LMFDB - Embedded newform 644.1.h.d.643.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [644,1,Mod(643,644)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(644, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("644.643");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 644 = 2^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	644.h (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$0.321397868136$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.4508.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.0.1869629888.4

Embedding invariants

Embedding label			643.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	644.643

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.41421 q^{10} +1.41421 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +1.41421 q^{17} -1.00000 q^{18} -1.41421 q^{20} +1.41421 q^{21} +1.00000 q^{23} -1.41421 q^{24} +1.00000 q^{25} +1.00000 q^{28} +2.00000 q^{30} -1.41421 q^{31} -1.00000 q^{32} -1.41421 q^{34} -1.41421 q^{35} +1.00000 q^{36} +1.41421 q^{40} -1.41421 q^{42} -2.00000 q^{43} -1.41421 q^{45} -1.00000 q^{46} -1.41421 q^{47} +1.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -1.00000 q^{56} +1.41421 q^{59} -2.00000 q^{60} -1.41421 q^{61} +1.41421 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.41421 q^{68} +1.41421 q^{69} +1.41421 q^{70} -1.00000 q^{72} +1.41421 q^{75} -1.41421 q^{80} -1.00000 q^{81} +1.41421 q^{84} -2.00000 q^{85} +2.00000 q^{86} +1.41421 q^{89} +1.41421 q^{90} +1.00000 q^{92} -2.00000 q^{93} +1.41421 q^{94} -1.41421 q^{96} -1.41421 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{14} - 4 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{23} + 2 q^{25} + 2 q^{28} + 4 q^{30} - 2 q^{32} + 2 q^{36} - 4 q^{43} - 2 q^{46} + 2 q^{49} - 2 q^{50} + 4 q^{51} - 2 q^{56} - 4 q^{60} + 2 q^{63} + 2 q^{64} - 2 q^{72} - 2 q^{81} - 4 q^{85} + 4 q^{86} + 2 q^{92} - 4 q^{93} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^14 - 4 * q^15 + 2 * q^16 - 2 * q^18 + 2 * q^23 + 2 * q^25 + 2 * q^28 + 4 * q^30 - 2 * q^32 + 2 * q^36 - 4 * q^43 - 2 * q^46 + 2 * q^49 - 2 * q^50 + 4 * q^51 - 2 * q^56 - 4 * q^60 + 2 * q^63 + 2 * q^64 - 2 * q^72 - 2 * q^81 - 4 * q^85 + 4 * q^86 + 2 * q^92 - 4 * q^93 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/644\mathbb{Z}\right)^\times$.

$n$	$185$	$281$	$323$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.00000	−1.00000
$2$	−1.00000	−1.00000
$3$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	1.00000	1.00000
$5$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$5$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$6$	−1.41421	−1.41421
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	−1.00000	−1.00000
$9$	1.00000	1.00000
$10$	1.41421	1.41421
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.41421	1.41421
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−1.00000	−1.00000
$15$	−2.00000	−2.00000
$16$	1.00000	1.00000
$17$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$17$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$18$	−1.00000	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.41421	−1.41421
$21$	1.41421	1.41421
$22$	0	0
$23$	1.00000	1.00000
$23$	1.00000	1.00000
$24$	−1.41421	−1.41421
$25$	1.00000	1.00000
$26$	0	0
$27$	0	0
$28$	1.00000	1.00000
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	2.00000	2.00000
$31$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$31$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	−1.41421	−1.41421
$35$	−1.41421	−1.41421
$36$	1.00000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	1.41421	1.41421
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.41421	−1.41421
$43$	−2.00000	−2.00000	−1.00000			$\pi$
$43$	−2.00000	−2.00000	−1.00000			$\pi$
$44$	0	0
$45$	−1.41421	−1.41421
$46$	−1.00000	−1.00000
$47$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$47$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$48$	1.41421	1.41421
$49$	1.00000	1.00000
$50$	−1.00000	−1.00000
$51$	2.00000	2.00000
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−1.00000
$57$	0	0
$58$	0	0
$59$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$59$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$60$	−2.00000	−2.00000
