LMFDB - Embedded newform 3332.1.g.i.883.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(883,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("3332.883");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3332.g (of order $2$, degree $1$, 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:	$1.66288462209$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.0.205346735104.4

Embedding invariants

Embedding label			883.5
Root			$-0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	3332.883
Dual form			3332.1.g.i.883.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.309017 - 0.951057i) q^{2} -1.90211 q^{3} +(-0.809017 - 0.587785i) q^{4} +1.61803i q^{5} +(-0.587785 + 1.80902i) q^{6} +(-0.809017 + 0.587785i) q^{8} +2.61803 q^{9} +O(q^{10})$	\(q+(0.309017 - 0.951057i) q^{2} -1.90211 q^{3} +(-0.809017 - 0.587785i) q^{4} +1.61803i q^{5} +(-0.587785 + 1.80902i) q^{6} +(-0.809017 + 0.587785i) q^{8} +2.61803 q^{9} +(1.53884 + 0.500000i) q^{10} +(1.53884 + 1.11803i) q^{12} -3.07768i q^{15} +(0.309017 + 0.951057i) q^{16} -1.00000i q^{17} +(0.809017 - 2.48990i) q^{18} +(0.951057 - 1.30902i) q^{20} +(1.53884 - 1.11803i) q^{24} -1.61803 q^{25} -3.07768 q^{27} +(-2.92705 - 0.951057i) q^{30} +1.17557 q^{31} +1.00000 q^{32} +(-0.951057 - 0.309017i) q^{34} +(-2.11803 - 1.53884i) q^{36} +(-0.951057 - 1.30902i) q^{40} +0.618034i q^{41} +1.90211i q^{43} +4.23607i q^{45} +(-0.587785 - 1.80902i) q^{48} +(-0.500000 + 1.53884i) q^{50} +1.90211i q^{51} -1.61803 q^{53} +(-0.951057 + 2.92705i) q^{54} +(-1.80902 + 2.48990i) q^{60} -0.618034i q^{61} +(0.363271 - 1.11803i) q^{62} +(0.309017 - 0.951057i) q^{64} +1.17557i q^{67} +(-0.587785 + 0.809017i) q^{68} +(-2.11803 + 1.53884i) q^{72} +0.618034i q^{73} +3.07768 q^{75} +(-1.53884 + 0.500000i) q^{80} +3.23607 q^{81} +(0.587785 + 0.190983i) q^{82} +1.61803 q^{85} +(1.80902 + 0.587785i) q^{86} +(4.02874 + 1.30902i) q^{90} -2.23607 q^{93} -1.90211 q^{96} +1.61803i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 - 2 * q^4 - 2 * q^8 + 12 * q^9	$ 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 10 q^{30} + 8 q^{32} - 8 q^{36} - 4 q^{50} - 4 q^{53} - 10 q^{60} - 2 q^{64} - 8 q^{72} + 8 q^{81} + 4 q^{85} + 10 q^{86}+O(q^{100}) $ 8 * q - 2 * q^2 - 2 * q^4 - 2 * q^8 + 12 * q^9 - 2 * q^16 + 2 * q^18 - 4 * q^25 - 10 * q^30 + 8 * q^32 - 8 * q^36 - 4 * q^50 - 4 * q^53 - 10 * q^60 - 2 * q^64 - 8 * q^72 + 8 * q^81 + 4 * q^85 + 10 * q^86

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$.

