LMFDB - Embedded newform 2175.1.b.a.2174.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,1,Mod(2174,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.2174");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2175.b (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:	$1.08546640248$
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:	no (minimal twist has level 87)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.87.1
Artin image:	$C_4\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			2174.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2175.2174
Dual form			2175.1.b.a.2174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -1.00000 q^{21} -1.00000i q^{22} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} -1.00000 q^{29} -1.00000i q^{33} -1.00000 q^{34} +1.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} +1.00000i q^{47} -1.00000i q^{48} -1.00000 q^{51} +1.00000 q^{54} -1.00000 q^{56} -1.00000i q^{58} -1.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} +1.00000i q^{67} -1.00000i q^{72} -1.00000i q^{77} +1.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} -1.00000i q^{87} -1.00000i q^{88} +1.00000 q^{89} +1.00000 q^{91} -1.00000 q^{94} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} - 2 q^{16} - 2 q^{21} - 2 q^{24} + 2 q^{26} - 2 q^{29} - 2 q^{34} + 2 q^{39} + 4 q^{41} - 2 q^{51} + 2 q^{54} - 2 q^{56} - 2 q^{64} + 2 q^{66} + 2 q^{81} + 2 q^{89} + 2 q^{91} - 2 q^{94} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^6 - 2 * q^9 - 2 * q^11 - 2 * q^14 - 2 * q^16 - 2 * q^21 - 2 * q^24 + 2 * q^26 - 2 * q^29 - 2 * q^34 + 2 * q^39 + 4 * q^41 - 2 * q^51 + 2 * q^54 - 2 * q^56 - 2 * q^64 + 2 * q^66 + 2 * q^81 + 2 * q^89 + 2 * q^91 - 2 * q^94 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$2$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	1.00000i	1.00000i
$3$	1.00000i	1.00000i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	−1.00000	−1.00000
$7$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$	1.00000i	1.00000i
$9$	−1.00000	−1.00000
$10$	0	0
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	0	0
$13$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$14$	−1.00000	−1.00000
$15$	0	0
$16$	−1.00000	−1.00000
$17$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$17$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$18$	− 1.00000i	− 1.00000i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−1.00000	−1.00000
$22$	− 1.00000i	− 1.00000i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−1.00000
$25$	0	0
$26$	1.00000	1.00000
$27$	− 1.00000i	− 1.00000i
$28$	0	0
$29$	−1.00000	−1.00000
$29$	−1.00000	−1.00000
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	− 1.00000i	− 1.00000i
$34$	−1.00000	−1.00000
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	1.00000	1.00000
$40$	0	0
$41$	2.00000	2.00000	1.00000			$0$
$41$	2.00000	2.00000	1.00000			$0$
$42$	− 1.00000i	− 1.00000i
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$47$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$48$	− 1.00000i	− 1.00000i
$49$	0	0
$50$	0	0
$51$	−1.00000	−1.00000
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	1.00000	1.00000
$55$	0	0
$56$	−1.00000	−1.00000
$57$	0	0
$58$	− 1.00000i	− 1.00000i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 1.00000i	− 1.00000i
$64$	−1.00000	−1.00000
$65$	0	0
$66$	1.00000	1.00000
$67$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$67$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 1.00000i	− 1.00000i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.00000i	− 1.00000i
$78$	1.00000i	1.00000i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	2.00000i	2.00000i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.00000i	− 1.00000i
$88$	− 1.00000i	− 1.00000i
$89$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$89$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$90$	0	0
$91$	1.00000	1.00000
$92$	0	0
$93$	0	0
$94$	−1.00000	−1.00000
