LMFDB - Embedded newform 2312.1.e.b.1155.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,1,Mod(1155,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, 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("2312.1155");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2312.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.15383830921$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.1680747204608.3

Embedding invariants

Embedding label			1155.2
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	2312.1155
Dual form			2312.1.e.b.1155.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.765367i q^{3} +1.00000 q^{4} -0.765367i q^{6} +1.00000 q^{8} +0.414214 q^{9} +O(q^{10})$	$q+1.00000 q^{2} -0.765367i q^{3} +1.00000 q^{4} -0.765367i q^{6} +1.00000 q^{8} +0.414214 q^{9} +1.84776i q^{11} -0.765367i q^{12} +1.00000 q^{16} +0.414214 q^{18} +1.84776i q^{22} -0.765367i q^{24} -1.00000 q^{25} -1.08239i q^{27} +1.00000 q^{32} +1.41421 q^{33} +0.414214 q^{36} -1.84776i q^{41} -1.41421 q^{43} +1.84776i q^{44} -0.765367i q^{48} -1.00000 q^{49} -1.00000 q^{50} -1.08239i q^{54} -1.41421 q^{59} +1.00000 q^{64} +1.41421 q^{66} -1.41421 q^{67} +0.414214 q^{72} -1.84776i q^{73} +0.765367i q^{75} -0.414214 q^{81} -1.84776i q^{82} +1.41421 q^{83} -1.41421 q^{86} +1.84776i q^{88} +1.41421 q^{89} -0.765367i q^{96} +0.765367i q^{97} -1.00000 q^{98} +0.765367i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 4 * q^9	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 4 q^{9} + 4 q^{16} - 4 q^{18} - 4 q^{25} + 4 q^{32} - 4 q^{36} - 4 q^{49} - 4 q^{50} + 4 q^{64} - 4 q^{72} + 4 q^{81} - 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 4 * q^9 + 4 * q^16 - 4 * q^18 - 4 * q^25 + 4 * q^32 - 4 * q^36 - 4 * q^49 - 4 * q^50 + 4 * q^64 - 4 * q^72 + 4 * q^81 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$3$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$4$	1.00000	1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	− 0.765367i	− 0.765367i
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	1.00000
$9$	0.414214	0.414214
$10$	0	0
$11$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$11$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$12$	− 0.765367i	− 0.765367i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	1.00000
$17$	0	0
$17$	0	0
$18$	0.414214	0.414214
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	1.84776i	1.84776i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	− 0.765367i	− 0.765367i
$25$	−1.00000	−1.00000
$26$	0	0
$27$	− 1.08239i	− 1.08239i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	1.00000
$33$	1.41421	1.41421
$34$	0	0
$35$	0	0
$36$	0.414214	0.414214
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$41$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$42$	0	0
$43$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$43$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$44$	1.84776i	1.84776i
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 0.765367i	− 0.765367i
$49$	−1.00000	−1.00000
$50$	−1.00000	−1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	− 1.08239i	− 1.08239i
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	1.41421	1.41421
$67$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$67$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0.414214	0.414214
$73$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$73$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$74$	0	0
$75$	0.765367i	0.765367i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−0.414214	−0.414214
$82$	− 1.84776i	− 1.84776i
$83$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$83$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$84$	0	0
$85$	0	0
$86$	−1.41421	−1.41421
$87$	0	0
$88$	1.84776i	1.84776i
$89$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$89$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	− 0.765367i	− 0.765367i
$97$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$97$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$98$	−1.00000	−1.00000
$99$	0.765367i	0.765367i
$100$	−1.00000	−1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$107$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$108$	− 1.08239i	− 1.08239i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$113$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.41421	−1.41421
$119$	0	0
$120$	0	0
$121$	−2.41421	−2.41421
$122$	0	0
$123$	−1.41421	−1.41421
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	1.08239i	1.08239i
$130$	0	0
$131$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$131$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$132$	1.41421	1.41421
$133$	0	0
$134$	−1.41421	−1.41421
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$139$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0.414214	0.414214
$145$	0	0
$146$	− 1.84776i	− 1.84776i
$147$	0.765367i	0.765367i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.765367i	0.765367i
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−0.414214	−0.414214