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0.309017	−	0.951057i	0.309017	−	0.951057i
$2$	0.309017	−	0.951057i	0.309017	−	0.951057i
$3$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$3$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$4$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$5$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$5$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$6$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$7$		0			0
$7$		0			0
$8$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$9$	2.61803			2.61803
$10$	1.53884	+	0.500000i	1.53884	+	0.500000i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	1.53884	+	1.11803i	1.53884	+	1.11803i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		−	3.07768i		−	3.07768i
$16$	0.309017	+	0.951057i	0.309017	+	0.951057i
$17$		−	1.00000i		−	1.00000i
$17$		−	1.00000i		−	1.00000i
$18$	0.809017	−	2.48990i	0.809017	−	2.48990i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.951057	−	1.30902i	0.951057	−	1.30902i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	1.53884	−	1.11803i	1.53884	−	1.11803i
$25$	−1.61803			−1.61803
$26$		0			0
$27$	−3.07768			−3.07768
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$31$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$31$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$32$	1.00000			1.00000
$33$		0			0
$34$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$35$		0			0
$36$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$41$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$41$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$42$		0			0
$43$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$43$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$44$		0			0
$45$			4.23607i			4.23607i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$49$		0			0
$50$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$51$			1.90211i			1.90211i
$52$		0			0
$53$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$53$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$54$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$	−1.80902	+	2.48990i	−1.80902	+	2.48990i
$61$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$61$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$62$	0.363271	−	1.11803i	0.363271	−	1.11803i
$63$		0			0
$64$	0.309017	−	0.951057i	0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$67$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$68$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$73$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$73$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$74$		0			0
$75$	3.07768			3.07768
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$81$	3.23607			3.23607
$82$	0.587785	+	0.190983i	0.587785	+	0.190983i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	1.61803			1.61803
$86$	1.80902	+	0.587785i	1.80902	+	0.587785i
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	4.02874	+	1.30902i	4.02874	+	1.30902i
$91$		0			0
$92$		0			0
$93$	−2.23607			−2.23607
$94$		0			0
$95$		0			0
$96$	−1.90211			−1.90211
$97$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$97$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$98$		0			0
$99$		0			0
$100$	1.30902	+	0.951057i	1.30902	+	0.951057i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	1.80902	+	0.587785i	1.80902	+	0.587785i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	2.48990	+	1.80902i	2.48990	+	1.80902i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	1.80902	+	2.48990i	1.80902	+	2.48990i
$121$	−1.00000			−1.00000
$122$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$123$		−	1.17557i		−	1.17557i
$124$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$125$		−	1.00000i		−	1.00000i
$126$		0			0
$127$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$127$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$128$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$129$		−	3.61803i		−	3.61803i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	1.11803	+	0.363271i	1.11803	+	0.363271i
$135$		−	4.97980i		−	4.97980i
$136$	0.587785	+	0.809017i	0.587785	+	0.809017i
$137$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$137$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$138$		0			0
$139$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$139$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.809017	+	2.48990i	0.809017	+	2.48990i
$145$		0			0
$146$	0.587785	+	0.190983i	0.587785	+	0.190983i
$147$		0			0
$148$		0			0
$149$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$149$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$150$	0.951057	−	2.92705i	0.951057	−	2.92705i