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	1.00000	1.00000
$100$	0	0
$101$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$102$	− 1.00000i	− 1.00000i
$103$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$103$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$109$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 1.00000i
$113$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$113$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000i	1.00000i
$118$	0	0
$119$	−1.00000	−1.00000
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2.00000i	2.00000i
$124$	0	0
$125$	0	0
$126$	1.00000	1.00000
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$132$	0	0
$133$	0	0
$134$	−1.00000	−1.00000
$135$	0	0
$136$	−1.00000	−1.00000
$137$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$137$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$138$	0	0
$139$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$140$	0	0
$141$	−1.00000	−1.00000
$142$	0	0
$143$	1.00000i	1.00000i
$144$	1.00000	1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	2.00000	2.00000	1.00000			$0$
$151$	2.00000	2.00000	1.00000			$0$
$152$	0	0
$153$	− 1.00000i	− 1.00000i
$154$	1.00000	1.00000
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	1.00000i	1.00000i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	− 1.00000i	− 1.00000i
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	1.00000	1.00000
$175$	0	0
$176$	1.00000	1.00000
$177$	0	0
$178$	1.00000i	1.00000i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$182$	1.00000i	1.00000i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.00000i	− 1.00000i
$188$	0	0
$189$	1.00000	1.00000
$190$	0	0
$191$	2.00000	2.00000	1.00000			$0$
$191$	2.00000	2.00000	1.00000			$0$
$192$	− 1.00000i	− 1.00000i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	1.00000i	1.00000i
$199$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$200$	0	0
$201$	−1.00000	−1.00000
$202$	− 1.00000i	− 1.00000i
$203$	− 1.00000i	− 1.00000i
$204$	0	0
$205$	0	0
$206$	−2.00000	−2.00000
$207$	0	0
$208$	1.00000i	1.00000i
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	1.00000	1.00000
$217$	0	0
$218$	1.00000i	1.00000i
$219$	0	0
$220$	0	0
$221$	1.00000	1.00000
$222$	0	0
$223$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$223$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$224$	0	0
$225$	0	0
$226$	1.00000	1.00000
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	1.00000	1.00000
$232$	− 1.00000i	− 1.00000i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−1.00000	−1.00000
$235$	0	0
$236$	0	0
$237$	0	0
$238$	− 1.00000i	− 1.00000i
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$242$	0	0
$243$	1.00000i	1.00000i
$244$	0	0
$245$	0	0
$246$	−2.00000	−2.00000
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.00000	1.00000
$262$	− 1.00000i	− 1.00000i
$263$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$263$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$264$	1.00000	1.00000
$265$	0	0
$266$	0	0
$267$	1.00000i	1.00000i
$268$	0	0
$269$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 1.00000i	− 1.00000i
$273$	1.00000i	1.00000i
$274$	2.00000	2.00000
$275$	0	0
$276$	0	0
$277$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$277$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$278$	1.00000i	1.00000i
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	− 1.00000i	− 1.00000i
$283$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$283$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	−1.00000	−1.00000
$287$	2.00000i	2.00000i
$288$	0	0
$289$	0	0
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$293$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000i	1.00000i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	2.00000i	2.00000i
$303$	− 1.00000i	− 1.00000i
$304$	0	0
$305$	0	0
$306$	1.00000	1.00000
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−2.00000	−2.00000
$310$	0	0
