LMFDB - Embedded newform 3212.1.bf.a.2335.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3212,1,Mod(291,3212)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3212, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 4, 5]))

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

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

chi := DirichletCharacter("3212.291");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3212 = 2^{2} \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3212.bf (of order $10$, degree $4$, 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:	$1.60299682058$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.1248350224.1

Embedding invariants

Embedding label			2335.1
Root			$0.809017 - 0.587785i$ of defining polynomial
Character	$\chi$	$=$	3212.2335
Dual form			3212.1.bf.a.2919.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.809017 + 0.587785i) q^{2} +(0.309017 - 0.951057i) q^{4} +(-0.190983 + 0.587785i) q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})$	\(q+(-0.809017 + 0.587785i) q^{2} +(0.309017 - 0.951057i) q^{4} +(-0.190983 + 0.587785i) q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.809017 + 0.587785i) q^{11} +(-0.190983 - 0.587785i) q^{14} +(-0.809017 - 0.587785i) q^{16} +(0.309017 - 0.951057i) q^{18} -1.00000 q^{22} +(0.309017 + 0.951057i) q^{25} +(0.500000 + 0.363271i) q^{28} +(-1.30902 + 0.951057i) q^{31} +1.00000 q^{32} +(0.309017 + 0.951057i) q^{36} +(0.190983 - 0.587785i) q^{37} +(-0.500000 - 1.53884i) q^{41} -2.00000 q^{43} +(0.809017 - 0.587785i) q^{44} +(0.500000 + 1.53884i) q^{47} +(0.500000 + 0.363271i) q^{49} +(-0.809017 - 0.587785i) q^{50} -0.618034 q^{56} +(-0.190983 + 0.587785i) q^{59} +(-0.500000 - 0.363271i) q^{61} +(0.500000 - 1.53884i) q^{62} +(-0.190983 - 0.587785i) q^{63} +(-0.809017 + 0.587785i) q^{64} +(-0.809017 - 0.587785i) q^{72} +(0.309017 - 0.951057i) q^{73} +(0.190983 + 0.587785i) q^{74} +(-0.500000 + 0.363271i) q^{77} +(0.309017 - 0.951057i) q^{81} +(1.30902 + 0.951057i) q^{82} +(0.500000 + 0.363271i) q^{83} +(1.61803 - 1.17557i) q^{86} +(-0.309017 + 0.951057i) q^{88} +0.618034 q^{89} +(-1.30902 - 0.951057i) q^{94} +(-1.61803 + 1.17557i) q^{97} -0.618034 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{4} - 3 q^{7} - q^{8} - q^{9}+O(q^{10}) $ 4 * q - q^2 - q^4 - 3 * q^7 - q^8 - q^9	$ 4 q - q^{2} - q^{4} - 3 q^{7} - q^{8} - q^{9} + q^{11} - 3 q^{14} - q^{16} - q^{18} - 4 q^{22} - q^{25} + 2 q^{28} - 3 q^{31} + 4 q^{32} - q^{36} + 3 q^{37} - 2 q^{41} - 8 q^{43} + q^{44} + 2 q^{47} + 2 q^{49} - q^{50} + 2 q^{56} - 3 q^{59} - 2 q^{61} + 2 q^{62} - 3 q^{63} - q^{64} - q^{72} - q^{73} + 3 q^{74} - 2 q^{77} - q^{81} + 3 q^{82} + 2 q^{83} + 2 q^{86} + q^{88} - 2 q^{89} - 3 q^{94} - 2 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) $ 4 * q - q^2 - q^4 - 3 * q^7 - q^8 - q^9 + q^11 - 3 * q^14 - q^16 - q^18 - 4 * q^22 - q^25 + 2 * q^28 - 3 * q^31 + 4 * q^32 - q^36 + 3 * q^37 - 2 * q^41 - 8 * q^43 + q^44 + 2 * q^47 + 2 * q^49 - q^50 + 2 * q^56 - 3 * q^59 - 2 * q^61 + 2 * q^62 - 3 * q^63 - q^64 - q^72 - q^73 + 3 * q^74 - 2 * q^77 - q^81 + 3 * q^82 + 2 * q^83 + 2 * q^86 + q^88 - 2 * q^89 - 3 * q^94 - 2 * q^97 + 2 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3212\mathbb{Z}\right)^\times$.