$151$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$151$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$152$		0			0
$153$		−	2.61803i		−	2.61803i
$154$		0			0
$155$			1.90211i			1.90211i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$	3.07768			3.07768
$160$			1.61803i			1.61803i
$161$		0			0
$162$	1.00000	−	3.07768i	1.00000	−	3.07768i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	0.363271	−	0.500000i	0.363271	−	0.500000i
$165$		0			0
$166$		0			0
$167$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$167$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	0.500000	−	1.53884i	0.500000	−	1.53884i
$171$		0			0
$172$	1.11803	−	1.53884i	1.11803	−	1.53884i
$173$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$173$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$179$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$180$	2.48990	−	3.42705i	2.48990	−	3.42705i
$181$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$181$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$182$		0			0
$183$			1.17557i			1.17557i
$184$		0			0
$185$		0			0
$186$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$191$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$192$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	1.53884	+	0.500000i	1.53884	+	0.500000i
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$199$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$200$	1.30902	−	0.951057i	1.30902	−	0.951057i
$201$		−	2.23607i		−	2.23607i
$202$		0			0
$203$		0			0
$204$	1.11803	−	1.53884i	1.11803	−	1.53884i
$205$	−1.00000			−1.00000
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	1.30902	+	0.951057i	1.30902	+	0.951057i
$213$		0			0
$214$		0			0
$215$	−3.07768			−3.07768
$216$	2.48990	−	1.80902i	2.48990	−	1.80902i
$217$		0			0
$218$		0			0
$219$		−	1.17557i		−	1.17557i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−4.23607			−4.23607
$226$		0			0
$227$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$227$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$239$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$240$	2.92705	−	0.951057i	2.92705	−	0.951057i
$241$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$241$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$242$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$243$	−3.07768			−3.07768
$244$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$245$		0			0
$246$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$247$		0			0
$248$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$249$		0			0
$250$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	1.11803	+	0.363271i	1.11803	+	0.363271i
$255$	−3.07768			−3.07768
$256$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	−3.44095	−	1.11803i	−3.44095	−	1.11803i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		−	2.61803i		−	2.61803i
$266$		0			0
$267$		0			0
$268$	0.690983	−	0.951057i	0.690983	−	0.951057i
$269$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$269$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$270$	−4.73607	−	1.53884i	−4.73607	−	1.53884i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	0.951057	−	0.309017i	0.951057	−	0.309017i
$273$		0			0
$274$	0.190983	−	0.587785i	0.190983	−	0.587785i
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	0.363271	−	1.11803i	0.363271	−	1.11803i
$279$	3.07768			3.07768
$280$		0			0
$281$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$281$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$282$		0			0
$283$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$283$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.61803			2.61803
$289$	−1.00000			−1.00000
$290$		0			0
$291$		−	3.07768i		−	3.07768i
$292$	0.363271	−	0.500000i	0.363271	−	0.500000i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	0.190983	−	0.587785i	0.190983	−	0.587785i
$299$		0			0
$300$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$301$		0			0
$302$	1.80902	+	0.587785i	1.80902	+	0.587785i
$303$		0			0
$304$		0			0
$305$	1.00000			1.00000
$306$	−2.48990	−	0.809017i	−2.48990	−	0.809017i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	1.80902	+	0.587785i	1.80902	+	0.587785i
$311$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$311$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$312$		0			0
$313$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$313$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$	0.951057	−	2.92705i	0.951057	−	2.92705i
$319$		0			0
$320$	1.53884	+	0.500000i	1.53884	+	0.500000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$329$		0			0
