LMFDB - Embedded newform 3192.1.eb.a.341.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,1,Mod(341,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5, 3]))

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

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

chi := DirichletCharacter("3192.341");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3192.eb (of order $6$, 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:	$1.59301552032$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			341.5
Root			$0.342020 - 0.939693i$ of defining polynomial
Character	$\chi$	$=$	3192.341
Dual form			3192.1.eb.a.2621.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.866025 + 0.500000i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.766044 + 0.642788i) q^{6} +(0.173648 + 0.984808i) q^{7} +1.00000i q^{8} +(-0.766044 - 0.642788i) q^{9} +(-0.984808 + 0.173648i) q^{12} +0.347296i q^{13} +(-0.342020 + 0.939693i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.939693 + 1.62760i) q^{17} +(-0.342020 - 0.939693i) q^{18} +(-0.866025 - 0.500000i) q^{19} +(-0.984808 - 0.173648i) q^{21} +(0.592396 + 0.342020i) q^{23} +(-0.939693 - 0.342020i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-0.173648 + 0.300767i) q^{26} +(0.866025 - 0.500000i) q^{27} +(-0.766044 + 0.642788i) q^{28} -1.53209i q^{29} +(-0.866025 + 0.500000i) q^{32} +1.87939i q^{34} +(0.173648 - 0.984808i) q^{36} +(0.866025 - 1.50000i) q^{37} +(-0.500000 - 0.866025i) q^{38} +(-0.326352 - 0.118782i) q^{39} +(-0.766044 - 0.642788i) q^{42} +(0.342020 + 0.592396i) q^{46} +(-0.500000 + 0.866025i) q^{47} +(-0.642788 - 0.766044i) q^{48} +(-0.939693 + 0.342020i) q^{49} -1.00000i q^{50} +(-1.85083 + 0.326352i) q^{51} +(-0.300767 + 0.173648i) q^{52} +(-1.62760 + 0.939693i) q^{53} +1.00000 q^{54} +(-0.984808 + 0.173648i) q^{56} +(0.766044 - 0.642788i) q^{57} +(0.766044 - 1.32683i) q^{58} +(0.342020 + 0.592396i) q^{59} +(0.500000 - 0.866025i) q^{63} -1.00000 q^{64} +(0.642788 + 1.11334i) q^{67} +(-0.939693 + 1.62760i) q^{68} +(-0.524005 + 0.439693i) q^{69} +(0.642788 - 0.766044i) q^{72} +(1.70574 - 0.984808i) q^{73} +(1.50000 - 0.866025i) q^{74} +(0.984808 - 0.173648i) q^{75} -1.00000i q^{76} +(-0.223238 - 0.266044i) q^{78} +(0.173648 + 0.984808i) q^{81} +(-0.342020 - 0.939693i) q^{84} +(1.43969 + 0.524005i) q^{87} +(-0.342020 + 0.0603074i) q^{91} +0.684040i q^{92} +(-0.866025 + 0.500000i) q^{94} +(-0.173648 - 0.984808i) q^{96} +(-0.984808 - 0.173648i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{4} - 6 q^{16} - 6 q^{25} - 6 q^{38} - 6 q^{39} - 6 q^{47} + 12 q^{54} + 6 q^{63} - 12 q^{64} + 18 q^{74} + 6 q^{87}+O(q^{100}) $ 12 * q + 6 * q^4 - 6 * q^16 - 6 * q^25 - 6 * q^38 - 6 * q^39 - 6 * q^47 + 12 * q^54 + 6 * q^63 - 12 * q^64 + 18 * q^74 + 6 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3192\mathbb{Z}\right)^\times$.