$n$	$585$	$881$	$1607$
$\chi(n)$	$e\left(\frac{4}{5}\right)$	$-1$	$-1$

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 $

$2$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$2$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$3$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$5$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$6$		0			0
$7$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$7$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$8$	0.309017	+	0.951057i	0.309017	+	0.951057i
$9$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$10$		0			0
$11$	0.809017	+	0.587785i	0.809017	+	0.587785i
$11$	0.809017	+	0.587785i	0.809017	+	0.587785i
$12$		0			0
$13$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$13$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$14$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$15$		0			0
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$	0.309017	−	0.951057i	0.309017	−	0.951057i
$19$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$19$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$20$		0			0
$21$		0			0
$22$	−1.00000			−1.00000
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.309017	+	0.951057i	0.309017	+	0.951057i
$26$		0			0
$27$		0			0
$28$	0.500000	+	0.363271i	0.500000	+	0.363271i
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$31$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$32$	1.00000			1.00000
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.309017	+	0.951057i	0.309017	+	0.951057i
$37$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$37$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$41$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$42$		0			0
$43$	−2.00000			−2.00000			−1.00000			$\pi$
$43$	−2.00000			−2.00000			−1.00000			$\pi$
$44$	0.809017	−	0.587785i	0.809017	−	0.587785i
$45$		0			0
$46$		0			0
$47$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$47$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$48$		0			0
$49$	0.500000	+	0.363271i	0.500000	+	0.363271i
$50$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$51$		0			0
$52$		0			0
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$		0			0
$55$		0			0
$56$	−0.618034			−0.618034
$57$		0			0
$58$		0			0
$59$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$59$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$60$		0			0
$61$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$61$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$62$	0.500000	−	1.53884i	0.500000	−	1.53884i
$63$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$73$	0.309017	−	0.951057i	0.309017	−	0.951057i
$73$	0.309017	−	0.951057i	0.309017	−	0.951057i
$74$	0.190983	+	0.587785i	0.190983	+	0.587785i
$75$		0			0
$76$		0			0
$77$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$		0			0
$81$	0.309017	−	0.951057i	0.309017	−	0.951057i
$82$	1.30902	+	0.951057i	1.30902	+	0.951057i
$83$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$83$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$84$		0			0
$85$		0			0
$86$	1.61803	−	1.17557i	1.61803	−	1.17557i
$87$		0			0
$88$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$89$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$89$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$95$		0			0
$96$		0			0
$97$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$97$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$98$	−0.618034			−0.618034
$99$	−1.00000			−1.00000
$100$	1.00000			1.00000
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$103$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$107$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$108$		0			0
$109$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$109$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$110$		0			0
$111$		0			0
$112$	0.500000	−	0.363271i	0.500000	−	0.363271i
$113$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$113$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$119$		0			0
$120$		0			0
$121$	0.309017	+	0.951057i	0.309017	+	0.951057i
$122$	0.618034			0.618034
$123$		0			0
$124$	0.500000	+	1.53884i	0.500000	+	1.53884i
$125$		0			0
$126$	0.500000	+	0.363271i	0.500000	+	0.363271i
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$	0.309017	−	0.951057i	0.309017	−	0.951057i
$129$		0			0
$130$		0			0
$131$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$131$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$137$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$138$		0			0
$139$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$139$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$	0.309017	+	0.951057i	0.309017	+	0.951057i
$147$		0			0
$148$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$149$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$149$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$151$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$152$		0			0
$153$		0			0
$154$	0.190983	−	0.587785i	0.190983	−	0.587785i
$155$		0			0
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.309017	+	0.951057i	0.309017	+	0.951057i
$163$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$163$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$164$	−1.61803			−1.61803
$165$		0			0
$166$	−0.618034			−0.618034
$167$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$167$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$168$		0			0
$169$	0.309017	−	0.951057i	0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$173$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$173$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$174$		0			0
$175$	−0.618034			−0.618034
