LMFDB - Embedded newform 1011.1.f.b.863.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1011,1,Mod(485,1011)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1011.485"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1011, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1011 = 3 \cdot 337 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1011.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.504554727772$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.114818259.2

Embedding invariants

Embedding label			863.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1011.863
Dual form			1011.1.f.b.485.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.707107 + 0.707107i) q^{5} +1.41421 q^{6} +1.00000i q^{7} -1.00000 q^{9} +(-1.00000 + 1.00000i) q^{10} +1.00000i q^{12} -1.00000 q^{13} -1.41421 q^{14} +(0.707107 - 0.707107i) q^{15} -1.00000 q^{16} +(0.707107 + 0.707107i) q^{17} -1.41421i q^{18} +(-0.707107 - 0.707107i) q^{20} +1.00000 q^{21} -1.41421i q^{26} +1.00000i q^{27} -1.00000i q^{28} +(0.707107 + 0.707107i) q^{29} +(1.00000 + 1.00000i) q^{30} +(1.00000 - 1.00000i) q^{31} -1.41421i q^{32} +(-1.00000 + 1.00000i) q^{34} +(-0.707107 + 0.707107i) q^{35} +1.00000 q^{36} +1.00000 q^{37} +1.00000i q^{39} +1.41421i q^{42} -1.00000i q^{43} +(-0.707107 - 0.707107i) q^{45} -1.41421 q^{47} +1.00000i q^{48} +(0.707107 - 0.707107i) q^{51} +1.00000 q^{52} +(-0.707107 + 0.707107i) q^{53} -1.41421 q^{54} +(-1.00000 + 1.00000i) q^{58} +(0.707107 - 0.707107i) q^{59} +(-0.707107 + 0.707107i) q^{60} +(1.41421 + 1.41421i) q^{62} -1.00000i q^{63} +1.00000 q^{64} +(-0.707107 - 0.707107i) q^{65} +(-1.00000 - 1.00000i) q^{67} +(-0.707107 - 0.707107i) q^{68} +(-1.00000 - 1.00000i) q^{70} +(-0.707107 + 0.707107i) q^{71} +1.41421i q^{74} -1.41421 q^{78} +1.00000 q^{79} +(-0.707107 - 0.707107i) q^{80} +1.00000 q^{81} +(0.707107 + 0.707107i) q^{83} -1.00000 q^{84} +1.00000i q^{85} +1.41421 q^{86} +(0.707107 - 0.707107i) q^{87} +(-0.707107 - 0.707107i) q^{89} +(1.00000 - 1.00000i) q^{90} -1.00000i q^{91} +(-1.00000 - 1.00000i) q^{93} -2.00000i q^{94} -1.41421 q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9} - 4 q^{10} - 4 q^{13} - 4 q^{16} + 4 q^{21} + 4 q^{30} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 4 q^{37} + 4 q^{52} - 4 q^{58} + 4 q^{64} - 4 q^{67} - 4 q^{70} + 4 q^{79} + 4 q^{81} - 4 q^{84}+ \cdots - 4 q^{93}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^10 - 4 * q^13 - 4 * q^16 + 4 * q^21 + 4 * q^30 + 4 * q^31 - 4 * q^34 + 4 * q^36 + 4 * q^37 + 4 * q^52 - 4 * q^58 + 4 * q^64 - 4 * q^67 - 4 * q^70 + 4 * q^79 + 4 * q^81 - 4 * q^84 + 4 * q^90 - 4 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1011\mathbb{Z}\right)^\times$.

