LMFDB - Embedded newform 812.1.l.a.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [812,1,Mod(307,812)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(812, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

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

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

chi := DirichletCharacter("812.307");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$812 = 2^{2} \cdot 7 \cdot 29$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	812.l (of order $4$ , degree $2$ , 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:	$0.405240790258$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.2731568.1

Embedding invariants

Embedding label			307.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	812.307
Dual form			812.1.l.a.447.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +1.00000i q^{9} +(1.00000 + 1.00000i) q^{11} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +(1.00000 - 1.00000i) q^{22} -2.00000i q^{23} -1.00000 q^{25} -1.00000i q^{28} +1.00000i q^{29} -1.00000i q^{32} -1.00000i q^{36} +(-1.00000 - 1.00000i) q^{37} +(1.00000 + 1.00000i) q^{43} +(-1.00000 - 1.00000i) q^{44} -2.00000 q^{46} -1.00000 q^{49} +1.00000i q^{50} +2.00000 q^{53} -1.00000 q^{56} +1.00000 q^{58} -1.00000 q^{63} -1.00000 q^{64} +2.00000 q^{71} -1.00000 q^{72} +(-1.00000 + 1.00000i) q^{74} +(-1.00000 + 1.00000i) q^{77} +(-1.00000 - 1.00000i) q^{79} -1.00000 q^{81} +(1.00000 - 1.00000i) q^{86} +(-1.00000 + 1.00000i) q^{88} +2.00000i q^{92} +1.00000i q^{98} +(-1.00000 + 1.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{4} + 2 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 2 q^{25} - 2 q^{37} + 2 q^{43} - 2 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{53} - 2 q^{56} + 2 q^{58} - 2 q^{63} - 2 q^{64} + 4 q^{71} - 2 q^{72}+ \cdots - 2 q^{99}+O(q^{100})$ 2 * q - 2 * q^4 + 2 * q^11 + 2 * q^14 + 2 * q^16 + 2 * q^18 + 2 * q^22 - 2 * q^25 - 2 * q^37 + 2 * q^43 - 2 * q^44 - 4 * q^46 - 2 * q^49 + 4 * q^53 - 2 * q^56 + 2 * q^58 - 2 * q^63 - 2 * q^64 + 4 * q^71 - 2 * q^72 - 2 * q^74 - 2 * q^77 - 2 * q^79 - 2 * q^81 + 2 * q^86 - 2 * q^88 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/812\mathbb{Z}\right)^\times$ .