$n$	$799$	$913$	$1009$	$1597$	$2129$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$	$-1$

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 $

$2$	0.866025	+	0.500000i	0.866025	+	0.500000i
$2$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$3$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$4$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$7$	0.173648	+	0.984808i	0.173648	+	0.984808i
$7$	0.173648	+	0.984808i	0.173648	+	0.984808i
$8$			1.00000i			1.00000i
$9$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$13$			0.347296i			0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$13$			0.347296i			0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$14$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$17$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$18$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$19$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$19$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$20$		0			0
$21$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$22$		0			0
$23$	0.592396	+	0.342020i	0.592396	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$23$	0.592396	+	0.342020i	0.592396	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$24$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$27$	0.866025	−	0.500000i	0.866025	−	0.500000i
$28$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$29$		−	1.53209i		−	1.53209i	−0.642788	−	0.766044i	$-0.722222\pi$
$29$		−	1.53209i		−	1.53209i	0.642788	−	0.766044i	$-0.277778\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$33$		0			0
$34$			1.87939i			1.87939i
$35$		0			0
$36$	0.173648	−	0.984808i	0.173648	−	0.984808i
$37$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$37$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$38$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$39$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	0.342020	+	0.592396i	0.342020	+	0.592396i
$47$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$47$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$48$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$49$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$50$		−	1.00000i		−	1.00000i
$51$	−1.85083	+	0.326352i	−1.85083	+	0.326352i
$52$	−0.300767	+	0.173648i	−0.300767	+	0.173648i
$53$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.642788	+	0.766044i	$0.722222\pi$
$53$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.984808	+	0.173648i	$0.944444\pi$
$54$	1.00000			1.00000
$55$		0			0
$56$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$57$	0.766044	−	0.642788i	0.766044	−	0.642788i
$58$	0.766044	−	1.32683i	0.766044	−	1.32683i
$59$	0.342020	+	0.592396i	0.342020	+	0.592396i	0.984808	−	0.173648i	$-0.0555556\pi$
$59$	0.342020	+	0.592396i	0.342020	+	0.592396i	−0.642788	+	0.766044i	$0.722222\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$	0.500000	−	0.866025i	0.500000	−	0.866025i
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	0.642788	+	1.11334i	0.642788	+	1.11334i	0.984808	+	0.173648i	$0.0555556\pi$
$67$	0.642788	+	1.11334i	0.642788	+	1.11334i	−0.342020	+	0.939693i	$0.611111\pi$
$68$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$69$	−0.524005	+	0.439693i	−0.524005	+	0.439693i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	0.642788	−	0.766044i	0.642788	−	0.766044i
$73$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$73$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$74$	1.50000	−	0.866025i	1.50000	−	0.866025i
$75$	0.984808	−	0.173648i	0.984808	−	0.173648i
$76$		−	1.00000i		−	1.00000i
$77$		0			0
$78$	−0.223238	−	0.266044i	−0.223238	−	0.266044i
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	0.173648	+	0.984808i	0.173648	+	0.984808i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$85$		0			0
$86$		0			0
$87$	1.43969	+	0.524005i	1.43969	+	0.524005i
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−0.342020	+	0.0603074i	−0.342020	+	0.0603074i
$92$			0.684040i			0.684040i
$93$		0			0
$94$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$95$		0			0
$96$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$99$		0			0
$100$	0.500000	−	0.866025i	0.500000	−	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$	−0.347296			−0.347296
$105$		0			0
$106$	−1.87939			−1.87939
$107$	0.300767	+	0.173648i	0.300767	+	0.173648i	0.642788	−	0.766044i	$-0.277778\pi$
$107$	0.300767	+	0.173648i	0.300767	+	0.173648i	−0.342020	+	0.939693i	$0.611111\pi$
$108$	0.866025	+	0.500000i	0.866025	+	0.500000i
$109$	−0.984808	−	1.70574i	−0.984808	−	1.70574i	−0.642788	−	0.766044i	$-0.722222\pi$
$109$	−0.984808	−	1.70574i	−0.984808	−	1.70574i	−0.342020	−	0.939693i	$-0.611111\pi$
$110$		0			0
$111$	1.11334	+	1.32683i	1.11334	+	1.32683i
$112$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$	0.984808	−	0.173648i	0.984808	−	0.173648i
$115$		0			0
$116$	1.32683	−	0.766044i	1.32683	−	0.766044i
$117$	0.223238	−	0.266044i	0.223238	−	0.266044i
$118$			0.684040i			0.684040i
$119$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$120$		0			0