$61$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$61$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$62$	1.41421	1.41421
$63$	1.00000	1.00000
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	1.41421	1.41421
$69$	1.41421	1.41421
$70$	1.41421	1.41421
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.00000	−1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.41421	1.41421
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.41421	−1.41421
$81$	−1.00000	−1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	1.41421	1.41421
$85$	−2.00000	−2.00000
$86$	2.00000	2.00000
$87$	0	0
$88$	0	0
$89$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$89$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$90$	1.41421	1.41421
$91$	0	0
$92$	1.00000	1.00000
$93$	−2.00000	−2.00000
$94$	1.41421	1.41421
$95$	0	0
$96$	−1.41421	−1.41421
$97$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$97$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$98$	−1.00000	−1.00000
$99$	0	0
$100$	1.00000	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−2.00000	−2.00000
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−2.00000	−2.00000
$106$	0	0
$107$	−2.00000	−2.00000	−1.00000			$\pi$
$107$	−2.00000	−2.00000	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	1.00000
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−1.41421	−1.41421
$116$	0	0
$117$	0	0
$118$	−1.41421	−1.41421
$119$	1.41421	1.41421
$120$	2.00000	2.00000
$121$	−1.00000	−1.00000
$122$	1.41421	1.41421
$123$	0	0
$124$	−1.41421	−1.41421
$125$	0	0
$126$	−1.00000	−1.00000
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	−2.82843	−2.82843
$130$	0	0
$131$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$131$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−1.41421	−1.41421
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.41421	−1.41421
$139$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$139$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$140$	−1.41421	−1.41421
$141$	−2.00000	−2.00000
$142$	0	0
$143$	0	0
$144$	1.00000	1.00000
$145$	0	0
$146$	0	0
$147$	1.41421	1.41421
$148$	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
$155$	2.00000	2.00000
$156$	0	0
$157$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$157$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$158$	0	0
$159$	0	0
$160$	1.41421	1.41421
$161$	1.00000	1.00000
$162$	1.00000	1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$167$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$168$	−1.41421	−1.41421
$169$	1.00000	1.00000
$170$	2.00000	2.00000
$171$	0	0
$172$	−2.00000	−2.00000
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.00000	1.00000
$176$	0	0
$177$	2.00000	2.00000
$178$	−1.41421	−1.41421
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.41421	−1.41421
$181$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$181$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$182$	0	0
$183$	−2.00000	−2.00000
$184$	−1.00000	−1.00000
$185$	0	0
$186$	2.00000	2.00000
$187$	0	0
$188$	−1.41421	−1.41421
$189$	0	0
$190$	0	0
$191$	−2.00000	−2.00000	−1.00000			$\pi$
$191$	−2.00000	−2.00000	−1.00000			$\pi$
$192$	1.41421	1.41421
$193$	−2.00000	−2.00000	−1.00000			$\pi$
$193$	−2.00000	−2.00000	−1.00000			$\pi$
$194$	1.41421	1.41421
$195$	0	0
$196$	1.00000	1.00000
$197$	2.00000	2.00000	1.00000			$0$
$197$	2.00000	2.00000	1.00000			$0$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.00000	−1.00000
$201$	0	0
$202$	0	0
$203$	0	0
$204$	2.00000	2.00000
$205$	0	0
$206$	0	0
$207$	1.00000	1.00000
$208$	0	0
$209$	0	0
$210$	2.00000	2.00000
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	2.00000	2.00000
$215$	2.82843	2.82843
$216$	0	0
$217$	−1.41421	−1.41421
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$223$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$224$	−1.00000	−1.00000
$225$	1.00000	1.00000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$229$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$230$	1.41421	1.41421
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	2.00000	2.00000
$236$	1.41421	1.41421
$237$	0	0
$238$	−1.41421	−1.41421