$176$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$177$		0			0
$178$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$179$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$179$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$180$		0			0
$181$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$181$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.61803			1.61803
$189$		0			0
$190$		0			0
$191$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$191$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$192$		0			0
$193$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$193$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$194$	0.618034	−	1.90211i	0.618034	−	1.90211i
$195$		0			0
$196$	0.500000	−	0.363271i	0.500000	−	0.363271i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.809017	−	0.587785i	0.809017	−	0.587785i
$199$	−2.00000			−2.00000			−1.00000			$\pi$
$199$	−2.00000			−2.00000			−1.00000			$\pi$
$200$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$211$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$212$		0			0
$213$		0			0
$214$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$215$		0			0
$216$		0			0
$217$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$218$	1.30902	−	0.951057i	1.30902	−	0.951057i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$225$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$226$		0			0
$227$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$227$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$		0			0
$236$	0.500000	+	0.363271i	0.500000	+	0.363271i
$237$		0			0
$238$		0			0
$239$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$239$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$243$		0			0
$244$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$249$		0			0
$250$		0			0
$251$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$	−0.618034			−0.618034
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$257$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$258$		0			0
$259$	0.309017	+	0.224514i	0.309017	+	0.224514i
$260$		0			0
$261$		0			0
$262$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$263$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$263$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$269$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$271$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$272$		0			0
$273$		0			0
$274$	−1.61803			−1.61803
$275$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$276$		0			0
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$279$	0.500000	−	1.53884i	0.500000	−	1.53884i
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.00000			1.00000
$288$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$293$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$293$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$294$		0			0
$295$		0			0
$296$	0.618034			0.618034
$297$		0			0
$298$	0.618034			0.618034
$299$		0			0
$300$		0			0
$301$	0.381966	−	1.17557i	0.381966	−	1.17557i
$302$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$307$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$308$	0.190983	+	0.587785i	0.190983	+	0.587785i
$309$		0			0
$310$		0			0
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$313$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$317$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$325$		0			0
$326$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$327$		0			0
$328$	1.30902	−	0.951057i	1.30902	−	0.951057i
$329$	−1.00000			−1.00000
$330$		0			0
$331$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$331$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$332$	0.500000	−	0.363271i	0.500000	−	0.363271i
$333$	0.190983	+	0.587785i	0.190983	+	0.587785i
$334$	0.500000	−	1.53884i	0.500000	−	1.53884i
$335$		0			0
$336$		0			0
$337$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$337$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$338$	0.309017	+	0.951057i	0.309017	+	0.951057i
$339$		0			0
$340$		0			0
$341$	−1.61803			−1.61803
$342$		0			0
$343$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$344$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$345$		0			0
$346$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$		0			0
$349$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$349$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$350$	0.500000	−	0.363271i	0.500000	−	0.363271i
$351$		0			0
$352$	0.809017	+	0.587785i	0.809017	+	0.587785i
$353$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$353$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$354$		0			0
$355$		0			0
$356$	0.190983	−	0.587785i	0.190983	−	0.587785i
$357$		0			0
$358$	1.61803	+	1.17557i	1.61803	+	1.17557i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$362$	−1.61803			−1.61803
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$367$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$368$		0			0
$369$	1.30902	+	0.951057i	1.30902	+	0.951057i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$373$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$374$		0			0
$375$		0			0
$376$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$377$		0			0
$378$		0			0
$379$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$379$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	0.500000	+	1.53884i	0.500000	+	1.53884i
$383$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$383$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.61803	−	1.17557i	1.61803	−	1.17557i
$388$	0.618034	+	1.90211i	0.618034	+	1.90211i
$389$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$389$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$390$		0			0