$121$	0.500000	−	0.866025i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.866025	−	0.500000i	0.866025	−	0.500000i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	0.342020	−	0.939693i	0.342020	−	0.939693i
$134$			1.28558i			1.28558i
$135$		0			0
$136$	−1.62760	+	0.939693i	−1.62760	+	0.939693i
$137$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$137$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$138$	−0.673648	+	0.118782i	−0.673648	+	0.118782i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$142$		0			0
$143$		0			0
$144$	0.939693	−	0.342020i	0.939693	−	0.342020i
$145$		0			0
$146$	1.96962			1.96962
$147$		−	1.00000i		−	1.00000i
$148$	1.73205			1.73205
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	0.939693	+	0.342020i	0.939693	+	0.342020i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	0.500000	−	0.866025i	0.500000	−	0.866025i
$153$	0.326352	−	1.85083i	0.326352	−	1.85083i
$154$		0			0
$155$		0			0
$156$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$		0			0
$159$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$160$		0			0
$161$	−0.233956	+	0.642788i	−0.233956	+	0.642788i
$162$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$	0.173648	−	0.984808i	0.173648	−	0.984808i
$169$	0.879385			0.879385
$170$		0			0
$171$	0.342020	+	0.939693i	0.342020	+	0.939693i
$172$		0			0
$173$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$173$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$174$	0.984808	+	1.17365i	0.984808	+	1.17365i
$175$	0.766044	−	0.642788i	0.766044	−	0.642788i
$176$		0			0
$177$	−0.673648	+	0.118782i	−0.673648	+	0.118782i
$178$		0			0
$179$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$179$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$181$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$182$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$183$		0			0
$184$	−0.342020	+	0.592396i	−0.342020	+	0.592396i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.00000			−1.00000
$189$	0.642788	+	0.766044i	0.642788	+	0.766044i
$190$		0			0
$191$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$191$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$192$	0.342020	−	0.939693i	0.342020	−	0.939693i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$199$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$200$	0.866025	−	0.500000i	0.866025	−	0.500000i
$201$	−1.26604	+	0.223238i	−1.26604	+	0.223238i
$202$		0			0
$203$	1.50881	−	0.266044i	1.50881	−	0.266044i
$204$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$205$		0			0
$206$		0			0
$207$	−0.233956	−	0.642788i	−0.233956	−	0.642788i
$208$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$209$		0			0
$210$		0			0
$211$	1.96962			1.96962			0.984808	−	0.173648i	$-0.0555556\pi$
$211$	1.96962			1.96962			0.984808	+	0.173648i	$0.0555556\pi$
$212$	−1.62760	−	0.939693i	−1.62760	−	0.939693i
$213$		0			0
$214$	0.173648	+	0.300767i	0.173648	+	0.300767i
$215$		0			0
$216$	0.500000	+	0.866025i	0.500000	+	0.866025i
$217$		0			0
$218$		−	1.96962i		−	1.96962i
$219$	0.342020	+	1.93969i	0.342020	+	1.93969i
$220$		0			0
$221$	−0.565258	+	0.326352i	−0.565258	+	0.326352i
$222$	0.300767	+	1.70574i	0.300767	+	1.70574i
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$225$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$226$		0			0
$227$	0.984808	+	1.70574i	0.984808	+	1.70574i	0.642788	+	0.766044i	$0.277778\pi$
$227$	0.984808	+	1.70574i	0.984808	+	1.70574i	0.342020	+	0.939693i	$0.388889\pi$
$228$	0.939693	+	0.342020i	0.939693	+	0.342020i
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	1.53209			1.53209
$233$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$233$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$234$	0.326352	−	0.118782i	0.326352	−	0.118782i
$235$		0			0
$236$	−0.342020	+	0.592396i	−0.342020	+	0.592396i
$237$		0			0
$238$	−1.85083	+	0.326352i	−1.85083	+	0.326352i
$239$			1.96962i			1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$239$			1.96962i			1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	0.866025	−	0.500000i	0.866025	−	0.500000i
$243$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.173648	−	0.300767i	0.173648	−	0.300767i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	1.00000			1.00000
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	1.62760	+	0.592396i	1.62760	+	0.592396i
$260$		0			0
$261$	−0.984808	+	1.17365i	−0.984808	+	1.17365i
$262$		0			0
$263$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$264$		0			0
$265$		0			0
$266$	0.766044	−	0.642788i	0.766044	−	0.642788i
$267$		0			0
$268$	−0.642788	+	1.11334i	−0.642788	+	1.11334i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$271$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$272$	−1.87939			−1.87939
$273$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$274$	1.28558			1.28558
$275$		0			0
$276$	−0.642788	−	0.233956i	−0.642788	−	0.233956i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.984808	+	0.173648i	0.984808	+	0.173648i
$289$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$290$		0			0
$291$		0			0
$292$	1.70574	+	0.984808i	1.70574	+	0.984808i
$293$	0.684040			0.684040			0.342020	−	0.939693i	$-0.388889\pi$
