\pi$
$607$	−2.00000	−2.00000	−1.00000			$\pi$
$608$	−1.41421	−1.41421
$609$	−1.41421	−1.41421
$610$	0	0
$611$	0	0
$612$	−1.41421	−1.41421
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−2.82843	−2.82843
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	2.00000	2.00000
$623$	0	0
$624$	0	0
$625$	−1.00000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−2.00000	−2.00000
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	−2.00000	−2.00000
$635$	0	0
$636$	−1.41421	−1.41421
$637$	0	0
$638$	0	0
$639$	1.41421	1.41421
$640$	0	0
$641$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$641$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$642$	2.00000	2.00000
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−2.00000	−2.00000
$647$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$647$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	2.00000	2.00000
$656$	0	0
$657$	0	0
$658$	2.00000	2.00000
$659$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$659$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.41421	−1.41421
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	−1.41421	−1.41421
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−1.00000	−1.00000
$676$	−1.00000	−1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	2.00000	2.00000
$679$	−2.00000	−2.00000
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$683$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$684$	1.00000	1.00000
$685$	0	0
$686$	−1.41421	−1.41421
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−1.00000	−1.00000
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	2.00000	2.00000
$706$	2.00000	2.00000
$707$	1.41421	1.41421
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	2.82843	2.82843
$711$	0	0
$712$	0	0
$713$	0	0
$714$	−2.00000	−2.00000
$715$	0	0
$716$	−1.41421	−1.41421
$717$	0	0
$718$	0	0
$719$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$719$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$720$	−1.41421	−1.41421
$721$	−2.00000	−2.00000
$722$	1.41421	1.41421
$723$	2.00000	2.00000
$724$	−2.00000	−2.00000
$725$	−1.41421	−1.41421
$726$	−1.41421	−1.41421
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	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
$735$	−1.41421	−1.41421
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	−2.00000	−2.00000
$743$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$743$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.41421	1.41421
$748$	0	0
$749$	1.41421	1.41421
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.41421	1.41421
$753$	1.41421	1.41421
$754$	0	0
$755$	0	0
$756$	1.00000	1.00000
$757$	−2.00000	−2.00000	−1.00000			$\pi$
$757$	−2.00000	−2.00000	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$761$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.00000	−2.00000
$766$	0	0
$767$	0	0
$768$	−1.00000	−1.00000
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	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
$783$	1.41421	1.41421
$784$	−1.00000	−1.00000
$785$	0	0
$786$	−2.00000	−2.00000
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.41421	1.41421
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−2.00000	−2.00000
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	1.41421	1.41421
$799$	2.00000	2.00000
$800$	−1.41421	−1.41421
$801$	0	0
$802$	2.00000	2.00000
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	2.00000	2.00000
$811$	2.00000	2.00000	1.00000			$0$
$811$	2.00000	2.00000	1.00000			$0$
$812$	1.41421	1.41421
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−1.41421	−1.41421
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$827$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$828$	0	0
$829$	2.00000	2.00000	1.00000			$0$
$829$	2.00000	2.00000	1.00000			$0$
$830$	2.82843	2.82843
$831$	0	0
$832$	0	0
$833$	−1.41421	−1.41421
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−2.00000	−2.00000
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	−1.41421	−1.41421
$844$	0	0
$845$	−1.41421	−1.41421
$846$	−2.00000	−2.00000
$847$	−1.00000	−1.00000
$848$	−1.41421	−1.41421
$849$	0	0
$850$	−2.00000	−2.00000
$851$	0	0
$852$	−1.41421	−1.41421
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	1.41421	1.41421
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−2.00000	−2.00000
$863$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$863$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$864$	1.41421	1.41421
$865$	0	0
$866$	−2.82843	−2.82843
$867$	−1.00000	−1.00000
$868$	0	0
$869$	0	0
$870$	2.82843	2.82843
$871$	0	0
$872$	0	0
$873$	2.00000	2.00000
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$881$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$882$	1.41421	1.41421
$883$	2.00000	2.00000	1.00000			$0$
$883$	2.00000	2.00000	1.00000			$0$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.41421	−1.41421
$894$	0	0
$895$	−2.00000	−2.00000
$896$	0	0
$897$	0	0
$898$	2.00000	2.00000
$899$	0	0
$900$	1.00000	1.00000
$901$	−2.00000	−2.00000
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.82843	−2.82843
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−1.41421	−1.41421