$n$	$407$	$465$	$785$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		−	1.00000i		−	1.00000i
$2$		−	1.00000i		−	1.00000i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$	−1.00000			−1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$			1.00000i			1.00000i
$9$			1.00000i			1.00000i
$10$		0			0
$11$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$11$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	1.00000			1.00000
$15$		0			0
$16$	1.00000			1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	1.00000			1.00000
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$	1.00000	−	1.00000i	1.00000	−	1.00000i
$23$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$23$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$24$		0			0
$25$	−1.00000			−1.00000
$26$		0			0
$27$		0			0
$28$		−	1.00000i		−	1.00000i
$29$			1.00000i			1.00000i
$29$			1.00000i			1.00000i
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$		−	1.00000i		−	1.00000i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		−	1.00000i		−	1.00000i
$37$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$37$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$43$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$44$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$45$		0			0
$46$	−2.00000			−2.00000
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$			1.00000i			1.00000i
$51$		0			0
$52$		0			0
$53$	2.00000			2.00000			1.00000			$0$
$53$	2.00000			2.00000			1.00000			$0$
$54$		0			0
$55$		0			0
$56$	−1.00000			−1.00000
$57$		0			0
$58$	1.00000			1.00000
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$	−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	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			2.00000			1.00000			$0$
$71$	2.00000			2.00000			1.00000			$0$
$72$	−1.00000			−1.00000
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$75$		0			0
$76$		0			0
$77$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$78$		0			0
$79$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$79$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$80$		0			0
$81$	−1.00000			−1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	1.00000	−	1.00000i	1.00000	−	1.00000i
$87$		0			0
$88$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$			2.00000i			2.00000i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$			1.00000i			1.00000i
$99$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$100$	1.00000			1.00000
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		−	2.00000i		−	2.00000i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$109$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$110$		0			0
$111$		0			0
$112$			1.00000i			1.00000i
$113$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$113$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		−	1.00000i		−	1.00000i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			1.00000i			1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$			1.00000i			1.00000i
$127$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$127$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$128$			1.00000i			1.00000i
$129$		0			0
$130$		0			0
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$137$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		−	2.00000i		−	2.00000i
$143$		0			0
$144$			1.00000i			1.00000i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.00000	+	1.00000i	1.00000	+	1.00000i
$149$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$149$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$	1.00000	+	1.00000i	1.00000	+	1.00000i
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$159$		0			0
$160$		0			0
$161$	2.00000			2.00000
$162$			1.00000i			1.00000i
$163$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$163$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		−	1.00000i		−	1.00000i
$176$	1.00000	+	1.00000i	1.00000	+	1.00000i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	2.00000			2.00000
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$191$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$192$		0			0
$193$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$193$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$194$		0			0
$195$		0			0
$196$	1.00000			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	1.00000	+	1.00000i	1.00000	+	1.00000i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		−	1.00000i		−	1.00000i
$201$		0			0
$202$		0			0
$203$	−1.00000			−1.00000
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.00000			2.00000
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$212$	−2.00000			−2.00000
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.00000			−2.00000
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	1.00000			1.00000
$225$		−	1.00000i		−	1.00000i
$226$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$	−1.00000			−1.00000
$233$	2.00000			2.00000			1.00000			$0$
$233$	2.00000			2.00000			1.00000			$0$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	1.00000			1.00000
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$	1.00000			1.00000
$253$	2.00000	−	2.00000i	2.00000	−	2.00000i
$254$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	1.00000	−	1.00000i	1.00000	−	1.00000i
$260$		0			0
$261$	−1.00000			−1.00000
$262$		0			0
$263$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$263$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$271$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$272$		0			0
$273$		0			0
$274$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$275$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$276$		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
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−2.00000			−2.00000
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.00000			1.00000
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$	1.00000	−	1.00000i	1.00000	−	1.00000i
$297$		0			0
$298$	2.00000			2.00000
$299$		0			0
$300$		0			0
$301$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$	1.00000	−	1.00000i	1.00000	−	1.00000i
$309$		0			0
$310$		0			0
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		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
$315$		0			0
$316$	1.00000	+	1.00000i	1.00000	+	1.00000i
$317$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$317$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$318$		0			0
$319$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$320$		0			0
$321$		0			0
$322$		−	2.00000i		−	2.00000i
$323$		0			0
$324$	1.00000			1.00000
$325$		0			0
$326$	1.00000	+	1.00000i	1.00000	+	1.00000i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$331$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$332$		0			0
$333$	1.00000	−	1.00000i	1.00000	−	1.00000i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$337$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$338$			1.00000i			1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$345$		0			0
$346$		0			0
$347$	2.00000			2.00000			1.00000			$0$
$347$	2.00000			2.00000			1.00000			$0$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−1.00000			−1.00000
$351$		0			0
$352$	1.00000	−	1.00000i	1.00000	−	1.00000i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$359$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		−	2.00000i		−	2.00000i
$369$		0			0
$370$		0			0
$371$			2.00000i			2.00000i
$372$		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
$379$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$379$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$387$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$388$		0			0
$389$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$389$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	1.00000i		−	1.00000i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	1.00000	−	1.00000i	1.00000	−	1.00000i
$397$		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			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$			1.00000i			1.00000i
$407$		−	2.00000i		−	2.00000i
$408$		0			0
$409$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$409$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		−	2.00000i		−	2.00000i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$421$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$422$	1.00000	+	1.00000i	1.00000	+	1.00000i
$423$		0			0
$424$			2.00000i			2.00000i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$			2.00000i			2.00000i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$443$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$449$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$450$	−1.00000			−1.00000
$451$		0			0
$452$	1.00000	+	1.00000i	1.00000	+	1.00000i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	−2.00000			−2.00000			−1.00000			$\pi$
$463$	−2.00000			−2.00000			−1.00000			$\pi$