$293$	0.684040			0.684040			0.342020	+	0.939693i	$0.388889\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$		0			0
$296$	1.50000	+	0.866025i	1.50000	+	0.866025i
$297$		0			0
$298$		0			0
$299$	−0.118782	+	0.205737i	−0.118782	+	0.205737i
$300$	0.642788	+	0.766044i	0.642788	+	0.766044i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.866025	−	0.500000i	0.866025	−	0.500000i
$305$		0			0
$306$	1.20805	−	1.43969i	1.20805	−	1.43969i
$307$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$307$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$311$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$312$	0.118782	−	0.326352i	0.118782	−	0.326352i
$313$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$313$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.300767	−	0.173648i	−0.300767	−	0.173648i	0.342020	−	0.939693i	$-0.388889\pi$
$317$	−0.300767	−	0.173648i	−0.300767	−	0.173648i	−0.642788	+	0.766044i	$0.722222\pi$
$318$	0.642788	−	1.76604i	0.642788	−	1.76604i
$319$		0			0
$320$		0			0
$321$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$322$	−0.524005	+	0.439693i	−0.524005	+	0.439693i
$323$		−	1.87939i		−	1.87939i
$324$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$325$	0.300767	−	0.173648i	0.300767	−	0.173648i
$326$		0			0
$327$	1.93969	−	0.342020i	1.93969	−	0.342020i
$328$		0			0
$329$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$330$		0			0
$331$	0.984808	−	1.70574i	0.984808	−	1.70574i	0.342020	−	0.939693i	$-0.388889\pi$
$331$	0.984808	−	1.70574i	0.984808	−	1.70574i	0.642788	−	0.766044i	$-0.277778\pi$
$332$		0			0
$333$	−1.62760	+	0.592396i	−1.62760	+	0.592396i
$334$		0			0
$335$		0			0
$336$	0.642788	−	0.766044i	0.642788	−	0.766044i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.761570	+	0.439693i	0.761570	+	0.439693i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$343$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$344$		0			0
$345$		0			0
$346$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$	0.266044	+	1.50881i	0.266044	+	1.50881i
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	0.984808	−	0.173648i	0.984808	−	0.173648i
$351$	0.173648	+	0.300767i	0.173648	+	0.300767i
$352$		0			0
$353$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$353$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$354$	−0.642788	−	0.233956i	−0.642788	−	0.233956i
$355$		0			0
$356$		0			0
$357$	−0.642788	−	1.76604i	−0.642788	−	1.76604i
$358$	1.00000			1.00000
$359$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$359$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$360$		0			0
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$363$	0.642788	+	0.766044i	0.642788	+	0.766044i
$364$	−0.223238	−	0.266044i	−0.223238	−	0.266044i
$365$		0			0
$366$		0			0
$367$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$368$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$369$		0			0
$370$		0			0
$371$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$372$		0			0
$373$	0.342020	−	0.592396i	0.342020	−	0.592396i	−0.642788	−	0.766044i	$-0.722222\pi$
$373$	0.342020	−	0.592396i	0.342020	−	0.592396i	0.984808	+	0.173648i	$0.0555556\pi$
$374$		0			0
$375$		0			0
$376$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$377$	0.532089			0.532089
$378$	0.173648	+	0.984808i	0.173648	+	0.984808i
$379$	−0.684040			−0.684040			−0.342020	−	0.939693i	$-0.611111\pi$
$379$	−0.684040			−0.684040			−0.342020	+	0.939693i	$0.611111\pi$
$380$		0			0
$381$		0			0
$382$	−0.642788	−	1.11334i	−0.642788	−	1.11334i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.766044	−	0.642788i	0.766044	−	0.642788i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$			1.28558i			1.28558i
$392$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$	0.684040			0.684040
$399$	0.766044	+	0.642788i	0.766044	+	0.642788i
$400$	1.00000			1.00000
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$	−1.20805	−	0.439693i	−1.20805	−	0.439693i
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.43969	+	0.524005i	1.43969	+	0.524005i
$407$		0			0
$408$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$	0.223238	+	1.26604i	0.223238	+	1.26604i
$412$		0			0
$413$	−0.524005	+	0.439693i	−0.524005	+	0.439693i
$414$	0.118782	−	0.673648i	0.118782	−	0.673648i
$415$		0			0
$416$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.28558			1.28558			0.642788	−	0.766044i	$-0.277778\pi$
$421$	1.28558			1.28558			0.642788	+	0.766044i	$0.277778\pi$
$422$	1.70574	+	0.984808i	1.70574	+	0.984808i
$423$	0.939693	−	0.342020i	0.939693	−	0.342020i
$424$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$425$	0.939693	−	1.62760i	0.939693	−	1.62760i
$426$		0			0
$427$		0			0
$428$			0.347296i			0.347296i
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$			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
$436$	0.984808	−	1.70574i	0.984808	−	1.70574i
$437$	−0.342020	−	0.592396i	−0.342020	−	0.592396i
$438$	−0.673648	+	1.85083i	−0.673648	+	1.85083i
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	0.939693	+	0.342020i	0.939693	+	0.342020i
$442$	−0.652704			−0.652704