$330$		0			0
$331$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$331$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$332$		0			0
$333$		0			0
$334$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$335$	−1.90211			−1.90211
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$339$		0			0
$340$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$345$		0			0
$346$	0.587785	+	0.190983i	0.587785	+	0.190983i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$359$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$359$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$360$	−2.48990	−	3.42705i	−2.48990	−	3.42705i
$361$	1.00000			1.00000
$362$	1.90211	+	0.618034i	1.90211	+	0.618034i
$363$	1.90211			1.90211
$364$		0			0
$365$	−1.00000			−1.00000
$366$	1.11803	+	0.363271i	1.11803	+	0.363271i
$367$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$367$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$368$		0			0
$369$			1.61803i			1.61803i
$370$		0			0
$371$		0			0
$372$	1.80902	+	1.31433i	1.80902	+	1.31433i
$373$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$373$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$374$		0			0
$375$			1.90211i			1.90211i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		−	2.23607i		−	2.23607i
$382$	1.11803	+	0.363271i	1.11803	+	0.363271i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	1.53884	+	1.11803i	1.53884	+	1.11803i
$385$		0			0
$386$		0			0
$387$			4.97980i			4.97980i
$388$	0.951057	−	1.30902i	0.951057	−	1.30902i
$389$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$389$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$397$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$398$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$399$		0			0
$400$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	−2.12663	−	0.690983i	−2.12663	−	0.690983i
$403$		0			0
$404$		0			0
$405$			5.23607i			5.23607i
$406$		0			0
$407$		0			0
$408$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$411$	−1.17557			−1.17557
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.23607			−2.23607
$418$		0			0
$419$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$419$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$421$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$422$		0			0
$423$		0			0
$424$	1.30902	−	0.951057i	1.30902	−	0.951057i
$425$			1.61803i			1.61803i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$439$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$439$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.17557			−1.17557
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.30902	+	4.02874i	−1.30902	+	4.02874i
$451$		0			0
$452$		0			0
$453$		−	3.61803i		−	3.61803i
$454$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$455$		0			0
$456$		0			0
$457$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$457$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$458$		0			0
$459$			3.07768i			3.07768i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$463$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$464$		0			0
$465$		−	3.61803i		−	3.61803i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.23607			−4.23607
$478$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$479$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$479$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$480$		−	3.07768i		−	3.07768i
$481$		0			0
$482$	1.53884	+	0.500000i	1.53884	+	0.500000i
$483$		0			0
$484$	0.809017	+	0.587785i	0.809017	+	0.587785i
$485$	−2.61803			−2.61803
$486$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	0.363271	+	0.500000i	0.363271	+	0.500000i
$489$		0			0
$490$		0			0
$491$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$491$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$492$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.363271	+	1.11803i	0.363271	+	1.11803i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$501$	3.61803			3.61803
$502$		0			0
$503$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$503$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.90211			1.90211
$508$	0.690983	−	0.951057i	0.690983	−	0.951057i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$511$		0			0
$512$	0.309017	+	0.951057i	0.309017	+	0.951057i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−2.12663	+	2.92705i	−2.12663	+	2.92705i
$517$		0			0
$518$		0			0
$519$		−	1.17557i		−	1.17557i
$520$		0			0
$521$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$521$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.17557i		−	1.17557i
$528$		0			0
$529$	−1.00000			−1.00000