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−2.00000	−2.00000
$241$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$241$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$242$	1.00000	1.00000
$243$	−1.41421	−1.41421
$244$	−1.41421	−1.41421
$245$	−1.41421	−1.41421
$246$	0	0
$247$	0	0
$248$	1.41421	1.41421
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.00000	1.00000
$253$	0	0
$254$	0	0
$255$	−2.82843	−2.82843
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	2.82843	2.82843
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.41421	1.41421
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.00000	2.00000
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$271$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$272$	1.41421	1.41421
$273$	0	0
$274$	0	0
$275$	0	0
$276$	1.41421	1.41421
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−1.41421	−1.41421
$279$	−1.41421	−1.41421
$280$	1.41421	1.41421
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	2.00000	2.00000
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.00000	−1.00000
$289$	1.00000	1.00000
$290$	0	0
$291$	−2.00000	−2.00000
$292$	0	0
$293$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$294$	−1.41421	−1.41421
$295$	−2.00000	−2.00000
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	1.41421	1.41421
$301$	−2.00000	−2.00000
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.00000	2.00000
$306$	−1.41421	−1.41421
$307$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$307$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$308$	0	0
$309$	0	0
$310$	−2.00000	−2.00000
$311$	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
$313$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$313$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$314$	−1.41421	−1.41421
$315$	−1.41421	−1.41421
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	−1.41421	−1.41421
$321$	−2.82843	−2.82843
$322$	−1.00000	−1.00000
$323$	0	0
$324$	−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			$\pi$
$332$	0	0
$333$	0	0
$334$	1.41421	1.41421
$335$	0	0
$336$	1.41421	1.41421
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.00000	−1.00000
$339$	0	0
$340$	−2.00000	−2.00000
$341$	0	0
$342$	0	0
$343$	1.00000	1.00000
$344$	2.00000	2.00000
$345$	−2.00000	−2.00000
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.00000	−1.00000
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−2.00000	−2.00000
$355$	0	0
$356$	1.41421	1.41421
$357$	2.00000	2.00000
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	1.41421	1.41421
$361$	1.00000	1.00000
$362$	−1.41421	−1.41421
$363$	−1.41421	−1.41421
$364$	0	0
$365$	0	0
$366$	2.00000	2.00000
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.00000	1.00000
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−2.00000	−2.00000
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	1.41421	1.41421
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	2.00000	2.00000
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.41421	−1.41421
$385$	0	0
$386$	2.00000	2.00000
$387$	−2.00000	−2.00000
$388$	−1.41421	−1.41421
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	1.41421	1.41421
$392$	−1.00000	−1.00000
$393$	−2.00000	−2.00000
$394$	−2.00000	−2.00000
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.41421	1.41421
$406$	0	0
$407$	0	0
$408$	−2.00000	−2.00000
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.41421	1.41421
$414$	−1.00000	−1.00000
$415$	0	0
$416$	0	0
$417$	2.00000	2.00000
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	−2.00000	−2.00000
$421$	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
$426$	0	0
$427$	−1.41421	−1.41421
$428$	−2.00000	−2.00000
$429$	0	0
$430$	−2.82843	−2.82843
$431$	2.00000	2.00000	1.00000			$0$
$431$	2.00000	2.00000	1.00000			$0$
$432$	0	0
$433$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$433$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$434$	1.41421	1.41421
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$439$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$440$	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
$445$	−2.00000	−2.00000
$446$	−1.41421	−1.41421
$447$	0	0