$311$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$312$	1.00000i	1.00000i
$313$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$313$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$317$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$318$	0	0
$319$	1.00000	1.00000
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.00000i	1.00000i
$328$	2.00000i	2.00000i
$329$	−1.00000	−1.00000
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	1.00000	1.00000
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	1.00000	1.00000
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−2.00000	−2.00000	−1.00000			$\pi$
$349$	−2.00000	−2.00000	−1.00000			$\pi$
$350$	0	0
$351$	−1.00000	−1.00000
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.00000i	− 1.00000i
$358$	0	0
$359$	−2.00000	−2.00000	−1.00000			$\pi$
$359$	−2.00000	−2.00000	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	− 1.00000i	− 1.00000i
$363$	0	0
$364$	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
$369$	−2.00000	−2.00000
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$373$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$374$	1.00000	1.00000
$375$	0	0
$376$	−1.00000	−1.00000
$377$	1.00000i	1.00000i
$378$	1.00000i	1.00000i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	2.00000i	2.00000i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 1.00000i	− 1.00000i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$397$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$398$	1.00000i	1.00000i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	− 1.00000i	− 1.00000i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	1.00000	1.00000
$407$	0	0
$408$	− 1.00000i	− 1.00000i
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	2.00000	2.00000
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.00000i	1.00000i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	− 1.00000i	− 1.00000i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.00000	−1.00000
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.00000i	1.00000i
$433$	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
$438$	0	0
$439$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$439$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$440$	0	0
$441$	0	0
$442$	1.00000i	1.00000i
$443$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$444$	0	0
$445$	0	0
$446$	1.00000	1.00000
$447$	0	0
$448$	− 1.00000i	− 1.00000i
$449$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$450$	0	0
$451$	−2.00000	−2.00000
$452$	0	0
$453$	2.00000i	2.00000i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$457$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$458$	0	0
$459$	1.00000	1.00000
$460$	0	0
$461$	2.00000	2.00000	1.00000			$0$
$461$	2.00000	2.00000	1.00000			$0$
$462$	1.00000i	1.00000i
$463$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$463$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$464$	1.00000	1.00000
$465$	0	0
$466$	0	0
$467$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$467$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$468$	0	0
$469$	−1.00000	−1.00000
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.00000	−2.00000	−1.00000			$\pi$
$479$	−2.00000	−2.00000	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	− 1.00000i	− 1.00000i
$483$	0	0
$484$	0	0
$485$	0	0
$486$	−1.00000	−1.00000
$487$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$487$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.00000	2.00000	1.00000			$0$
$491$	2.00000	2.00000	1.00000			$0$
$492$	0	0
$493$	− 1.00000i	− 1.00000i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$499$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$500$	0	0
$501$	0	0
$502$	− 1.00000i	− 1.00000i
$503$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$503$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$504$	1.00000	1.00000
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.00000i	− 1.00000i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.00000i	1.00000i