$530$	−2.48990	−	0.809017i	−2.48990	−	0.809017i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$537$			3.61803i			3.61803i
$538$	1.90211	+	0.618034i	1.90211	+	0.618034i
$539$		0			0
$540$	−2.92705	+	4.02874i	−2.92705	+	4.02874i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	3.80423i		−	3.80423i
$544$		−	1.00000i		−	1.00000i
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$549$		−	1.61803i		−	1.61803i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$558$	0.951057	−	2.92705i	0.951057	−	2.92705i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.500000	−	1.53884i	0.500000	−	1.53884i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$	0.363271	−	1.11803i	0.363271	−	1.11803i
$567$		0			0
$568$		0			0
$569$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$569$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	2.23607i		−	2.23607i
$574$		0			0
$575$		0			0
$576$	0.809017	−	2.48990i	0.809017	−	2.48990i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$583$		0			0
$584$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$597$	3.61803			3.61803
$598$		0			0
$599$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$599$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$600$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$601$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$601$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$602$		0			0
$603$			3.07768i			3.07768i
$604$	1.11803	−	1.53884i	1.11803	−	1.53884i
$605$		−	1.61803i		−	1.61803i
$606$		0			0
$607$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$607$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$	0.309017	−	0.951057i	0.309017	−	0.951057i
$611$		0			0
$612$	−1.53884	+	2.11803i	−1.53884	+	2.11803i
$613$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$613$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$614$		0			0
$615$	1.90211			1.90211
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	1.11803	−	1.53884i	1.11803	−	1.53884i
$621$		0			0
$622$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$623$		0			0
$624$		0			0
$625$		0			0
$626$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$631$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.90211			−1.90211
$636$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.951057	−	1.30902i	0.951057	−	1.30902i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$643$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$644$		0			0
$645$	5.85410			5.85410
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−2.61803	+	1.90211i	−2.61803	+	1.90211i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$657$			1.61803i			1.61803i
$658$		0			0
$659$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$659$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	1.11803	+	0.363271i	1.11803	+	0.363271i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	1.53884	+	1.11803i	1.53884	+	1.11803i
$669$		0			0
$670$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	4.97980			4.97980
$676$	0.809017	+	0.587785i	0.809017	+	0.587785i
$677$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$677$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$678$		0			0
$679$		0			0
$680$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$681$	2.23607			2.23607
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$			1.00000i			1.00000i
$686$		0			0
$687$		0			0
$688$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$689$		0			0
$690$		0			0
$691$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$691$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$692$	0.363271	−	0.500000i	0.363271	−	0.500000i
$693$		0			0
$694$		0			0
$695$			1.90211i			1.90211i
$696$		0			0
$697$	0.618034			0.618034
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.00000			2.00000			1.00000			$0$
$701$	2.00000			2.00000			1.00000			$0$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$717$			3.61803i			3.61803i
$718$	1.11803	+	0.363271i	1.11803	+	0.363271i
$719$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$719$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$720$	−4.02874	+	1.30902i	−4.02874	+	1.30902i
$721$		0			0
$722$	0.309017	−	0.951057i	0.309017	−	0.951057i
$723$		−	3.07768i		−	3.07768i
$724$	1.17557	−	1.61803i	1.17557	−	1.61803i
$725$		0			0
$726$	0.587785	−	1.80902i	0.587785	−	1.80902i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	2.61803			2.61803
$730$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$731$	1.90211			1.90211
$732$	0.690983	−	0.951057i	0.690983	−	0.951057i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	0.587785	−	1.80902i	0.587785	−	1.80902i