$448$	1.00000	1.00000
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	−1.00000	−1.00000
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−1.41421	−1.41421
$459$	0	0
$460$	−1.41421	−1.41421
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	2.82843	2.82843
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	−2.00000	−2.00000
$471$	2.00000	2.00000
$472$	−1.41421	−1.41421
$473$	0	0
$474$	0	0
$475$	0	0
$476$	1.41421	1.41421
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	2.00000	2.00000
$481$	0	0
$482$	−1.41421	−1.41421
$483$	1.41421	1.41421
$484$	−1.00000	−1.00000
$485$	2.00000	2.00000
$486$	1.41421	1.41421
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.41421	1.41421
$489$	0	0
$490$	1.41421	1.41421
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.41421	−1.41421
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−2.00000	−2.00000
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−1.00000	−1.00000
$505$	0	0
$506$	0	0
$507$	1.41421	1.41421
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	2.82843	2.82843
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−2.82843	−2.82843
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$521$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.41421	−1.41421
$525$	1.41421	1.41421
$526$	0	0
$527$	−2.00000	−2.00000
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	1.41421	1.41421
$532$	0	0
$533$	0	0
$534$	−2.00000	−2.00000
$535$	2.82843	2.82843
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	−1.41421	−1.41421
$543$	2.00000	2.00000
$544$	−1.41421	−1.41421
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−1.41421	−1.41421
$550$	0	0
$551$	0	0
$552$	−1.41421	−1.41421
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.41421	1.41421
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.41421	1.41421
$559$	0	0
$560$	−1.41421	−1.41421
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−2.00000	−2.00000
$565$	0	0
$566$	0	0
$567$	−1.00000	−1.00000
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−2.00000	−2.00000	−1.00000			$\pi$
$571$	−2.00000	−2.00000	−1.00000			$\pi$
$572$	0	0
$573$	−2.82843	−2.82843
$574$	0	0
$575$	1.00000	1.00000
$576$	1.00000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.00000	−1.00000
$579$	−2.82843	−2.82843
$580$	0	0
$581$	0	0
$582$	2.00000	2.00000
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1.41421	1.41421
$587$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$587$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$588$	1.41421	1.41421
$589$	0	0
$590$	2.00000	2.00000
$591$	2.82843	2.82843
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−2.00000	−2.00000
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.41421	−1.41421
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	2.00000	2.00000
$603$	0	0
$604$	0	0
$605$	1.41421	1.41421
$606$	0	0
$607$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$608$	0	0
$609$	0	0
$610$	−2.00000	−2.00000
$611$	0	0
$612$	1.41421	1.41421
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.41421	1.41421
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	2.00000	2.00000
$621$	0	0
$622$	−1.41421	−1.41421
$623$	1.41421	1.41421
$624$	0	0
$625$	−1.00000	−1.00000
$626$	−1.41421	−1.41421
$627$	0	0
$628$	1.41421	1.41421
$629$	0	0
$630$	1.41421	1.41421
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.41421	1.41421
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	2.82843	2.82843
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	1.00000	1.00000
$645$	4.00000	4.00000
$646$	0	0
$647$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$647$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$648$	1.00000	1.00000
$649$	0	0
$650$	0	0
$651$	−2.00000	−2.00000
$652$	0	0
$653$	2.00000	2.00000	1.00000			$0$
$653$	2.00000	2.00000	1.00000			$0$
$654$	0	0
$655$	2.00000	2.00000
$656$	0	0
$657$	0	0
$658$	1.41421	1.41421
$659$	2.00000	2.00000	1.00000			$0$
$659$	2.00000	2.00000	1.00000			$0$
$660$	0	0
$661$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$661$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.41421	−1.41421
$669$	2.00000	2.00000
$670$	0	0
$671$	0	0
$672$	−1.41421	−1.41421