$910$	0	0
$911$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$911$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$912$	1.00000	1.00000
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.41421	−1.41421
$918$	2.00000	2.00000
$919$	−2.00000	−2.00000	−1.00000			$\pi$
$919$	−2.00000	−2.00000	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−2.00000	−2.00000
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.82843	2.82843
$927$	2.00000	2.00000
$928$	2.00000	2.00000
$929$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$929$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$930$	0	0
$931$	1.00000	1.00000
$932$	0	0
$933$	−1.41421	−1.41421
$934$	2.00000	2.00000
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−2.00000	−2.00000
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.41421	1.41421
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	1.41421	1.41421
$951$	1.41421	1.41421
$952$	0	0
$953$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$953$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$954$	2.00000	2.00000
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−2.00000	−2.00000
$959$	0	0
$960$	1.41421	1.41421
$961$	−1.00000	−1.00000
$962$	0	0
$963$	−1.41421	−1.41421
$964$	−2.00000	−2.00000
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	1.41421	1.41421
$970$	4.00000	4.00000
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.00000	−1.00000
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$977$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$978$	0	0
$979$	0	0
$980$	1.41421	1.41421
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	2.82843	2.82843
$987$	−1.41421	−1.41421
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	−2.00000	−2.00000
$995$	0	0
$996$	−1.41421	−1.41421
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	2.82843	2.82843
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.1.h.c.398.2	yes	2
3.2	odd	2	inner	399.1.h.c.398.1	✓	2
7.2	even	3		2793.1.s.d.227.1		4
7.3	odd	6		2793.1.s.c.1538.1		4
7.4	even	3		2793.1.s.d.1538.1		4
7.5	odd	6		2793.1.s.c.227.1		4
7.6	odd	2		399.1.h.d.398.2	yes	2
19.18	odd	2		399.1.h.d.398.1	yes	2
21.2	odd	6		2793.1.s.d.227.2		4
21.5	even	6		2793.1.s.c.227.2		4
21.11	odd	6		2793.1.s.d.1538.2		4
21.17	even	6		2793.1.s.c.1538.2		4
21.20	even	2		399.1.h.d.398.1	yes	2
57.56	even	2		399.1.h.d.398.2	yes	2
133.18	odd	6		2793.1.s.c.1538.2		4
133.37	odd	6		2793.1.s.c.227.2		4
133.75	even	6		2793.1.s.d.227.2		4
133.94	even	6		2793.1.s.d.1538.2		4
133.132	even	2	inner	399.1.h.c.398.1	✓	2
399.170	even	6		2793.1.s.c.227.1		4
399.227	odd	6		2793.1.s.d.1538.1		4
399.284	even	6		2793.1.s.c.1538.1		4
399.341	odd	6		2793.1.s.d.227.1		4
399.398	odd	2	CM	399.1.h.c.398.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.1.h.c.398.1	✓	2	3.2	odd	2	inner
399.1.h.c.398.1	✓	2	133.132	even	2	inner
399.1.h.c.398.2	yes	2	1.1	even	1	trivial
399.1.h.c.398.2	yes	2	399.398	odd	2	CM
399.1.h.d.398.1	yes	2	19.18	odd	2
399.1.h.d.398.1	yes	2	21.20	even	2
399.1.h.d.398.2	yes	2	7.6	odd	2
399.1.h.d.398.2	yes	2	57.56	even	2
2793.1.s.c.227.1		4	7.5	odd	6
2793.1.s.c.227.1		4	399.170	even	6
2793.1.s.c.227.2		4	21.5	even	6
2793.1.s.c.227.2		4	133.37	odd	6
2793.1.s.c.1538.1		4	7.3	odd	6
2793.1.s.c.1538.1		4	399.284	even	6
2793.1.s.c.1538.2		4	21.17	even	6
2793.1.s.c.1538.2		4	133.18	odd	6
2793.1.s.d.227.1		4	7.2	even	3
2793.1.s.d.227.1		4	399.341	odd	6
2793.1.s.d.227.2		4	21.2	odd	6
2793.1.s.d.227.2		4	133.75	even	6
2793.1.s.d.1538.1		4	7.4	even	3
2793.1.s.d.1538.1		4	399.227	odd	6
2793.1.s.d.1538.2		4	21.11	odd	6
2793.1.s.d.1538.2		4	133.94	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	399.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -1.41421 q^{14} -1.41421 q^{15} -1.00000 q^{16} -1.41421 q^{17} +1.41421 q^{18} +1.00000 q^{19} +1.41421 q^{20} +1.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} -1.41421 q^{29} -2.00000 q^{30} -1.41421 q^{32} -2.00000 q^{34} -1.41421 q^{35} +1.00000 q^{36} +1.41421 q^{38} +1.41421 q^{42} +1.41421 q^{45} -1.41421 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.41421 q^{50} +1.41421 q^{51} +1.41421 q^{53} -1.41421 q^{54} -1.00000 q^{57} -2.00000 q^{58} -1.41421 q^{60} -1.00000 q^{63} -1.00000 q^{64} -1.41421 q^{68} -2.00000 q^{70} +1.41421 q^{71} -1.00000 q^{75} +1.00000 q^{76} -1.41421 q^{80} +1.00000 q^{81} +1.41421 q^{83} +1.00000 q^{84} -2.00000 q^{85} +1.41421 q^{87} +2.00000 q^{90} -2.00000 q^{94} +1.41421 q^{95} +1.41421 q^{96} +2.00000 q^{97} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{12} - 2 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 2 q^{28} - 4 q^{30} - 4 q^{34} + 2 q^{36} + 2 q^{48} + 2 q^{49} - 2 q^{57} - 4 q^{58}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^7 + 2 * q^9 + 4 * q^10 - 2 * q^12 - 2 * q^16 + 2 * q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 2 * q^28 - 4 * q^30 - 4 * q^34 + 2 * q^36 + 2 * q^48 + 2 * q^49 - 2 * q^57 - 4 * q^58 - 2 * q^63 - 2 * q^64 - 4 * q^70 - 2 * q^75 + 2 * q^76 + 2 * q^81 + 2 * q^84 - 4 * q^85 + 4 * q^90 - 4 * q^94 + 4 * q^97

Introduction

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