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.53884	+	0.500000i	1.53884	+	0.500000i
$739$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$739$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	1.80902	−	1.31433i	1.80902	−	1.31433i
$745$			1.00000i			1.00000i
$746$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	1.80902	+	0.587785i	1.80902	+	0.587785i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.07768			−3.07768
$756$		0			0
$757$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$757$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	−2.12663	−	0.690983i	−2.12663	−	0.690983i
$763$		0			0
$764$	0.690983	−	0.951057i	0.690983	−	0.951057i
$765$	4.23607			4.23607
$766$		0			0
$767$		0			0
$768$	1.53884	−	1.11803i	1.53884	−	1.11803i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	4.73607	+	1.53884i	4.73607	+	1.53884i
$775$	−1.90211			−1.90211
$776$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$777$		0			0
$778$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$795$			4.97980i			4.97980i
$796$	1.53884	+	1.11803i	1.53884	+	1.11803i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−1.61803			−1.61803
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.31433	+	1.80902i	−1.31433	+	1.80902i
$805$		0			0
$806$		0			0
$807$		−	3.80423i		−	3.80423i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	4.97980	+	1.61803i	4.97980	+	1.61803i
$811$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$811$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	0.809017	+	0.587785i	0.809017	+	0.587785i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$835$		−	3.07768i		−	3.07768i
$836$		0			0
$837$	−3.61803			−3.61803
$838$	0.363271	−	1.11803i	0.363271	−	1.11803i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$843$	−3.07768			−3.07768
$844$		0			0
$845$		−	1.61803i		−	1.61803i
$846$		0			0
$847$		0			0
$848$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$849$	−2.23607			−2.23607
$850$	1.53884	+	0.500000i	1.53884	+	0.500000i
$851$		0			0
$852$		0			0
$853$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$853$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$857$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	2.48990	+	1.80902i	2.48990	+	1.80902i
$861$		0			0
$862$		0			0
$863$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$863$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$864$	−3.07768			−3.07768
$865$	−1.00000			−1.00000
$866$		0			0
$867$	1.90211			1.90211
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$			4.23607i			4.23607i
$874$		0			0
$875$		0			0
$876$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	0.587785	−	1.80902i	0.587785	−	1.80902i
$879$		0			0
$880$		0			0
$881$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$881$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$882$		0			0
$883$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$883$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$887$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$895$	3.07768			3.07768
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	3.42705	+	2.48990i	3.42705	+	2.48990i
$901$			1.61803i			1.61803i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.23607			−3.23607
$906$	−3.44095	−	1.11803i	−3.44095	−	1.11803i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	0.951057	+	0.690983i	0.951057	+	0.690983i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$	0.500000	−	1.53884i	0.500000	−	1.53884i
$915$	−1.90211			−1.90211
$916$		0			0
$917$		0			0
$918$	2.92705	+	0.951057i	2.92705	+	0.951057i
$919$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$919$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$927$		0			0
$928$		0			0
$929$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$929$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$930$	−3.44095	−	1.11803i	−3.44095	−	1.11803i
$931$		0			0
$932$		0			0
$933$	2.23607			2.23607
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$			3.07768i			3.07768i
$940$		0			0
$941$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$941$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$953$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$954$	−1.30902	+	4.02874i	−1.30902	+	4.02874i
$955$	−1.90211			−1.90211
$956$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$957$		0			0
$958$	0.587785	−	1.80902i	0.587785	−	1.80902i
$959$		0			0
$960$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$961$	0.381966			0.381966
$962$		0			0
$963$		0			0
$964$	0.951057	−	1.30902i	0.951057	−	1.30902i
$965$		0			0
$966$		0			0