$163$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$163$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$164$	− 1.84776i	− 1.84776i
$165$	0	0
$166$	1.41421	1.41421
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−1.41421	−1.41421
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	1.84776i	1.84776i
$177$	1.08239i	1.08239i
$178$	1.41421	1.41421
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 0.765367i	− 0.765367i
$193$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$193$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$194$	0.765367i	0.765367i
$195$	0	0
$196$	−1.00000	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0.765367i	0.765367i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−1.00000
$201$	1.08239i	1.08239i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$211$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$212$	0	0
$213$	0	0
$214$	0.765367i	0.765367i
$215$	0	0
$216$	− 1.08239i	− 1.08239i
$217$	0	0
$218$	0	0
$219$	−1.41421	−1.41421
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−0.414214	−0.414214
$226$	− 0.765367i	− 0.765367i
$227$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$227$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$233$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$234$	0	0
$235$	0	0
$236$	−1.41421	−1.41421
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$241$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$242$	−2.41421	−2.41421
$243$	− 0.765367i	− 0.765367i
$244$	0	0
$245$	0	0
$246$	−1.41421	−1.41421
$247$	0	0
$248$	0	0
$249$	− 1.08239i	− 1.08239i
$250$	0	0
$251$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$251$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$257$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$258$	1.08239i	1.08239i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.84776i	1.84776i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.41421	1.41421
$265$	0	0
$266$	0	0
$267$	− 1.08239i	− 1.08239i
$268$	−1.41421	−1.41421
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.84776i	− 1.84776i
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0.765367i	0.765367i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$283$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.414214	0.414214
$289$	0	0
$290$	0	0
$291$	0.585786	0.585786
$292$	− 1.84776i	− 1.84776i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.765367i	0.765367i
$295$	0	0
$296$	0	0
$297$	2.00000	2.00000
$298$	0	0
$299$	0	0
$300$	0.765367i	0.765367i
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.00000	−2.00000	−1.00000			$\pi$
$307$	−2.00000	−2.00000	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$313$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0.585786	0.585786
$322$	0	0
$323$	0	0
$324$	−0.414214	−0.414214
$325$	0	0
$326$	− 0.765367i	− 0.765367i
$327$	0	0
$328$	− 1.84776i	− 1.84776i
$329$	0	0
$330$	0	0
$331$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$331$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$332$	1.41421	1.41421
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$337$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$338$	1.00000	1.00000
$339$	−0.585786	−0.585786
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−1.41421	−1.41421
$345$	0	0
$346$	0	0
$347$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$347$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	1.84776i	1.84776i
$353$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$353$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$354$	1.08239i	1.08239i
$355$	0	0
$356$	1.41421	1.41421
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−1.00000	−1.00000
$362$	0	0
$363$	1.84776i	1.84776i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	− 0.765367i	− 0.765367i
$370$	0	0
$371$	0	0
$372$	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
$378$	0	0
$379$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$379$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	− 0.765367i	− 0.765367i
$385$	0	0
$386$	0.765367i	0.765367i
$387$	−0.585786	−0.585786
$388$	0.765367i	0.765367i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−1.00000
$393$	1.41421	1.41421
$394$	0	0
$395$	0	0
$396$	0.765367i	0.765367i
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−1.00000
$401$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$401$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$402$	1.08239i	1.08239i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$409$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.585786	0.585786
$418$	0	0
$419$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$419$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.84776i	1.84776i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0.765367i	0.765367i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.08239i	− 1.08239i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−1.41421	−1.41421
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−0.414214	−0.414214