$523$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$523$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$524$	0	0
$525$	0	0
$526$	−2.00000	−2.00000
$527$	0	0
$528$	1.00000i	1.00000i
$529$	−1.00000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.00000i	− 2.00000i
$534$	−1.00000	−1.00000
$535$	0	0
$536$	−1.00000	−1.00000
$537$	0	0
$538$	1.00000i	1.00000i
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	− 1.00000i	− 1.00000i
$544$	0	0
$545$	0	0
$546$	−1.00000	−1.00000
$547$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$547$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−1.00000	−1.00000
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.00000	1.00000
$562$	0	0
$563$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$563$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$564$	0	0
$565$	0	0
$566$	−2.00000	−2.00000
$567$	1.00000i	1.00000i
$568$	0	0
$569$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$569$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$570$	0	0
$571$	2.00000	2.00000	1.00000			$0$
$571$	2.00000	2.00000	1.00000			$0$
$572$	0	0
$573$	2.00000i	2.00000i
$574$	−2.00000	−2.00000
$575$	0	0
$576$	1.00000	1.00000
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1.00000	1.00000
$587$	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
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	−1.00000	−1.00000
$595$	0	0
$596$	0	0
$597$	1.00000i	1.00000i
$598$	0	0
$599$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	− 1.00000i	− 1.00000i
$604$	0	0
$605$	0	0
$606$	1.00000	1.00000
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	1.00000	1.00000
$610$	0	0
$611$	1.00000	1.00000
$612$	0	0
$613$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$613$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$614$	0	0
$615$	0	0
$616$	1.00000	1.00000
$617$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$617$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$618$	− 2.00000i	− 2.00000i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	− 1.00000i	− 1.00000i
$623$	1.00000i	1.00000i
$624$	−1.00000	−1.00000
$625$	0	0
$626$	1.00000	1.00000
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	0	0
$633$	0	0
$634$	−1.00000	−1.00000
$635$	0	0
$636$	0	0
$637$	0	0
$638$	1.00000i	1.00000i
$639$	0	0
$640$	0	0
$641$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$642$	0	0
$643$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$643$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000i	1.00000i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$653$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$654$	−1.00000	−1.00000
$655$	0	0
$656$	−2.00000	−2.00000
$657$	0	0
$658$	− 1.00000i	− 1.00000i
$659$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$659$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$660$	0	0
$661$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$662$	0	0
$663$	1.00000i	1.00000i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.00000	1.00000
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$673$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$677$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$678$	1.00000i	1.00000i
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−1.00000
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$692$	0	0
$693$	1.00000i	1.00000i
$694$	0	0
$695$	0	0
$696$	1.00000	1.00000
$697$	2.00000i	2.00000i
$698$	− 2.00000i	− 2.00000i
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	− 1.00000i	− 1.00000i
$703$	0	0
$704$	1.00000	1.00000
$705$	0	0
$706$	0	0
$707$	− 1.00000i	− 1.00000i
$708$	0	0
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.00000i	1.00000i
$713$	0	0
$714$	1.00000	1.00000
$715$	0	0
$716$	0	0
$717$	0	0
$718$	− 2.00000i	− 2.00000i
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−2.00000	−2.00000
$722$	1.00000i	1.00000i