$464$			1.00000i			1.00000i
$465$		0			0
$466$		−	2.00000i		−	2.00000i
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.00000i			2.00000i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			2.00000i			2.00000i
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		−	1.00000i		−	1.00000i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$491$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.00000i			2.00000i
$498$		0			0
$499$	2.00000			2.00000			1.00000			$0$
$499$	2.00000			2.00000			1.00000			$0$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		−	1.00000i		−	1.00000i
$505$		0			0
$506$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$507$		0			0
$508$	1.00000	+	1.00000i	1.00000	+	1.00000i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$			1.00000i			1.00000i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$527$		0			0
$528$		0			0
$529$	−3.00000			−3.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$540$		0			0
$541$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$541$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$549$		0			0
$550$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$551$		0			0
$552$		0			0
$553$	1.00000	−	1.00000i	1.00000	−	1.00000i
$554$		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
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	1.00000i		−	1.00000i
$568$			2.00000i			2.00000i
$569$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$569$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$570$		0			0
$571$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$571$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2.00000i			2.00000i
$576$		−	1.00000i		−	1.00000i
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−1.00000			−1.00000
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.00000	+	2.00000i	2.00000	+	2.00000i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		−	2.00000i		−	2.00000i
$597$		0			0
$598$		0			0
$599$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$599$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$601$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$602$	1.00000	+	1.00000i	1.00000	+	1.00000i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$617$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$617$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	1.00000	−	1.00000i	1.00000	−	1.00000i
$633$		0			0
$634$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.00000	+	1.00000i	1.00000	+	1.00000i
$639$			2.00000i			2.00000i
$640$		0			0
$641$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$641$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−2.00000			−2.00000
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		−	1.00000i		−	1.00000i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.00000	−	1.00000i	1.00000	−	1.00000i
$653$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$653$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$659$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$667$	2.00000			2.00000
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$673$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$674$	1.00000	−	1.00000i	1.00000	−	1.00000i
$675$		0			0
$676$	1.00000			1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$683$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$684$		0			0
$685$		0			0
$686$	−1.00000			−1.00000
$687$		0			0
$688$	1.00000	+	1.00000i	1.00000	+	1.00000i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$694$		−	2.00000i		−	2.00000i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$			1.00000i			1.00000i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$710$		0			0
$711$	1.00000	−	1.00000i	1.00000	−	1.00000i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	1.00000	−	1.00000i	1.00000	−	1.00000i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	1.00000			1.00000
$723$		0			0
$724$		0			0
$725$		−	1.00000i		−	1.00000i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	−2.00000			−2.00000
$737$		0			0
$738$		0			0
$739$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$739$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$	2.00000			2.00000
$743$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$743$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$757$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$758$	1.00000	−	1.00000i	1.00000	−	1.00000i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	2.00000			2.00000
$764$	1.00000	+	1.00000i	1.00000	+	1.00000i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$769$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$770$		0			0
$771$		0			0
$772$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$	1.00000	+	1.00000i	1.00000	+	1.00000i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.00000	−	1.00000i	1.00000	−	1.00000i
$779$		0			0
$780$		0			0
$781$	2.00000	+	2.00000i	2.00000	+	2.00000i
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.00000	−	1.00000i	1.00000	−	1.00000i
$792$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$			1.00000i			1.00000i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$809$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	1.00000			1.00000
$813$		0			0
$814$	−2.00000			−2.00000
$815$		0			0
$816$		0			0
$817$		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
$823$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$823$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$827$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$828$	−2.00000			−2.00000
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$843$		0			0
$844$	1.00000	−	1.00000i	1.00000	−	1.00000i
$845$		0			0
$846$		0			0
$847$	−1.00000			−1.00000
$848$	2.00000			2.00000
$849$		0			0
$850$		0			0
$851$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.00000			−2.00000			−1.00000			$\pi$
$863$	−2.00000			−2.00000			−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	2.00000i		−	2.00000i
$870$		0			0
$871$		0			0
$872$	2.00000			2.00000
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.00000			2.00000			1.00000			$0$
$877$	2.00000			2.00000			1.00000			$0$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$881$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$882$	−1.00000			−1.00000
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	1.00000	+	1.00000i	1.00000	+	1.00000i
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$	1.00000	−	1.00000i	1.00000	−	1.00000i
$890$		0			0
$891$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−1.00000			−1.00000
$897$		0			0
$898$	1.00000	+	1.00000i	1.00000	+	1.00000i
$899$		0			0
$900$			1.00000i			1.00000i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.00000	−	1.00000i	1.00000	−	1.00000i
$905$		0			0
$906$		0			0
$907$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$907$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$911$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.00000	+	1.00000i	1.00000	+	1.00000i
$926$			2.00000i			2.00000i
$927$		0			0
$928$	1.00000			1.00000
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−2.00000			−2.00000
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	2.00000			2.00000
$947$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$947$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$948$		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	$0.5\pi$
$954$	2.00000			2.00000
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.00000	+	1.00000i	1.00000	+	1.00000i
$960$		0			0
$961$			1.00000i			1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$967$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$968$	−1.00000			−1.00000
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.00000			2.00000
$982$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.00000	−	2.00000i	2.00000	−	2.00000i
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$	2.00000			2.00000
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		−	2.00000i		−	2.00000i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	812.1.l.a.307.1	✓	2
4.3	odd	2		812.1.l.b.307.1	yes	2
7.6	odd	2	CM	812.1.l.a.307.1	✓	2
28.27	even	2		812.1.l.b.307.1	yes	2
29.12	odd	4		812.1.l.b.447.1	yes	2
116.99	even	4	inner	812.1.l.a.447.1	yes	2
203.41	even	4		812.1.l.b.447.1	yes	2
812.447	odd	4	inner	812.1.l.a.447.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
812.1.l.a.307.1	✓	2	1.1	even	1	trivial
812.1.l.a.307.1	✓	2	7.6	odd	2	CM
812.1.l.a.447.1	yes	2	116.99	even	4	inner
812.1.l.a.447.1	yes	2	812.447	odd	4	inner
812.1.l.b.307.1	yes	2	4.3	odd	2
812.1.l.b.307.1	yes	2	28.27	even	2
812.1.l.b.447.1	yes	2	29.12	odd	4
812.1.l.b.447.1	yes	2	203.41	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	812.1.l.a.307.1

Level	$812$
Weight	$1$
Character	812.307
Analytic conductor	$0.405$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{4}$
CM discriminant	-7
Inner twists	$4$