$442$	0	0
$443$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$443$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$449$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$450$	−0.414214	−0.414214
$451$	3.41421	3.41421
$452$	− 0.765367i	− 0.765367i
$453$	0	0
$454$	− 0.765367i	− 0.765367i
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	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
$465$	0	0
$466$	− 0.765367i	− 0.765367i
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.41421	−1.41421
$473$	− 2.61313i	− 2.61313i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	1.84776i	1.84776i
$483$	0	0
$484$	−2.41421	−2.41421
$485$	0	0
$486$	− 0.765367i	− 0.765367i
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−0.585786	−0.585786
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−1.41421	−1.41421
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	− 1.08239i	− 1.08239i
$499$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$499$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$500$	0	0
$501$	0	0
$502$	−1.41421	−1.41421
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.765367i	− 0.765367i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	1.41421	1.41421
$515$	0	0
$516$	1.08239i	1.08239i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$521$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$522$	0	0
$523$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$523$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$524$	1.84776i	1.84776i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	1.41421	1.41421
$529$	−1.00000	−1.00000
$530$	0	0
$531$	−0.585786	−0.585786
$532$	0	0
$533$	0	0
$534$	− 1.08239i	− 1.08239i
$535$	0	0
$536$	−1.41421	−1.41421
$537$	0	0
$538$	0	0
$539$	− 1.84776i	− 1.84776i
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$547$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$548$	0	0
$549$	0	0
$550$	− 1.84776i	− 1.84776i
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0.765367i	0.765367i
$557$	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
$563$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$563$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$564$	0	0
$565$	0	0
$566$	− 1.84776i	− 1.84776i
$567$	0	0
$568$	0	0
$569$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$569$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$570$	0	0
$571$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$571$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0.414214	0.414214
$577$	−2.00000	−2.00000	−1.00000			$\pi$
$577$	−2.00000	−2.00000	−1.00000			$\pi$
$578$	0	0
$579$	0.585786	0.585786
$580$	0	0
$581$	0	0
$582$	0.585786	0.585786
$583$	0	0
$584$	− 1.84776i	− 1.84776i
$585$	0	0
$586$	0	0
$587$	−2.00000	−2.00000	−1.00000			$\pi$
$587$	−2.00000	−2.00000	−1.00000			$\pi$
$588$	0.765367i	0.765367i
$589$	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	$0.250000\pi$
$594$	2.00000	2.00000
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.765367i	0.765367i
$601$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$601$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$602$	0	0
$603$	−0.585786	−0.585786
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−2.00000	−2.00000
$615$	0	0
$616$	0	0
$617$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$617$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$618$	0	0
$619$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$619$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	0.765367i	0.765367i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	1.41421	1.41421
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$641$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$642$	0.585786	0.585786
$643$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$643$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.414214	−0.414214
$649$	− 2.61313i	− 2.61313i
$650$	0	0
$651$	0	0
$652$	− 0.765367i	− 0.765367i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	− 1.84776i	− 1.84776i
$657$	− 0.765367i	− 0.765367i
$658$	0	0
$659$	2.00000	2.00000	1.00000			$0$
$659$	2.00000	2.00000	1.00000			$0$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.41421	1.41421
$663$	0	0
$664$	1.41421	1.41421
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$673$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$674$	0.765367i	0.765367i
$675$	1.08239i	1.08239i
$676$	1.00000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−0.585786	−0.585786
$679$	0	0
$680$	0	0
$681$	−0.585786	−0.585786
$682$	0	0
$683$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$683$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.41421	−1.41421
$689$	0	0
$690$	0	0
$691$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$691$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$692$	0	0
$693$	0	0
$694$	− 1.84776i	− 1.84776i
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−0.585786	−0.585786
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.84776i	1.84776i