$723$	− 1.00000i	− 1.00000i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	1.00000i	1.00000i
$729$	−1.00000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.00000i	− 1.00000i
$738$	− 2.00000i	− 2.00000i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$744$	0	0
$745$	0	0
$746$	−2.00000	−2.00000
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 1.00000i	− 1.00000i
$753$	− 1.00000i	− 1.00000i
$754$	−1.00000	−1.00000
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1.00000i	1.00000i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$773$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	1.00000i	1.00000i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.00000i	1.00000i
$784$	0	0
$785$	0	0
$786$	1.00000	1.00000
$787$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$787$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$788$	0	0
$789$	−2.00000	−2.00000
$790$	0	0
$791$	1.00000	1.00000
$792$	1.00000i	1.00000i
$793$	0	0
$794$	2.00000	2.00000
$795$	0	0
$796$	0	0
$797$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$797$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$798$	0	0
$799$	−1.00000	−1.00000
$800$	0	0
$801$	−1.00000	−1.00000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.00000i	1.00000i
$808$	− 1.00000i	− 1.00000i
$809$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$810$	0	0
$811$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	1.00000	1.00000
$817$	0	0
$818$	0	0
$819$	−1.00000	−1.00000
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.00000i	2.00000i
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−2.00000	−2.00000
$825$	0	0
$826$	0	0
$827$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$827$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−1.00000	−1.00000
$832$	1.00000i	1.00000i
$833$	0	0
$834$	−1.00000	−1.00000
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$839$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	1.00000	1.00000
$847$	0	0
$848$	0	0
$849$	−2.00000	−2.00000
$850$	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	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	− 1.00000i	− 1.00000i
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	−2.00000	−2.00000
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.00000	1.00000
$872$	1.00000i	1.00000i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$877$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$878$	1.00000i	1.00000i
$879$	1.00000	1.00000
$880$	0	0
$881$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$882$	0	0
$883$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$883$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	1.00000	1.00000
$887$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$887$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−1.00000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	1.00000
$897$	0	0
$898$	1.00000i	1.00000i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 2.00000i	− 2.00000i
$903$	0	0
$904$	1.00000	1.00000
$905$	0	0
$906$	−2.00000	−2.00000
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	1.00000	1.00000
$910$	0	0
$911$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$911$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$912$	0	0
$913$	0	0
$914$	−1.00000	−1.00000
$915$	0	0
$916$	0	0
$917$	− 1.00000i	− 1.00000i
$918$	1.00000i	1.00000i
$919$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$919$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$920$	0	0
$921$	0	0
$922$	2.00000i	2.00000i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1.00000	1.00000
$927$	− 2.00000i	− 2.00000i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 1.00000i	− 1.00000i
$934$	2.00000	2.00000
$935$	0	0
$936$	−1.00000	−1.00000
$937$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$937$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$938$	− 1.00000i	− 1.00000i
$939$	1.00000	1.00000
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$947$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−1.00000	−1.00000