$391$		0			0
$392$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$397$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$397$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$398$	1.61803	−	1.17557i	1.61803	−	1.17557i
$399$		0			0
$400$	0.309017	−	0.951057i	0.309017	−	0.951057i
$401$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$401$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.500000	−	0.363271i	0.500000	−	0.363271i
$408$		0			0
$409$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$409$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$		0			0
$412$	1.61803	+	1.17557i	1.61803	+	1.17557i
$413$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.309017	−	0.224514i	0.309017	−	0.224514i
$428$	1.61803			1.61803
$429$		0			0
$430$		0			0
$431$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$431$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$432$		0			0
$433$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$433$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$434$	0.809017	+	0.587785i	0.809017	+	0.587785i
$435$		0			0
$436$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−0.618034			−0.618034
$442$		0			0
$443$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$443$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$	1.00000			1.00000
$451$	0.500000	−	1.53884i	0.500000	−	1.53884i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$457$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$461$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$467$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.618034			−0.618034
$473$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.61803	+	1.17557i	1.61803	+	1.17557i
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$	0.190983	−	0.587785i	0.190983	−	0.587785i
$489$		0			0
$490$		0			0
$491$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$491$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.61803			1.61803
$497$		0			0
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$	0.500000	−	0.363271i	0.500000	−	0.363271i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$509$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$	0.500000	+	0.363271i	0.500000	+	0.363271i
$512$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$513$		0			0
$514$	0.190983	+	0.587785i	0.190983	+	0.587785i
$515$		0			0
$516$		0			0
$517$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$518$	−0.381966			−0.381966
$519$		0			0
$520$		0			0
$521$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$521$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$522$		0			0
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$	0.500000	−	1.53884i	0.500000	−	1.53884i
$525$		0			0
$526$	0.500000	−	0.363271i	0.500000	−	0.363271i
$527$		0			0
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.61803			−1.61803
$539$	0.190983	+	0.587785i	0.190983	+	0.587785i
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$	0.500000	+	1.53884i	0.500000	+	1.53884i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$	1.30902	−	0.951057i	1.30902	−	0.951057i
$549$	0.618034			0.618034
$550$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.500000	+	0.363271i	0.500000	+	0.363271i
$557$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$557$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$558$	0.500000	+	1.53884i	0.500000	+	1.53884i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$563$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.500000	+	0.363271i	0.500000	+	0.363271i
$568$		0			0
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$571$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$572$		0			0
$573$		0			0
$574$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$575$		0			0
$576$	0.309017	−	0.951057i	0.309017	−	0.951057i
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$579$		0			0
$580$		0			0
$581$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$582$		0			0
$583$		0			0
$584$	1.00000			1.00000
$585$		0			0
$586$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$587$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$587$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$593$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$593$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$594$		0			0
$595$		0			0
$596$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$597$		0			0
$598$		0			0
$599$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$599$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$600$		0			0
$601$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$601$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$602$	0.381966	+	1.17557i	0.381966	+	1.17557i
$603$		0			0
$604$	1.61803			1.61803
$605$		0			0
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$615$		0			0
$616$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$624$		0			0
$625$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$631$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$641$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$642$		0			0
$643$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$643$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$647$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$648$	1.00000			1.00000
$649$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$650$		0			0
$651$		0			0
$652$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$653$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$653$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$654$		0			0
$655$		0			0
$656$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$657$	0.309017	+	0.951057i	0.309017	+	0.951057i