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	1.00000	1.00000
$677$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$677$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$678$	0	0
$679$	−1.41421	−1.41421
$680$	2.00000	2.00000
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−1.00000
$687$	2.00000	2.00000
$688$	−2.00000	−2.00000
$689$	0	0
$690$	2.00000	2.00000
$691$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$691$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.00000	−2.00000
$696$	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
$705$	2.82843	2.82843
$706$	0	0
$707$	0	0
$708$	2.00000	2.00000
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−1.41421	−1.41421
$713$	−1.41421	−1.41421
$714$	−2.00000	−2.00000
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$719$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$720$	−1.41421	−1.41421
$721$	0	0
$722$	−1.00000	−1.00000
$723$	2.00000	2.00000
$724$	1.41421	1.41421
$725$	0	0
$726$	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
$731$	−2.82843	−2.82843
$732$	−2.00000	−2.00000
$733$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$733$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$734$	0	0
$735$	−2.00000	−2.00000
$736$	−1.00000	−1.00000
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	2.00000	2.00000
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.00000	−2.00000
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−1.41421	−1.41421
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−2.00000	−2.00000
$765$	−2.00000	−2.00000
$766$	0	0
$767$	0	0
$768$	1.41421	1.41421
$769$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$769$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−2.00000
$773$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$773$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$774$	2.00000	2.00000
$775$	−1.41421	−1.41421
$776$	1.41421	1.41421
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−1.41421	−1.41421
$783$	0	0
$784$	1.00000	1.00000
$785$	−2.00000	−2.00000
$786$	2.00000	2.00000
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	2.00000	2.00000
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$797$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$798$	0	0
$799$	−2.00000	−2.00000
$800$	−1.00000	−1.00000
$801$	1.41421	1.41421
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.41421	−1.41421
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.00000	−2.00000	−1.00000			$\pi$
$809$	−2.00000	−2.00000	−1.00000			$\pi$
$810$	−1.41421	−1.41421
$811$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$811$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$812$	0	0
$813$	2.00000	2.00000
$814$	0	0
$815$	0	0
$816$	2.00000	2.00000
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	−1.41421	−1.41421
$827$	2.00000	2.00000	1.00000			$0$
$827$	2.00000	2.00000	1.00000			$0$
$828$	1.00000	1.00000
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.41421	1.41421
$834$	−2.00000	−2.00000
$835$	2.00000	2.00000
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	2.00000	2.00000
$841$	−1.00000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.41421	−1.41421
$846$	1.41421	1.41421
$847$	−1.00000	−1.00000
$848$	0	0
$849$	0	0
$850$	−1.41421	−1.41421
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	1.41421	1.41421
$855$	0	0
$856$	2.00000	2.00000
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$859$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$860$	2.82843	2.82843
$861$	0	0
$862$	−2.00000	−2.00000
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	1.41421	1.41421
$867$	1.41421	1.41421
$868$	−1.41421	−1.41421
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.41421	−1.41421
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−1.41421	−1.41421
$879$	−2.00000	−2.00000
$880$	0	0
$881$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$881$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$882$	−1.00000	−1.00000
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−2.82843	−2.82843
$886$	0	0
$887$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$887$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$888$	0	0
$889$	0	0