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$	−0.592396	+	1.62760i	−0.592396	+	1.62760i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$451$		0			0
$452$		0			0
$453$		0			0
$454$			1.96962i			1.96962i
$455$		0			0
$456$	0.642788	+	0.766044i	0.642788	+	0.766044i
$457$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$457$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$458$		0			0
$459$	1.62760	+	0.939693i	1.62760	+	0.939693i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$464$	1.32683	+	0.766044i	1.32683	+	0.766044i
$465$		0			0
$466$	0.866025	+	1.50000i	0.866025	+	1.50000i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	0.342020	+	0.0603074i	0.342020	+	0.0603074i
$469$	−0.984808	+	0.826352i	−0.984808	+	0.826352i
$470$		0			0
$471$		0			0
$472$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$473$		0			0
$474$		0			0
$475$			1.00000i			1.00000i
$476$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$477$	1.85083	+	0.326352i	1.85083	+	0.326352i
$478$	−0.984808	+	1.70574i	−0.984808	+	1.70574i
$479$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$480$		0			0
$481$	0.520945	+	0.300767i	0.520945	+	0.300767i
$482$		0			0
$483$	−0.524005	−	0.439693i	−0.524005	−	0.439693i
$484$	1.00000			1.00000
$485$		0			0
$486$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	2.49362	−	1.43969i	2.49362	−	1.43969i
$494$	0.300767	−	0.173648i	0.300767	−	0.173648i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$503$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$504$	0.866025	+	0.500000i	0.866025	+	0.500000i
$505$		0			0
$506$		0			0
$507$	−0.300767	+	0.826352i	−0.300767	+	0.826352i
$508$		0			0
$509$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$509$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$510$		0			0
$511$	1.26604	+	1.50881i	1.26604	+	1.50881i
$512$		−	1.00000i		−	1.00000i
$513$	−1.00000			−1.00000
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.11334	+	1.32683i	1.11334	+	1.32683i
$519$	−1.11334	−	1.32683i	−1.11334	−	1.32683i
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$523$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.984808	−	0.173648i	$-0.944444\pi$
$523$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$524$		0			0
$525$	0.342020	+	0.939693i	0.342020	+	0.939693i
$526$	1.73205			1.73205
$527$		0			0
$528$		0			0
$529$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$530$		0			0
$531$	0.118782	−	0.673648i	0.118782	−	0.673648i
$532$	0.984808	−	0.173648i	0.984808	−	0.173648i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.11334	+	0.642788i	−1.11334	+	0.642788i
$537$	0.173648	+	0.984808i	0.173648	+	0.984808i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	−0.984808	−	1.70574i	−0.984808	−	1.70574i
$543$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$544$	−1.62760	−	0.939693i	−1.62760	−	0.939693i
$545$		0			0
$546$	0.223238	−	0.266044i	0.223238	−	0.266044i
$547$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$547$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$548$	1.11334	+	0.642788i	1.11334	+	0.642788i
$549$		0			0
$550$		0			0
$551$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$552$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$	0.342020	−	0.939693i	0.342020	−	0.939693i
$565$		0			0
$566$		0			0
$567$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$	0.984808	−	0.826352i	0.984808	−	0.826352i
$574$		0			0
$575$		−	0.684040i		−	0.684040i
$576$	0.766044	+	0.642788i	0.766044	+	0.642788i
$577$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$577$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$578$	−2.19285	+	1.26604i	−2.19285	+	1.26604i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.984808	+	1.70574i	0.984808	+	1.70574i
$585$		0			0
$586$	0.592396	+	0.342020i	0.592396	+	0.342020i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.866025	−	0.500000i	0.866025	−	0.500000i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.866025	+	1.50000i	0.866025	+	1.50000i
$593$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$593$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.118782	+	0.673648i	0.118782	+	0.673648i
$598$	−0.205737	+	0.118782i	−0.205737	+	0.118782i
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$	0.173648	+	0.984808i	0.173648	+	0.984808i
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	0.223238	−	1.26604i	0.223238	−	1.26604i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$	1.00000			1.00000
$609$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$610$		0			0
$611$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$612$	1.76604	−	0.642788i	1.76604	−	0.642788i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$615$		0			0
$616$		0			0
$617$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$617$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$	0.684040			0.684040
$622$			1.53209i			1.53209i
$623$		0			0
$624$	0.266044	−	0.223238i	0.266044	−	0.223238i
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$	−0.642788	−	1.11334i	−0.642788	−	1.11334i
$627$		0			0
$628$		0			0
$629$	3.25519			3.25519
$630$		0			0