$952$	− 1.00000i	− 1.00000i
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.00000i	1.00000i
$958$	− 2.00000i	− 2.00000i
$959$	2.00000	2.00000
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.00000	2.00000	1.00000			$0$
$971$	2.00000	2.00000	1.00000			$0$
$972$	0	0
$973$	1.00000i	1.00000i
$974$	2.00000	2.00000
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−1.00000	−1.00000
$980$	0	0
$981$	−1.00000	−1.00000
$982$	2.00000i	2.00000i
$983$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$983$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$984$	−2.00000	−2.00000
$985$	0	0
$986$	1.00000	1.00000
$987$	− 1.00000i	− 1.00000i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	1.00000i	1.00000i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.1.b.a.2174.2		2
3.2	odd	2		2175.1.b.b.2174.1		2
5.2	odd	4		87.1.d.a.86.1	✓	1
5.3	odd	4		2175.1.h.b.1826.1		1
5.4	even	2	inner	2175.1.b.a.2174.1		2
15.2	even	4		87.1.d.b.86.1	yes	1
15.8	even	4		2175.1.h.a.1826.1		1
15.14	odd	2		2175.1.b.b.2174.2		2
20.7	even	4		1392.1.i.a.1217.1		1
29.28	even	2		2175.1.b.b.2174.1		2
45.2	even	12		2349.1.h.a.782.1		2
45.7	odd	12		2349.1.h.b.782.1		2
45.22	odd	12		2349.1.h.b.1565.1		2
45.32	even	12		2349.1.h.a.1565.1		2
60.47	odd	4		1392.1.i.b.1217.1		1
87.86	odd	2	CM	2175.1.b.a.2174.2		2
145.2	even	28		2523.1.j.b.605.1		12
145.7	odd	28		2523.1.h.b.1952.1		6
145.12	even	4		2523.1.b.b.842.1		2
145.17	even	4		2523.1.b.b.842.2		2
145.22	odd	28		2523.1.h.a.1952.1		6
145.27	even	28		2523.1.j.b.605.2		12
145.28	odd	4		2175.1.h.a.1826.1		1
145.32	even	28		2523.1.j.b.1412.1		12
145.37	even	28		2523.1.j.b.1415.1		12
145.42	odd	28		2523.1.h.a.1745.1		6
145.47	even	28		2523.1.j.b.1619.2		12
145.52	odd	28		2523.1.h.b.1037.1		6
145.57	odd	4		87.1.d.b.86.1	yes	1
145.62	odd	28		2523.1.h.a.2333.1		6
145.67	odd	28		2523.1.h.a.1949.1		6
145.72	even	28		2523.1.j.b.2327.2		12
145.77	even	28		2523.1.j.b.1031.2		12
145.82	odd	28		2523.1.h.b.236.1		6
145.92	odd	28		2523.1.h.a.236.1		6
145.97	even	28		2523.1.j.b.1031.1		12
145.102	even	28		2523.1.j.b.2327.1		12
145.107	odd	28		2523.1.h.b.1949.1		6
145.112	odd	28		2523.1.h.b.2333.1		6
145.122	odd	28		2523.1.h.a.1037.1		6
145.127	even	28		2523.1.j.b.1619.1		12
145.132	odd	28		2523.1.h.b.1745.1		6
145.137	even	28		2523.1.j.b.1415.2		12
145.142	even	28		2523.1.j.b.1412.2		12
145.144	even	2		2175.1.b.b.2174.2		2
435.2	odd	28		2523.1.j.b.605.2		12
435.17	odd	4		2523.1.b.b.842.1		2
435.32	odd	28		2523.1.j.b.1412.2		12
435.47	odd	28		2523.1.j.b.1619.1		12
435.62	even	28		2523.1.h.b.2333.1		6
435.77	odd	28		2523.1.j.b.1031.1		12
435.92	even	28		2523.1.h.b.236.1		6
435.107	even	28		2523.1.h.a.1949.1		6
435.122	even	28		2523.1.h.b.1037.1		6
435.137	odd	28		2523.1.j.b.1415.1		12
435.152	even	28		2523.1.h.a.1952.1		6
435.167	even	28		2523.1.h.b.1952.1		6
435.173	even	4		2175.1.h.b.1826.1		1
435.182	odd	28		2523.1.j.b.1415.2		12
435.197	even	28		2523.1.h.a.1037.1		6
435.212	even	28		2523.1.h.b.1949.1		6
435.227	even	28		2523.1.h.a.236.1		6
435.242	odd	28		2523.1.j.b.1031.2		12
435.257	even	28		2523.1.h.a.2333.1		6
435.272	odd	28		2523.1.j.b.1619.2		12
435.287	odd	28		2523.1.j.b.1412.1		12
435.302	odd	4		2523.1.b.b.842.2		2
435.317	odd	28		2523.1.j.b.605.1		12
435.332	even	28		2523.1.h.b.1745.1		6
435.347	even	4		87.1.d.a.86.1	✓	1
435.362	odd	28		2523.1.j.b.2327.1		12
435.392	odd	28		2523.1.j.b.2327.2		12
435.422	even	28		2523.1.h.a.1745.1		6
435.434	odd	2	inner	2175.1.b.a.2174.1		2
580.347	even	4		1392.1.i.b.1217.1		1
1305.202	odd	12		2349.1.h.a.1565.1		2
1305.347	even	12		2349.1.h.b.1565.1		2
1305.637	odd	12		2349.1.h.a.782.1		2
1305.1217	even	12		2349.1.h.b.782.1		2
1740.347	odd	4		1392.1.i.a.1217.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.1.d.a.86.1	✓	1	5.2	odd	4
87.1.d.a.86.1	✓	1	435.347	even	4
87.1.d.b.86.1	yes	1	15.2	even	4
87.1.d.b.86.1	yes	1	145.57	odd	4
1392.1.i.a.1217.1		1	20.7	even	4