$658$	0.809017	−	0.587785i	0.809017	−	0.587785i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$661$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$662$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$663$		0			0
$664$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$665$		0			0
$666$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$667$		0			0
$668$	0.500000	+	1.53884i	0.500000	+	1.53884i
$669$		0			0
$670$		0			0
$671$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$672$		0			0
$673$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$673$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$674$		0			0
$675$		0			0
$676$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$		0			0
$679$	−0.381966	−	1.17557i	−0.381966	−	1.17557i
$680$		0			0
$681$		0			0
$682$	1.30902	−	0.951057i	1.30902	−	0.951057i
$683$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$683$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$		0			0
$686$	0.309017	−	0.951057i	0.309017	−	0.951057i
$687$		0			0
$688$	1.61803	+	1.17557i	1.61803	+	1.17557i
$689$		0			0
$690$		0			0
$691$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$691$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$692$	0.618034			0.618034
$693$	0.190983	−	0.587785i	0.190983	−	0.587785i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	1.30902	+	0.951057i	1.30902	+	0.951057i
$699$		0			0
$700$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$	−1.00000			−1.00000
$705$		0			0
$706$	1.30902	−	0.951057i	1.30902	−	0.951057i
$707$		0			0
$708$		0			0
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$	0.190983	+	0.587785i	0.190983	+	0.587785i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−2.00000			−2.00000
$717$		0			0
$718$		0			0
$719$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$719$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$720$		0			0
$721$	−1.00000	−	0.726543i	−1.00000	−	0.726543i
$722$	0.309017	−	0.951057i	0.309017	−	0.951057i
$723$		0			0
$724$	1.30902	−	0.951057i	1.30902	−	0.951057i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$733$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.61803			−1.61803
$739$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$739$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$743$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$747$	−0.618034			−0.618034
$748$		0			0
$749$	−1.00000			−1.00000
$750$		0			0
$751$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$751$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$752$	0.500000	−	1.53884i	0.500000	−	1.53884i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$757$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$758$	1.61803			1.61803
$759$		0			0
$760$		0			0
$761$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$761$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$	0.309017	−	0.951057i	0.309017	−	0.951057i
$764$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$765$		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
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$775$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$776$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$777$		0			0
$778$	0.618034	+	1.90211i	0.618034	+	1.90211i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$785$		0			0
$786$		0			0
$787$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$787$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$793$		0			0
$794$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$795$		0			0
$796$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$797$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$797$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$		0			0
$800$	0.309017	+	0.951057i	0.309017	+	0.951057i
$801$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$802$	−1.61803			−1.61803
$803$	0.809017	−	0.587785i	0.809017	−	0.587785i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$809$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$810$		0			0
$811$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$811$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$821$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$823$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$824$	−2.00000			−2.00000
$825$		0			0
$826$	0.381966			0.381966
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$		0			0
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	1.61803			1.61803
$847$	−0.618034			−0.618034
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$853$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$854$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$855$		0			0
$856$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$857$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$857$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$858$		0			0
$859$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$859$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$860$		0			0
$861$		0			0
$862$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$863$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$863$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−1.00000			−1.00000
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$873$	0.618034	−	1.90211i	0.618034	−	1.90211i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$877$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.500000	−	0.363271i	0.500000	−	0.363271i
$883$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$883$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$887$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$887$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.809017	−	0.587785i	0.809017	−	0.587785i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.500000	+	0.363271i	0.500000	+	0.363271i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$901$		0			0