$n$	$10$	$338$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$		−	1.00000i		−	1.00000i
$3$		−	1.00000i		−	1.00000i
$4$	−1.00000			−1.00000
$5$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$5$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$6$	1.41421			1.41421
$7$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$	−1.00000			−1.00000
$10$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$			1.00000i			1.00000i
$13$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$14$	−1.41421			−1.41421
$15$	0.707107	−	0.707107i	0.707107	−	0.707107i
$16$	−1.00000			−1.00000
$17$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$17$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$18$		−	1.41421i		−	1.41421i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$21$	1.00000			1.00000
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$		0			0
$26$		−	1.41421i		−	1.41421i
$27$			1.00000i			1.00000i
$28$		−	1.00000i		−	1.00000i
$29$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$29$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$30$	1.00000	+	1.00000i	1.00000	+	1.00000i
$31$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$31$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$32$		−	1.41421i		−	1.41421i
$33$		0			0
$34$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$35$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$36$	1.00000			1.00000
$37$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$37$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$			1.00000i			1.00000i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$			1.41421i			1.41421i
$43$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$43$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$44$		0			0
$45$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$46$		0			0
$47$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$47$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$48$			1.00000i			1.00000i
$49$		0			0
$50$		0			0
$51$	0.707107	−	0.707107i	0.707107	−	0.707107i
$52$	1.00000			1.00000
$53$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$53$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$54$	−1.41421			−1.41421
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$59$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$59$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$60$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$	1.41421	+	1.41421i	1.41421	+	1.41421i
$63$		−	1.00000i		−	1.00000i
$64$	1.00000			1.00000
$65$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$66$		0			0
$67$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$67$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$68$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$69$		0			0
$70$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$71$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$71$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$			1.41421i			1.41421i
$75$		0			0
$76$		0			0
$77$		0			0
$78$	−1.41421			−1.41421
$79$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$79$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$80$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$81$	1.00000			1.00000
$82$		0			0
$83$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$83$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$84$	−1.00000			−1.00000
$85$			1.00000i			1.00000i
$86$	1.41421			1.41421
$87$	0.707107	−	0.707107i	0.707107	−	0.707107i
$88$		0			0
$89$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$89$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$90$	1.00000	−	1.00000i	1.00000	−	1.00000i
$91$		−	1.00000i		−	1.00000i
$92$		0			0
$93$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$94$		−	2.00000i		−	2.00000i
$95$		0			0
$96$	−1.41421			−1.41421
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$101$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$	1.00000	+	1.00000i	1.00000	+	1.00000i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$	0.707107	+	0.707107i	0.707107	+	0.707107i
$106$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		−	1.00000i		−	1.00000i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		−	1.00000i		−	1.00000i
$112$		−	1.00000i		−	1.00000i
$113$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$113$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$117$	1.00000			1.00000
$118$	1.00000	+	1.00000i	1.00000	+	1.00000i
$119$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$120$		0			0
$121$			1.00000i			1.00000i
$122$		0			0
$123$		0			0
$124$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$125$	0.707107	−	0.707107i	0.707107	−	0.707107i
$126$	1.41421			1.41421
$127$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$127$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$128$		0			0
$129$	−1.00000			−1.00000
$130$	1.00000	−	1.00000i	1.00000	−	1.00000i
$131$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$131$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$132$		0			0
$133$		0			0
$134$	1.41421	−	1.41421i	1.41421	−	1.41421i
$135$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$136$		0			0
$137$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$137$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$140$	0.707107	−	0.707107i	0.707107	−	0.707107i
$141$			1.41421i			1.41421i
$142$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$143$		0			0
$144$	1.00000			1.00000
$145$			1.00000i			1.00000i
$146$		0			0
$147$		0			0
$148$	−1.00000			−1.00000
$149$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$149$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$154$		0			0
$155$	1.41421			1.41421
$156$		−	1.00000i		−	1.00000i
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$			1.41421i			1.41421i
$159$	0.707107	+	0.707107i	0.707107	+	0.707107i
$160$	1.00000	−	1.00000i	1.00000	−	1.00000i
$161$		0			0
$162$			1.41421i			1.41421i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$167$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$167$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		0			0
$170$	−1.41421			−1.41421
$171$		0			0
$172$			1.00000i			1.00000i
$173$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$173$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$174$	1.00000	+	1.00000i	1.00000	+	1.00000i
$175$		0			0
$176$		0			0
$177$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$178$	1.00000	−	1.00000i	1.00000	−	1.00000i
$179$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$180$	0.707107	+	0.707107i	0.707107	+	0.707107i
$181$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$181$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$182$	1.41421			1.41421
$183$		0			0
$184$		0			0
$185$	0.707107	+	0.707107i	0.707107	+	0.707107i
$186$	1.41421	−	1.41421i	1.41421	−	1.41421i
$187$		0			0
$188$	1.41421			1.41421
$189$	−1.00000			−1.00000
$190$		0			0
$191$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$191$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$192$		−	1.00000i		−	1.00000i
$193$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$196$		0			0
$197$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$197$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$199$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$200$		0			0
$201$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$202$	2.00000	+	2.00000i	2.00000	+	2.00000i
$203$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$204$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.00000			1.00000
$209$		0			0
$210$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.707107	−	0.707107i	0.707107	−	0.707107i
$213$	0.707107	+	0.707107i	0.707107	+	0.707107i
$214$		0			0
$215$	0.707107	−	0.707107i	0.707107	−	0.707107i
$216$		0			0
$217$	1.00000	+	1.00000i	1.00000	+	1.00000i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$222$	1.41421			1.41421