$631$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−0.673648	+	1.85083i	−0.673648	+	1.85083i
$634$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$635$		0			0
$636$	1.43969	−	1.20805i	1.43969	−	1.20805i
$637$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$	−0.342020	+	0.0603074i	−0.342020	+	0.0603074i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	−0.673648	+	0.118782i	−0.673648	+	0.118782i
$645$		0			0
$646$	0.939693	−	1.62760i	0.939693	−	1.62760i
$647$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$647$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$648$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$649$		0			0
$650$	0.347296			0.347296
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$	1.85083	+	0.673648i	1.85083	+	0.673648i
$655$		0			0
$656$		0			0
$657$	−1.93969	−	0.342020i	−1.93969	−	0.342020i
$658$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$659$			0.347296i			0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$659$			0.347296i			0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$660$		0			0
$661$	1.32683	−	0.766044i	1.32683	−	0.766044i	0.342020	−	0.939693i	$-0.388889\pi$
$661$	1.32683	−	0.766044i	1.32683	−	0.766044i	0.984808	+	0.173648i	$0.0555556\pi$
$662$	1.70574	−	0.984808i	1.70574	−	0.984808i
$663$	−0.113341	−	0.642788i	−0.113341	−	0.642788i
$664$		0			0
$665$		0			0
$666$	−1.70574	−	0.300767i	−1.70574	−	0.300767i
$667$	0.524005	−	0.907604i	0.524005	−	0.907604i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	0.939693	−	0.342020i	0.939693	−	0.342020i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$676$	0.439693	+	0.761570i	0.439693	+	0.761570i
$677$	0.642788	−	1.11334i	0.642788	−	1.11334i	−0.342020	−	0.939693i	$-0.611111\pi$
$677$	0.642788	−	1.11334i	0.642788	−	1.11334i	0.984808	−	0.173648i	$-0.0555556\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.93969	+	0.342020i	−1.93969	+	0.342020i
$682$		0			0
$683$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$683$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$684$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$685$		0			0
$686$		−	1.00000i		−	1.00000i
$687$		0			0
$688$		0			0
$689$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$	−1.73205			−1.73205
$693$		0			0
$694$		0			0
$695$		0			0
$696$	−0.524005	+	1.43969i	−0.524005	+	1.43969i
$697$		0			0
$698$		0			0
$699$	−1.32683	+	1.11334i	−1.32683	+	1.11334i
$700$	0.939693	+	0.342020i	0.939693	+	0.342020i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$			0.347296i			0.347296i
$703$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$704$		0			0
$705$		0			0
$706$		−	1.53209i		−	1.53209i
$707$		0			0
$708$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$	0.326352	−	1.85083i	0.326352	−	1.85083i
$715$		0			0
$716$	0.866025	+	0.500000i	0.866025	+	0.500000i
$717$	−1.85083	−	0.673648i	−1.85083	−	0.673648i
$718$	−0.984808	−	1.70574i	−0.984808	−	1.70574i
$719$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$719$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$720$		0			0
$721$		0			0
$722$			1.00000i			1.00000i
$723$		0			0
$724$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$725$	−1.32683	+	0.766044i	−1.32683	+	0.766044i
$726$	0.173648	+	0.984808i	0.173648	+	0.984808i
$727$		−	1.28558i		−	1.28558i	−0.766044	−	0.642788i	$-0.777778\pi$
$727$		−	1.28558i		−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$728$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$729$	0.500000	−	0.866025i	0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$	1.73205			1.73205
$735$		0			0
$736$	−0.684040			−0.684040
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$	0.223238	+	0.266044i	0.223238	+	0.266044i
$742$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$	0.592396	−	0.342020i	0.592396	−	0.342020i
$747$		0			0
$748$		0			0
$749$	−0.118782	+	0.326352i	−0.118782	+	0.326352i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$753$		0			0
$754$	0.460802	+	0.266044i	0.460802	+	0.266044i
$755$		0			0
$756$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.592396	−	0.342020i	−0.592396	−	0.342020i
$759$		0			0
$760$		0			0
$761$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$761$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$762$		0			0
$763$	1.50881	−	1.26604i	1.50881	−	1.26604i
$764$		−	1.28558i		−	1.28558i
$765$		0			0
$766$		0			0
$767$	−0.205737	+	0.118782i	−0.205737	+	0.118782i
$768$	0.984808	−	0.173648i	0.984808	−	0.173648i
$769$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$769$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.984808	−	1.70574i	−0.984808	−	1.70574i	−0.642788	−	0.766044i	$-0.722222\pi$
$773$	−0.984808	−	1.70574i	−0.984808	−	1.70574i	−0.342020	−	0.939693i	$-0.611111\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.11334	+	1.32683i	−1.11334	+	1.32683i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.642788	+	1.11334i	−0.642788	+	1.11334i
$783$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$784$	0.173648	−	0.984808i	0.173648	−	0.984808i
$785$		0			0
$786$		0			0
$787$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.642788	+	0.766044i	$0.722222\pi$