1392.1.i.a.1217.1		1	1740.347	odd	4
1392.1.i.b.1217.1		1	60.47	odd	4
1392.1.i.b.1217.1		1	580.347	even	4
2175.1.b.a.2174.1		2	5.4	even	2	inner
2175.1.b.a.2174.1		2	435.434	odd	2	inner
2175.1.b.a.2174.2		2	1.1	even	1	trivial
2175.1.b.a.2174.2		2	87.86	odd	2	CM
2175.1.b.b.2174.1		2	3.2	odd	2
2175.1.b.b.2174.1		2	29.28	even	2
2175.1.b.b.2174.2		2	15.14	odd	2
2175.1.b.b.2174.2		2	145.144	even	2
2175.1.h.a.1826.1		1	15.8	even	4
2175.1.h.a.1826.1		1	145.28	odd	4
2175.1.h.b.1826.1		1	5.3	odd	4
2175.1.h.b.1826.1		1	435.173	even	4
2349.1.h.a.782.1		2	45.2	even	12
2349.1.h.a.782.1		2	1305.637	odd	12
2349.1.h.a.1565.1		2	45.32	even	12
2349.1.h.a.1565.1		2	1305.202	odd	12
2349.1.h.b.782.1		2	45.7	odd	12
2349.1.h.b.782.1		2	1305.1217	even	12
2349.1.h.b.1565.1		2	45.22	odd	12
2349.1.h.b.1565.1		2	1305.347	even	12
2523.1.b.b.842.1		2	145.12	even	4
2523.1.b.b.842.1		2	435.17	odd	4
2523.1.b.b.842.2		2	145.17	even	4
2523.1.b.b.842.2		2	435.302	odd	4
2523.1.h.a.236.1		6	145.92	odd	28
2523.1.h.a.236.1		6	435.227	even	28
2523.1.h.a.1037.1		6	145.122	odd	28
2523.1.h.a.1037.1		6	435.197	even	28
2523.1.h.a.1745.1		6	145.42	odd	28
2523.1.h.a.1745.1		6	435.422	even	28
2523.1.h.a.1949.1		6	145.67	odd	28
2523.1.h.a.1949.1		6	435.107	even	28
2523.1.h.a.1952.1		6	145.22	odd	28
2523.1.h.a.1952.1		6	435.152	even	28
2523.1.h.a.2333.1		6	145.62	odd	28
2523.1.h.a.2333.1		6	435.257	even	28
2523.1.h.b.236.1		6	145.82	odd	28
2523.1.h.b.236.1		6	435.92	even	28
2523.1.h.b.1037.1		6	145.52	odd	28
2523.1.h.b.1037.1		6	435.122	even	28
2523.1.h.b.1745.1		6	145.132	odd	28
2523.1.h.b.1745.1		6	435.332	even	28
2523.1.h.b.1949.1		6	145.107	odd	28
2523.1.h.b.1949.1		6	435.212	even	28
2523.1.h.b.1952.1		6	145.7	odd	28
2523.1.h.b.1952.1		6	435.167	even	28
2523.1.h.b.2333.1		6	145.112	odd	28
2523.1.h.b.2333.1		6	435.62	even	28
2523.1.j.b.605.1		12	145.2	even	28
2523.1.j.b.605.1		12	435.317	odd	28
2523.1.j.b.605.2		12	145.27	even	28
2523.1.j.b.605.2		12	435.2	odd	28
2523.1.j.b.1031.1		12	145.97	even	28
2523.1.j.b.1031.1		12	435.77	odd	28
2523.1.j.b.1031.2		12	145.77	even	28
2523.1.j.b.1031.2		12	435.242	odd	28
2523.1.j.b.1412.1		12	145.32	even	28
2523.1.j.b.1412.1		12	435.287	odd	28
2523.1.j.b.1412.2		12	145.142	even	28
2523.1.j.b.1412.2		12	435.32	odd	28
2523.1.j.b.1415.1		12	145.37	even	28
2523.1.j.b.1415.1		12	435.137	odd	28
2523.1.j.b.1415.2		12	145.137	even	28
2523.1.j.b.1415.2		12	435.182	odd	28
2523.1.j.b.1619.1		12	145.127	even	28
2523.1.j.b.1619.1		12	435.47	odd	28
2523.1.j.b.1619.2		12	145.47	even	28
2523.1.j.b.1619.2		12	435.272	odd	28
2523.1.j.b.2327.1		12	145.102	even	28
2523.1.j.b.2327.1		12	435.362	odd	28
2523.1.j.b.2327.2		12	145.72	even	28
2523.1.j.b.2327.2		12	435.392	odd	28

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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -1.00000 q^{21} -1.00000i q^{22} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} -1.00000 q^{29} -1.00000i q^{33} -1.00000 q^{34} +1.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} +1.00000i q^{47} -1.00000i q^{48} -1.00000 q^{51} +1.00000 q^{54} -1.00000 q^{56} -1.00000i q^{58} -1.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} +1.00000i q^{67} -1.00000i q^{72} -1.00000i q^{77} +1.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} -1.00000i q^{87} -1.00000i q^{88} +1.00000 q^{89} +1.00000 q^{91} -1.00000 q^{94} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} - 2 q^{16} - 2 q^{21} - 2 q^{24} + 2 q^{26} - 2 q^{29} - 2 q^{34} + 2 q^{39} + 4 q^{41} - 2 q^{51} + 2 q^{54} - 2 q^{56} - 2 q^{64} + 2 q^{66} + 2 q^{81} + 2 q^{89} + 2 q^{91} - 2 q^{94} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^6 - 2 * q^9 - 2 * q^11 - 2 * q^14 - 2 * q^16 - 2 * q^21 - 2 * q^24 + 2 * q^26 - 2 * q^29 - 2 * q^34 + 2 * q^39 + 4 * q^41 - 2 * q^51 + 2 * q^54 - 2 * q^56 - 2 * q^64 + 2 * q^66 + 2 * q^81 + 2 * q^89 + 2 * q^91 - 2 * q^94 + 2 * q^99

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