$705$	0	0
$706$	1.41421	1.41421
$707$	0	0
$708$	1.08239i	1.08239i
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	1.41421	1.41421
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	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
$723$	1.41421	1.41421
$724$	0	0
$725$	0	0
$726$	1.84776i	1.84776i
$727$	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
$735$	0	0
$736$	0	0
$737$	− 2.61313i	− 2.61313i
$738$	− 0.765367i	− 0.765367i
$739$	2.00000	2.00000	1.00000			$0$
$739$	2.00000	2.00000	1.00000			$0$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.585786	0.585786
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	1.08239i	1.08239i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	1.84776i	1.84776i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 0.765367i	− 0.765367i
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	− 1.08239i	− 1.08239i
$772$	0.765367i	0.765367i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−0.585786	−0.585786
$775$	0	0
$776$	0.765367i	0.765367i
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.00000	−1.00000
$785$	0	0
$786$	1.41421	1.41421
$787$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$787$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.765367i	0.765367i
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−1.00000
$801$	0.585786	0.585786
$802$	0.765367i	0.765367i
$803$	3.41421	3.41421
$804$	1.08239i	1.08239i
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$809$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$810$	0	0
$811$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$811$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1.41421	−1.41421
$819$	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		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−1.41421	−1.41421
$826$	0	0
$827$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$827$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0.585786	0.585786
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 0.765367i	− 0.765367i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−1.00000	−1.00000
$842$	0	0
$843$	0	0
$844$	1.84776i	1.84776i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.41421	−1.41421
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0.765367i	0.765367i
$857$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$857$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$858$	0	0
$859$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$859$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	− 1.08239i	− 1.08239i
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0.317025i	0.317025i
$874$	0	0
$875$	0	0
$876$	−1.41421	−1.41421
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$881$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$882$	−0.414214	−0.414214
$883$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$883$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$884$	0	0
$885$	0	0
$886$	1.41421	1.41421
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 0.765367i	− 0.765367i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 0.765367i	− 0.765367i
$899$	0	0
$900$	−0.414214	−0.414214
$901$	0	0
$902$	3.41421	3.41421
$903$	0	0
$904$	− 0.765367i	− 0.765367i
$905$	0	0
$906$	0	0
$907$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$907$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$908$	− 0.765367i	− 0.765367i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	2.61313i	2.61313i
$914$	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
$921$	1.53073i	1.53073i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$929$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$930$	0	0
$931$	0	0
$932$	− 0.765367i	− 0.765367i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	2.00000	1.00000			$0$
$937$	2.00000	2.00000	1.00000			$0$
$938$	0	0
$939$	0.585786	0.585786
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	−1.41421	−1.41421
$945$	0	0
$946$	− 2.61313i	− 2.61313i
$947$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$947$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$953$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.00000	−1.00000
$962$	0	0
$963$	0.317025i	0.317025i
$964$	1.84776i	1.84776i
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.41421	−2.41421
$969$	0	0
$970$	0	0
$971$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$971$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$972$	− 0.765367i	− 0.765367i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$977$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$978$	−0.585786	−0.585786
$979$	2.61313i	2.61313i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−1.41421	−1.41421
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	− 1.08239i	− 1.08239i
$994$	0	0
$995$	0	0
$996$	− 1.08239i	− 1.08239i
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 1.84776i	− 1.84776i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.1.e.b.1155.2		4