$890$	2.00000	2.00000
$891$	0	0
$892$	1.41421	1.41421
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.00000	−1.00000
$897$	0	0
$898$	0	0
$899$	0	0
$900$	1.00000	1.00000
$901$	0	0
$902$	0	0
$903$	−2.82843	−2.82843
$904$	0	0
$905$	−2.00000	−2.00000
$906$	0	0
$907$	2.00000	2.00000	1.00000			$0$
$907$	2.00000	2.00000	1.00000			$0$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.82843	2.82843
$916$	1.41421	1.41421
$917$	−1.41421	−1.41421
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	1.41421	1.41421
$921$	−2.00000	−2.00000
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	−2.82843	−2.82843
$931$	0	0
$932$	0	0
$933$	2.00000	2.00000
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$937$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$938$	0	0
$939$	2.00000	2.00000
$940$	2.00000	2.00000
$941$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$941$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$942$	−2.00000	−2.00000
$943$	0	0
$944$	1.41421	1.41421
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−1.41421	−1.41421
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	2.82843	2.82843
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−2.00000	−2.00000
$961$	1.00000	1.00000
$962$	0	0
$963$	−2.00000	−2.00000
$964$	1.41421	1.41421
$965$	2.82843	2.82843
$966$	−1.41421	−1.41421
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.00000	1.00000
$969$	0	0
$970$	−2.00000	−2.00000
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.41421	−1.41421
$973$	1.41421	1.41421
$974$	0	0
$975$	0	0
$976$	−1.41421	−1.41421
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	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
$985$	−2.82843	−2.82843
$986$	0	0
$987$	−2.00000	−2.00000
$988$	0	0
$989$	−2.00000	−2.00000
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	1.41421	1.41421
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	644.1.h.d.643.2	yes	2
4.3	odd	2		644.1.h.c.643.1	✓	2
7.6	odd	2	inner	644.1.h.d.643.1	yes	2
23.22	odd	2		644.1.h.c.643.2	yes	2
28.27	even	2		644.1.h.c.643.2	yes	2
92.91	even	2	inner	644.1.h.d.643.1	yes	2
161.160	even	2		644.1.h.c.643.1	✓	2
644.643	odd	2	CM	644.1.h.d.643.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.1.h.c.643.1	✓	2	4.3	odd	2
644.1.h.c.643.1	✓	2	161.160	even	2
644.1.h.c.643.2	yes	2	23.22	odd	2
644.1.h.c.643.2	yes	2	28.27	even	2
644.1.h.d.643.1	yes	2	7.6	odd	2	inner
644.1.h.d.643.1	yes	2	92.91	even	2	inner
644.1.h.d.643.2	yes	2	1.1	even	1	trivial
644.1.h.d.643.2	yes	2	644.643	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	644.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.41421 q^{10} +1.41421 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +1.41421 q^{17} -1.00000 q^{18} -1.41421 q^{20} +1.41421 q^{21} +1.00000 q^{23} -1.41421 q^{24} +1.00000 q^{25} +1.00000 q^{28} +2.00000 q^{30} -1.41421 q^{31} -1.00000 q^{32} -1.41421 q^{34} -1.41421 q^{35} +1.00000 q^{36} +1.41421 q^{40} -1.41421 q^{42} -2.00000 q^{43} -1.41421 q^{45} -1.00000 q^{46} -1.41421 q^{47} +1.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -1.00000 q^{56} +1.41421 q^{59} -2.00000 q^{60} -1.41421 q^{61} +1.41421 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.41421 q^{68} +1.41421 q^{69} +1.41421 q^{70} -1.00000 q^{72} +1.41421 q^{75} -1.41421 q^{80} -1.00000 q^{81} +1.41421 q^{84} -2.00000 q^{85} +2.00000 q^{86} +1.41421 q^{89} +1.41421 q^{90} +1.00000 q^{92} -2.00000 q^{93} +1.41421 q^{94} -1.41421 q^{96} -1.41421 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{14} - 4 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{23} + 2 q^{25} + 2 q^{28} + 4 q^{30} - 2 q^{32} + 2 q^{36} - 4 q^{43} - 2 q^{46} + 2 q^{49} - 2 q^{50} + 4 q^{51} - 2 q^{56} - 4 q^{60} + 2 q^{63} + 2 q^{64} - 2 q^{72} - 2 q^{81} - 4 q^{85} + 4 q^{86} + 2 q^{92} - 4 q^{93} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^14 - 4 * q^15 + 2 * q^16 - 2 * q^18 + 2 * q^23 + 2 * q^25 + 2 * q^28 + 4 * q^30 - 2 * q^32 + 2 * q^36 - 4 * q^43 - 2 * q^46 + 2 * q^49 - 2 * q^50 + 4 * q^51 - 2 * q^56 - 4 * q^60 + 2 * q^63 + 2 * q^64 - 2 * q^72 - 2 * q^81 - 4 * q^85 + 4 * q^86 + 2 * q^92 - 4 * q^93 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more