$787$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.984808	+	0.173648i	$0.944444\pi$
$788$		0			0
$789$	0.300767	+	1.70574i	0.300767	+	1.70574i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.592396	+	0.342020i	0.592396	+	0.342020i
$797$	−1.28558			−1.28558			−0.642788	−	0.766044i	$-0.722222\pi$
$797$	−1.28558			−1.28558			−0.642788	+	0.766044i	$0.722222\pi$
$798$	0.342020	+	0.939693i	0.342020	+	0.939693i
$799$	−1.87939			−1.87939
$800$	0.866025	+	0.500000i	0.866025	+	0.500000i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.826352	−	0.984808i	−0.826352	−	0.984808i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$809$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$810$		0			0
$811$		−	1.87939i		−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$811$		−	1.87939i		−	1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$812$	0.984808	+	1.17365i	0.984808	+	1.17365i
$813$	1.50881	−	1.26604i	1.50881	−	1.26604i
$814$		0			0
$815$		0			0
$816$	0.642788	−	1.76604i	0.642788	−	1.76604i
$817$		0			0
$818$		0			0
$819$	0.300767	+	0.173648i	0.300767	+	0.173648i
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$	−0.439693	+	1.20805i	−0.439693	+	1.20805i
$823$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$823$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$824$		0			0
$825$		0			0
$826$	−0.673648	+	0.118782i	−0.673648	+	0.118782i
$827$			1.53209i			1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$827$			1.53209i			1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$828$	0.439693	−	0.524005i	0.439693	−	0.524005i
$829$	−1.32683	+	0.766044i	−1.32683	+	0.766044i	−0.984808	−	0.173648i	$-0.944444\pi$
$829$	−1.32683	+	0.766044i	−1.32683	+	0.766044i	−0.342020	+	0.939693i	$0.611111\pi$
$830$		0			0
$831$		0			0
$832$		−	0.347296i		−	0.347296i
$833$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−1.34730			−1.34730
$842$	1.11334	+	0.642788i	1.11334	+	0.642788i
$843$		0			0
$844$	0.984808	+	1.70574i	0.984808	+	1.70574i
$845$		0			0
$846$	0.984808	+	0.173648i	0.984808	+	0.173648i
$847$	0.939693	+	0.342020i	0.939693	+	0.342020i
$848$		−	1.87939i		−	1.87939i
$849$		0			0
$850$	1.62760	−	0.939693i	1.62760	−	0.939693i
$851$	1.02606	−	0.592396i	1.02606	−	0.592396i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$865$		0			0
$866$		0			0
$867$	−1.62760	−	1.93969i	−1.62760	−	1.93969i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−0.386659	+	0.223238i	−0.386659	+	0.223238i
$872$	1.70574	−	0.984808i	1.70574	−	0.984808i
$873$		0			0
$874$		−	0.684040i		−	0.684040i
$875$		0			0
$876$	−1.50881	+	1.26604i	−1.50881	+	1.26604i
$877$	−0.984808	+	1.70574i	−0.984808	+	1.70574i	−0.342020	+	0.939693i	$0.611111\pi$
$877$	−0.984808	+	1.70574i	−0.984808	+	1.70574i	−0.642788	+	0.766044i	$0.722222\pi$
$878$		0			0
$879$	−0.233956	+	0.642788i	−0.233956	+	0.642788i
$880$		0			0
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$	0.642788	+	0.766044i	0.642788	+	0.766044i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.565258	−	0.326352i	−0.565258	−	0.326352i
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$	−1.32683	+	1.11334i	−1.32683	+	1.11334i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.866025	−	0.500000i	0.866025	−	0.500000i
$894$		0			0
$895$		0			0
$896$	0.342020	−	0.939693i	0.342020	−	0.939693i
$897$	−0.152704	−	0.181985i	−0.152704	−	0.181985i
$898$		0			0
$899$		0			0
$900$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$901$	−3.05888	−	1.76604i	−3.05888	−	1.76604i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.342020	−	0.592396i	−0.342020	−	0.592396i	0.642788	−	0.766044i	$-0.277778\pi$
$907$	−0.342020	−	0.592396i	−0.342020	−	0.592396i	−0.984808	+	0.173648i	$0.944444\pi$
$908$	−0.984808	+	1.70574i	−0.984808	+	1.70574i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	0.173648	+	0.984808i	0.173648	+	0.984808i
$913$		0			0
$914$	−1.62760	+	0.939693i	−1.62760	+	0.939693i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.939693	+	1.62760i	0.939693	+	1.62760i
$919$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$919$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$920$		0			0
$921$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.73205			−1.73205
$926$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$927$		0			0
$928$	0.766044	+	1.32683i	0.766044	+	1.32683i
$929$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$929$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$930$		0			0
$931$	0.984808	+	0.173648i	0.984808	+	0.173648i
$932$			1.73205i			1.73205i
$933$	−1.50881	+	0.266044i	−1.50881	+	0.266044i
$934$		0			0
$935$		0			0
$936$	0.266044	+	0.223238i	0.266044	+	0.223238i
$937$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$937$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$938$	−1.26604	+	0.223238i	−1.26604	+	0.223238i
$939$	0.984808	−	0.826352i	0.984808	−	0.826352i
$940$		0			0
$941$	0.642788	+	1.11334i	0.642788	+	1.11334i	0.984808	+	0.173648i	$0.0555556\pi$
$941$	0.642788	+	1.11334i	0.642788	+	1.11334i	−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$	−0.684040			−0.684040