8.3	odd	2	CM	2312.1.e.b.1155.2		4
17.2	even	8		2312.1.j.c.251.2		8
17.3	odd	16		2312.1.p.d.1555.1		4
17.4	even	4		2312.1.f.c.579.2		4
17.5	odd	16		136.1.p.a.43.1	yes	4
17.6	odd	16		2312.1.p.a.1579.1		4
17.7	odd	16		2312.1.p.b.155.1		4
17.8	even	8		2312.1.j.c.1483.2		8
17.9	even	8		2312.1.j.c.1483.3		8
17.10	odd	16		136.1.p.a.19.1	✓	4
17.11	odd	16		2312.1.p.d.1579.1		4
17.12	odd	16		2312.1.p.b.179.1		4
17.13	even	4		2312.1.f.c.579.3		4
17.14	odd	16		2312.1.p.a.1555.1		4
17.15	even	8		2312.1.j.c.251.3		8
17.16	even	2	inner	2312.1.e.b.1155.3		4
51.5	even	16		1224.1.bv.a.451.1		4
51.44	even	16		1224.1.bv.a.19.1		4
68.27	even	16		544.1.bl.a.495.1		4
68.39	even	16		544.1.bl.a.111.1		4
85.22	even	16		3400.1.br.b.2899.1		4
85.27	even	16		3400.1.br.a.699.1		4
85.39	odd	16		3400.1.ce.a.451.1		4
85.44	odd	16		3400.1.ce.a.1651.1		4
85.73	even	16		3400.1.br.a.2899.1		4
85.78	even	16		3400.1.br.b.699.1		4
136.3	even	16		2312.1.p.d.1555.1		4
136.5	odd	16		544.1.bl.a.111.1		4
136.11	even	16		2312.1.p.d.1579.1		4
136.19	odd	8		2312.1.j.c.251.2		8
136.27	even	16		136.1.p.a.19.1	✓	4
136.43	odd	8		2312.1.j.c.1483.3		8
136.59	odd	8		2312.1.j.c.1483.2		8
136.61	odd	16		544.1.bl.a.495.1		4
136.67	odd	2	inner	2312.1.e.b.1155.3		4
136.75	even	16		2312.1.p.b.155.1		4
136.83	odd	8		2312.1.j.c.251.3		8
136.91	even	16		2312.1.p.a.1579.1		4
136.99	even	16		2312.1.p.a.1555.1		4
136.107	even	16		136.1.p.a.43.1	yes	4
136.115	odd	4		2312.1.f.c.579.3		4
136.123	odd	4		2312.1.f.c.579.2		4
136.131	even	16		2312.1.p.b.179.1		4
408.107	odd	16		1224.1.bv.a.451.1		4
408.299	odd	16		1224.1.bv.a.19.1		4
680.27	odd	16		3400.1.br.a.699.1		4
680.107	odd	16		3400.1.br.b.2899.1		4
680.163	odd	16		3400.1.br.b.699.1		4
680.243	odd	16		3400.1.br.a.2899.1		4
680.299	even	16		3400.1.ce.a.1651.1		4
680.379	even	16		3400.1.ce.a.451.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.1.p.a.19.1	✓	4	17.10	odd	16
136.1.p.a.19.1	✓	4	136.27	even	16
136.1.p.a.43.1	yes	4	17.5	odd	16
136.1.p.a.43.1	yes	4	136.107	even	16
544.1.bl.a.111.1		4	68.39	even	16
544.1.bl.a.111.1		4	136.5	odd	16
544.1.bl.a.495.1		4	68.27	even	16
544.1.bl.a.495.1		4	136.61	odd	16
1224.1.bv.a.19.1		4	51.44	even	16
1224.1.bv.a.19.1		4	408.299	odd	16
1224.1.bv.a.451.1		4	51.5	even	16
1224.1.bv.a.451.1		4	408.107	odd	16
2312.1.e.b.1155.2		4	1.1	even	1	trivial
2312.1.e.b.1155.2		4	8.3	odd	2	CM
2312.1.e.b.1155.3		4	17.16	even	2	inner
2312.1.e.b.1155.3		4	136.67	odd	2	inner
2312.1.f.c.579.2		4	17.4	even	4
2312.1.f.c.579.2		4	136.123	odd	4
2312.1.f.c.579.3		4	17.13	even	4
2312.1.f.c.579.3		4	136.115	odd	4
2312.1.j.c.251.2		8	17.2	even	8
2312.1.j.c.251.2		8	136.19	odd	8
2312.1.j.c.251.3		8	17.15	even	8
2312.1.j.c.251.3		8	136.83	odd	8
2312.1.j.c.1483.2		8	17.8	even	8
2312.1.j.c.1483.2		8	136.59	odd	8
2312.1.j.c.1483.3		8	17.9	even	8
2312.1.j.c.1483.3		8	136.43	odd	8
2312.1.p.a.1555.1		4	17.14	odd	16
2312.1.p.a.1555.1		4	136.99	even	16
2312.1.p.a.1579.1		4	17.6	odd	16
2312.1.p.a.1579.1		4	136.91	even	16
2312.1.p.b.155.1		4	17.7	odd	16
2312.1.p.b.155.1		4	136.75	even	16
2312.1.p.b.179.1		4	17.12	odd	16
2312.1.p.b.179.1		4	136.131	even	16
2312.1.p.d.1555.1		4	17.3	odd	16
2312.1.p.d.1555.1		4	136.3	even	16
2312.1.p.d.1579.1		4	17.11	odd	16
2312.1.p.d.1579.1		4	136.11	even	16
3400.1.br.a.699.1		4	85.27	even	16
3400.1.br.a.699.1		4	680.27	odd	16
3400.1.br.a.2899.1		4	85.73	even	16
3400.1.br.a.2899.1		4	680.243	odd	16
3400.1.br.b.699.1		4	85.78	even	16
3400.1.br.b.699.1		4	680.163	odd	16
3400.1.br.b.2899.1		4	85.22	even	16
3400.1.br.b.2899.1		4	680.107	odd	16
3400.1.ce.a.451.1		4	85.39	odd	16
3400.1.ce.a.451.1		4	680.379	even	16
3400.1.ce.a.1651.1		4	85.44	odd	16
3400.1.ce.a.1651.1		4	680.299	even	16

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2312.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.765367i q^{3} +1.00000 q^{4} -0.765367i q^{6} +1.00000 q^{8} +0.414214 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.765367i q^{3} +1.00000 q^{4} -0.765367i q^{6} +1.00000 q^{8} +0.414214 q^{9} +1.84776i q^{11} -0.765367i q^{12} +1.00000 q^{16} +0.414214 q^{18} +1.84776i q^{22} -0.765367i q^{24} -1.00000 q^{25} -1.08239i q^{27} +1.00000 q^{32} +1.41421 q^{33} +0.414214 q^{36} -1.84776i q^{41} -1.41421 q^{43} +1.84776i q^{44} -0.765367i q^{48} -1.00000 q^{49} -1.00000 q^{50} -1.08239i q^{54} -1.41421 q^{59} +1.00000 q^{64} +1.41421 q^{66} -1.41421 q^{67} +0.414214 q^{72} -1.84776i q^{73} +0.765367i q^{75} -0.414214 q^{81} -1.84776i q^{82} +1.41421 q^{83} -1.41421 q^{86} +1.84776i q^{88} +1.41421 q^{89} -0.765367i q^{96} +0.765367i q^{97} -1.00000 q^{98} +0.765367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 4 q^{9} + 4 q^{16} - 4 q^{18} - 4 q^{25} + 4 q^{32} - 4 q^{36} - 4 q^{49} - 4 q^{50} + 4 q^{64} - 4 q^{72} + 4 q^{81} - 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 4 * q^9 + 4 * q^16 - 4 * q^18 - 4 * q^25 + 4 * q^32 - 4 * q^36 - 4 * q^49 - 4 * q^50 + 4 * q^64 - 4 * q^72 + 4 * q^81 - 4 * q^98

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