$967$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$967$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$968$	0.809017	−	0.587785i	0.809017	−	0.587785i
$969$		0			0
$970$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	2.48990	+	1.80902i	2.48990	+	1.80902i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.587785	−	0.190983i	0.587785	−	0.190983i
$977$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$977$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$983$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$983$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$984$	0.690983	+	0.951057i	0.690983	+	0.951057i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	1.17557			1.17557
$993$		−	2.23607i		−	2.23607i
$994$		0			0
$995$		−	3.07768i		−	3.07768i
$996$		0			0
$997$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$997$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.1.g.i.883.5	✓	8
4.3	odd	2	inner	3332.1.g.i.883.8	yes	8
7.2	even	3		3332.1.o.g.67.2		16
7.3	odd	6		3332.1.o.g.2039.5		16
7.4	even	3		3332.1.o.g.2039.6		16
7.5	odd	6		3332.1.o.g.67.1		16
7.6	odd	2	inner	3332.1.g.i.883.6	yes	8
17.16	even	2	inner	3332.1.g.i.883.6	yes	8
28.3	even	6		3332.1.o.g.2039.2		16
28.11	odd	6		3332.1.o.g.2039.1		16
28.19	even	6		3332.1.o.g.67.6		16
28.23	odd	6		3332.1.o.g.67.5		16
28.27	even	2	inner	3332.1.g.i.883.7	yes	8
68.67	odd	2	inner	3332.1.g.i.883.7	yes	8
119.16	even	6		3332.1.o.g.67.1		16
119.33	odd	6		3332.1.o.g.67.2		16
119.67	even	6		3332.1.o.g.2039.5		16
119.101	odd	6		3332.1.o.g.2039.6		16
119.118	odd	2	CM	3332.1.g.i.883.5	✓	8
476.67	odd	6		3332.1.o.g.2039.2		16
476.135	odd	6		3332.1.o.g.67.6		16
476.271	even	6		3332.1.o.g.67.5		16
476.339	even	6		3332.1.o.g.2039.1		16
476.475	even	2	inner	3332.1.g.i.883.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.1.g.i.883.5	✓	8	1.1	even	1	trivial
3332.1.g.i.883.5	✓	8	119.118	odd	2	CM
3332.1.g.i.883.6	yes	8	7.6	odd	2	inner
3332.1.g.i.883.6	yes	8	17.16	even	2	inner
3332.1.g.i.883.7	yes	8	28.27	even	2	inner
3332.1.g.i.883.7	yes	8	68.67	odd	2	inner
3332.1.g.i.883.8	yes	8	4.3	odd	2	inner
3332.1.g.i.883.8	yes	8	476.475	even	2	inner
3332.1.o.g.67.1		16	7.5	odd	6
3332.1.o.g.67.1		16	119.16	even	6
3332.1.o.g.67.2		16	7.2	even	3
3332.1.o.g.67.2		16	119.33	odd	6
3332.1.o.g.67.5		16	28.23	odd	6
3332.1.o.g.67.5		16	476.271	even	6
3332.1.o.g.67.6		16	28.19	even	6
3332.1.o.g.67.6		16	476.135	odd	6
3332.1.o.g.2039.1		16	28.11	odd	6
3332.1.o.g.2039.1		16	476.339	even	6
3332.1.o.g.2039.2		16	28.3	even	6
3332.1.o.g.2039.2		16	476.67	odd	6
3332.1.o.g.2039.5		16	7.3	odd	6
3332.1.o.g.2039.5		16	119.67	even	6
3332.1.o.g.2039.6		16	7.4	even	3
3332.1.o.g.2039.6		16	119.101	odd	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3332.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.309017 - 0.951057i) q^{2} -1.90211 q^{3} +(-0.809017 - 0.587785i) q^{4} +1.61803i q^{5} +(-0.587785 + 1.80902i) q^{6} +(-0.809017 + 0.587785i) q^{8} +2.61803 q^{9} +O(q^{10})\)	\(q+(0.309017 - 0.951057i) q^{2} -1.90211 q^{3} +(-0.809017 - 0.587785i) q^{4} +1.61803i q^{5} +(-0.587785 + 1.80902i) q^{6} +(-0.809017 + 0.587785i) q^{8} +2.61803 q^{9} +(1.53884 + 0.500000i) q^{10} +(1.53884 + 1.11803i) q^{12} -3.07768i q^{15} +(0.309017 + 0.951057i) q^{16} -1.00000i q^{17} +(0.809017 - 2.48990i) q^{18} +(0.951057 - 1.30902i) q^{20} +(1.53884 - 1.11803i) q^{24} -1.61803 q^{25} -3.07768 q^{27} +(-2.92705 - 0.951057i) q^{30} +1.17557 q^{31} +1.00000 q^{32} +(-0.951057 - 0.309017i) q^{34} +(-2.11803 - 1.53884i) q^{36} +(-0.951057 - 1.30902i) q^{40} +0.618034i q^{41} +1.90211i q^{43} +4.23607i q^{45} +(-0.587785 - 1.80902i) q^{48} +(-0.500000 + 1.53884i) q^{50} +1.90211i q^{51} -1.61803 q^{53} +(-0.951057 + 2.92705i) q^{54} +(-1.80902 + 2.48990i) q^{60} -0.618034i q^{61} +(0.363271 - 1.11803i) q^{62} +(0.309017 - 0.951057i) q^{64} +1.17557i q^{67} +(-0.587785 + 0.809017i) q^{68} +(-2.11803 + 1.53884i) q^{72} +0.618034i q^{73} +3.07768 q^{75} +(-1.53884 + 0.500000i) q^{80} +3.23607 q^{81} +(0.587785 + 0.190983i) q^{82} +1.61803 q^{85} +(1.80902 + 0.587785i) q^{86} +(4.02874 + 1.30902i) q^{90} -2.23607 q^{93} -1.90211 q^{96} +1.61803i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 - 2 * q^4 - 2 * q^8 + 12 * q^9	\( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 10 q^{30} + 8 q^{32} - 8 q^{36} - 4 q^{50} - 4 q^{53} - 10 q^{60} - 2 q^{64} - 8 q^{72} + 8 q^{81} + 4 q^{85} + 10 q^{86}+O(q^{100}) \) 8 * q - 2 * q^2 - 2 * q^4 - 2 * q^8 + 12 * q^9 - 2 * q^16 + 2 * q^18 - 4 * q^25 - 10 * q^30 + 8 * q^32 - 8 * q^36 - 4 * q^50 - 4 * q^53 - 10 * q^60 - 2 * q^64 - 8 * q^72 + 8 * q^81 + 4 * q^85 + 10 * q^86

Introduction

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