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	0.342020	+	0.592396i	0.342020	+	0.592396i
$950$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$951$	0.266044	−	0.223238i	0.266044	−	0.223238i
$952$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	1.43969	+	1.20805i	1.43969	+	1.20805i
$955$		0			0
$956$	−1.70574	+	0.984808i	−1.70574	+	0.984808i
$957$		0			0
$958$		−	2.00000i		−	2.00000i
$959$	0.826352	+	0.984808i	0.826352	+	0.984808i
$960$		0			0
$961$	0.500000	−	0.866025i	0.500000	−	0.866025i
$962$	0.300767	+	0.520945i	0.300767	+	0.520945i
$963$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$964$		0			0
$965$		0			0
$966$	−0.233956	−	0.642788i	−0.233956	−	0.642788i
$967$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$968$	0.866025	+	0.500000i	0.866025	+	0.500000i
$969$	1.76604	+	0.642788i	1.76604	+	0.642788i
$970$		0			0
$971$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$971$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$972$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$973$		0			0
$974$		0			0
$975$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.342020	+	1.93969i	−0.342020	+	1.93969i
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$	2.87939			2.87939
$987$	0.642788	−	0.766044i	0.642788	−	0.766044i
$988$	0.347296			0.347296
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	1.26604	+	1.50881i	1.26604	+	1.50881i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$		−	1.73205i		−	1.73205i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.1.eb.a.341.5	yes	12
3.2	odd	2		3192.1.eb.b.341.1	yes	12
7.3	odd	6		3192.1.eb.b.2621.1	yes	12
8.5	even	2	inner	3192.1.eb.a.341.2	✓	12
19.18	odd	2	inner	3192.1.eb.a.341.2	✓	12
21.17	even	6	inner	3192.1.eb.a.2621.5	yes	12
24.5	odd	2		3192.1.eb.b.341.6	yes	12
56.45	odd	6		3192.1.eb.b.2621.6	yes	12
57.56	even	2		3192.1.eb.b.341.6	yes	12
133.94	even	6		3192.1.eb.b.2621.6	yes	12
152.37	odd	2	CM	3192.1.eb.a.341.5	yes	12
168.101	even	6	inner	3192.1.eb.a.2621.2	yes	12
399.227	odd	6	inner	3192.1.eb.a.2621.2	yes	12
456.341	even	2		3192.1.eb.b.341.1	yes	12
1064.493	even	6		3192.1.eb.b.2621.1	yes	12
3192.2621	odd	6	inner	3192.1.eb.a.2621.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.1.eb.a.341.2	✓	12	8.5	even	2	inner
3192.1.eb.a.341.2	✓	12	19.18	odd	2	inner
3192.1.eb.a.341.5	yes	12	1.1	even	1	trivial
3192.1.eb.a.341.5	yes	12	152.37	odd	2	CM
3192.1.eb.a.2621.2	yes	12	168.101	even	6	inner
3192.1.eb.a.2621.2	yes	12	399.227	odd	6	inner
3192.1.eb.a.2621.5	yes	12	21.17	even	6	inner
3192.1.eb.a.2621.5	yes	12	3192.2621	odd	6	inner
3192.1.eb.b.341.1	yes	12	3.2	odd	2
3192.1.eb.b.341.1	yes	12	456.341	even	2
3192.1.eb.b.341.6	yes	12	24.5	odd	2
3192.1.eb.b.341.6	yes	12	57.56	even	2
3192.1.eb.b.2621.1	yes	12	7.3	odd	6
3192.1.eb.b.2621.1	yes	12	1064.493	even	6
3192.1.eb.b.2621.6	yes	12	56.45	odd	6
3192.1.eb.b.2621.6	yes	12	133.94	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3192.eb (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.766044 + 0.642788i) q^{6} +(0.173648 + 0.984808i) q^{7} +1.00000i q^{8} +(-0.766044 - 0.642788i) q^{9} +(-0.984808 + 0.173648i) q^{12} +0.347296i q^{13} +(-0.342020 + 0.939693i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.939693 + 1.62760i) q^{17} +(-0.342020 - 0.939693i) q^{18} +(-0.866025 - 0.500000i) q^{19} +(-0.984808 - 0.173648i) q^{21} +(0.592396 + 0.342020i) q^{23} +(-0.939693 - 0.342020i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-0.173648 + 0.300767i) q^{26} +(0.866025 - 0.500000i) q^{27} +(-0.766044 + 0.642788i) q^{28} -1.53209i q^{29} +(-0.866025 + 0.500000i) q^{32} +1.87939i q^{34} +(0.173648 - 0.984808i) q^{36} +(0.866025 - 1.50000i) q^{37} +(-0.500000 - 0.866025i) q^{38} +(-0.326352 - 0.118782i) q^{39} +(-0.766044 - 0.642788i) q^{42} +(0.342020 + 0.592396i) q^{46} +(-0.500000 + 0.866025i) q^{47} +(-0.642788 - 0.766044i) q^{48} +(-0.939693 + 0.342020i) q^{49} -1.00000i q^{50} +(-1.85083 + 0.326352i) q^{51} +(-0.300767 + 0.173648i) q^{52} +(-1.62760 + 0.939693i) q^{53} +1.00000 q^{54} +(-0.984808 + 0.173648i) q^{56} +(0.766044 - 0.642788i) q^{57} +(0.766044 - 1.32683i) q^{58} +(0.342020 + 0.592396i) q^{59} +(0.500000 - 0.866025i) q^{63} -1.00000 q^{64} +(0.642788 + 1.11334i) q^{67} +(-0.939693 + 1.62760i) q^{68} +(-0.524005 + 0.439693i) q^{69} +(0.642788 - 0.766044i) q^{72} +(1.70574 - 0.984808i) q^{73} +(1.50000 - 0.866025i) q^{74} +(0.984808 - 0.173648i) q^{75} -1.00000i q^{76} +(-0.223238 - 0.266044i) q^{78} +(0.173648 + 0.984808i) q^{81} +(-0.342020 - 0.939693i) q^{84} +(1.43969 + 0.524005i) q^{87} +(-0.342020 + 0.0603074i) q^{91} +0.684040i q^{92} +(-0.866025 + 0.500000i) q^{94} +(-0.173648 - 0.984808i) q^{96} +(-0.984808 - 0.173648i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{4} - 6 q^{16} - 6 q^{25} - 6 q^{38} - 6 q^{39} - 6 q^{47} + 12 q^{54} + 6 q^{63} - 12 q^{64} + 18 q^{74} + 6 q^{87}+O(q^{100}) \) 12 * q + 6 * q^4 - 6 * q^16 - 6 * q^25 - 6 * q^38 - 6 * q^39 - 6 * q^47 + 12 * q^54 + 6 * q^63 - 12 * q^64 + 18 * q^74 + 6 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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