$902$	0.500000	+	1.53884i	0.500000	+	1.53884i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$907$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$911$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$912$		0			0
$913$	0.190983	+	0.587785i	0.190983	+	0.587785i
$914$	0.618034			0.618034
$915$		0			0
$916$		0			0
$917$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$918$		0			0
$919$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$919$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$923$		0			0
$924$		0			0
$925$	0.618034			0.618034
$926$		0			0
$927$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$928$		0			0
$929$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−2.00000			−2.00000
$935$		0			0
$936$		0			0
$937$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$937$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$941$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$	0.500000	−	0.363271i	0.500000	−	0.363271i
$945$		0			0
$946$	2.00000			2.00000
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$953$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$954$		0			0
$955$		0			0
$956$	−2.00000			−2.00000
$957$		0			0
$958$		0			0
$959$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$960$		0			0
$961$	0.500000	−	1.53884i	0.500000	−	1.53884i
$962$		0			0
$963$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$969$		0			0
$970$		0			0
$971$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$971$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$972$		0			0
$973$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$974$		0			0
$975$		0			0
$976$	0.190983	+	0.587785i	0.190983	+	0.587785i
$977$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$977$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$978$		0			0
$979$	0.500000	+	0.363271i	0.500000	+	0.363271i
$980$		0			0
$981$	1.30902	−	0.951057i	1.30902	−	0.951057i
$982$	0.500000	+	1.53884i	0.500000	+	1.53884i
$983$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$983$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$991$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$992$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$997$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3212.1.bf.a.2335.1	✓	4
4.3	odd	2		3212.1.bf.b.2335.1	yes	4
11.4	even	5	inner	3212.1.bf.a.2919.1	yes	4
44.15	odd	10		3212.1.bf.b.2919.1	yes	4
73.72	even	2		3212.1.bf.b.2335.1	yes	4
292.291	odd	2	CM	3212.1.bf.a.2335.1	✓	4
803.510	even	10		3212.1.bf.b.2919.1	yes	4
3212.2919	odd	10	inner	3212.1.bf.a.2919.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3212.1.bf.a.2335.1	✓	4	1.1	even	1	trivial
3212.1.bf.a.2335.1	✓	4	292.291	odd	2	CM
3212.1.bf.a.2919.1	yes	4	11.4	even	5	inner
3212.1.bf.a.2919.1	yes	4	3212.2919	odd	10	inner
3212.1.bf.b.2335.1	yes	4	4.3	odd	2
3212.1.bf.b.2335.1	yes	4	73.72	even	2
3212.1.bf.b.2919.1	yes	4	44.15	odd	10
3212.1.bf.b.2919.1	yes	4	803.510	even	10

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:	\( N \)	\(=\)	\( 3212 = 2^{2} \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3212.bf (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.809017 + 0.587785i) q^{2} +(0.309017 - 0.951057i) q^{4} +(-0.190983 + 0.587785i) q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\)	\(q+(-0.809017 + 0.587785i) q^{2} +(0.309017 - 0.951057i) q^{4} +(-0.190983 + 0.587785i) q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.809017 + 0.587785i) q^{11} +(-0.190983 - 0.587785i) q^{14} +(-0.809017 - 0.587785i) q^{16} +(0.309017 - 0.951057i) q^{18} -1.00000 q^{22} +(0.309017 + 0.951057i) q^{25} +(0.500000 + 0.363271i) q^{28} +(-1.30902 + 0.951057i) q^{31} +1.00000 q^{32} +(0.309017 + 0.951057i) q^{36} +(0.190983 - 0.587785i) q^{37} +(-0.500000 - 1.53884i) q^{41} -2.00000 q^{43} +(0.809017 - 0.587785i) q^{44} +(0.500000 + 1.53884i) q^{47} +(0.500000 + 0.363271i) q^{49} +(-0.809017 - 0.587785i) q^{50} -0.618034 q^{56} +(-0.190983 + 0.587785i) q^{59} +(-0.500000 - 0.363271i) q^{61} +(0.500000 - 1.53884i) q^{62} +(-0.190983 - 0.587785i) q^{63} +(-0.809017 + 0.587785i) q^{64} +(-0.809017 - 0.587785i) q^{72} +(0.309017 - 0.951057i) q^{73} +(0.190983 + 0.587785i) q^{74} +(-0.500000 + 0.363271i) q^{77} +(0.309017 - 0.951057i) q^{81} +(1.30902 + 0.951057i) q^{82} +(0.500000 + 0.363271i) q^{83} +(1.61803 - 1.17557i) q^{86} +(-0.309017 + 0.951057i) q^{88} +0.618034 q^{89} +(-1.30902 - 0.951057i) q^{94} +(-1.61803 + 1.17557i) q^{97} -0.618034 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} - 3 q^{7} - q^{8} - q^{9}+O(q^{10}) \) 4 * q - q^2 - q^4 - 3 * q^7 - q^8 - q^9	\( 4 q - q^{2} - q^{4} - 3 q^{7} - q^{8} - q^{9} + q^{11} - 3 q^{14} - q^{16} - q^{18} - 4 q^{22} - q^{25} + 2 q^{28} - 3 q^{31} + 4 q^{32} - q^{36} + 3 q^{37} - 2 q^{41} - 8 q^{43} + q^{44} + 2 q^{47} + 2 q^{49} - q^{50} + 2 q^{56} - 3 q^{59} - 2 q^{61} + 2 q^{62} - 3 q^{63} - q^{64} - q^{72} - q^{73} + 3 q^{74} - 2 q^{77} - q^{81} + 3 q^{82} + 2 q^{83} + 2 q^{86} + q^{88} - 2 q^{89} - 3 q^{94} - 2 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q - q^2 - q^4 - 3 * q^7 - q^8 - q^9 + q^11 - 3 * q^14 - q^16 - q^18 - 4 * q^22 - q^25 + 2 * q^28 - 3 * q^31 + 4 * q^32 - q^36 + 3 * q^37 - 2 * q^41 - 8 * q^43 + q^44 + 2 * q^47 + 2 * q^49 - q^50 + 2 * q^56 - 3 * q^59 - 2 * q^61 + 2 * q^62 - 3 * q^63 - q^64 - q^72 - q^73 + 3 * q^74 - 2 * q^77 - q^81 + 3 * q^82 + 2 * q^83 + 2 * q^86 + q^88 - 2 * q^89 - 3 * q^94 - 2 * q^97 + 2 * q^98 - 4 * q^99

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