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$	1.41421			1.41421
$225$		0			0
$226$			2.00000i			2.00000i
$227$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$227$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$229$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$			1.41421i			1.41421i
$235$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$236$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$237$		−	1.00000i		−	1.00000i
$238$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$241$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$241$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$242$	−1.41421			−1.41421
$243$		−	1.00000i		−	1.00000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.707107	−	0.707107i	0.707107	−	0.707107i
$250$	1.00000	+	1.00000i	1.00000	+	1.00000i
$251$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$251$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$252$			1.00000i			1.00000i
$253$		0			0
$254$	1.41421	+	1.41421i	1.41421	+	1.41421i
$255$	1.00000			1.00000
$256$	1.00000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		−	1.41421i		−	1.41421i
$259$			1.00000i			1.00000i
$260$	0.707107	+	0.707107i	0.707107	+	0.707107i
$261$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$262$	2.00000			2.00000
$263$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$263$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	−1.00000			−1.00000
$266$		0			0
$267$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$268$	1.00000	+	1.00000i	1.00000	+	1.00000i
$269$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$269$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$270$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$271$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$271$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$272$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$273$	−1.00000			−1.00000
$274$		−	2.00000i		−	2.00000i
$275$		0			0
$276$		0			0
$277$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$277$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$278$	1.41421	−	1.41421i	1.41421	−	1.41421i
$279$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−2.00000			−2.00000
$283$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$283$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$284$	0.707107	−	0.707107i	0.707107	−	0.707107i
$285$		0			0
$286$		0			0
$287$		0			0
$288$			1.41421i			1.41421i
$289$		0			0
$290$	−1.41421			−1.41421
$291$		0			0
$292$		0			0
$293$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$293$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$294$		0			0
$295$	1.00000			1.00000
$296$		0			0
$297$		0			0
$298$		−	2.00000i		−	2.00000i
$299$		0			0
$300$		0			0
$301$	1.00000			1.00000
$302$		0			0
$303$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$304$		0			0
$305$		0			0
$306$	1.00000	−	1.00000i	1.00000	−	1.00000i
$307$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$307$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$			2.00000i			2.00000i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$313$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$	0.707107	−	0.707107i	0.707107	−	0.707107i
$316$	−1.00000			−1.00000
$317$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$317$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$318$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$319$		0			0
$320$	0.707107	+	0.707107i	0.707107	+	0.707107i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.00000			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.41421i		−	1.41421i
$330$		0			0
$331$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$331$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$332$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$333$	−1.00000			−1.00000
$334$			2.00000i			2.00000i
$335$		−	1.41421i		−	1.41421i
$336$	−1.00000			−1.00000
$337$			1.00000i			1.00000i
$337$			1.00000i			1.00000i
$338$		0			0
$339$		−	1.41421i		−	1.41421i
$340$		−	1.00000i		−	1.00000i
$341$		0			0
$342$		0			0
$343$			1.00000i			1.00000i
$344$		0			0
$345$		0			0
$346$	−2.00000			−2.00000
$347$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$347$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$348$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		−	1.00000i		−	1.00000i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$	1.00000	−	1.00000i	1.00000	−	1.00000i
$355$	−1.00000			−1.00000
$356$	0.707107	+	0.707107i	0.707107	+	0.707107i
$357$	0.707107	+	0.707107i	0.707107	+	0.707107i
$358$	2.00000			2.00000
$359$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$359$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$	−1.41421			−1.41421
$363$	1.00000			1.00000
$364$			1.00000i			1.00000i
$365$		0			0
$366$		0			0
$367$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$367$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$371$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$372$	1.00000	+	1.00000i	1.00000	+	1.00000i
$373$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$376$		0			0
$377$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$378$		−	1.41421i		−	1.41421i
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$382$	1.00000	−	1.00000i	1.00000	−	1.00000i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		−	1.41421i		−	1.41421i
$387$			1.00000i			1.00000i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$391$		0			0
$392$		0			0
$393$	−1.41421			−1.41421
$394$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$395$	0.707107	+	0.707107i	0.707107	+	0.707107i
$396$		0			0
$397$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$397$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$403$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$404$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$405$	0.707107	+	0.707107i	0.707107	+	0.707107i
$406$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$			1.41421i			1.41421i
$412$		0			0
$413$	0.707107	+	0.707107i	0.707107	+	0.707107i
$414$		0			0
$415$			1.00000i			1.00000i
$416$			1.41421i			1.41421i
$417$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$418$		0			0
$419$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$419$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$420$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$421$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$	1.41421			1.41421
$424$		0			0
$425$		0			0
$426$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$427$		0			0
$428$		0			0
$429$		0			0
$430$	1.00000	+	1.00000i	1.00000	+	1.00000i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		−	1.00000i		−	1.00000i
$433$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$433$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$434$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$435$	1.00000			1.00000
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$439$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$	1.00000	−	1.00000i	1.00000	−	1.00000i
$443$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$443$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$444$			1.00000i			1.00000i
$445$		−	1.00000i		−	1.00000i
$446$		0			0
$447$			1.41421i			1.41421i
$448$			1.00000i			1.00000i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−1.41421			−1.41421
$453$		0			0
$454$	−2.00000			−2.00000
$455$	0.707107	−	0.707107i	0.707107	−	0.707107i
$456$		0			0
$457$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$457$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$458$	1.41421			1.41421
$459$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$460$		0			0
$461$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$461$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$465$		−	1.41421i		−	1.41421i
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−1.00000			−1.00000
$469$	1.00000	−	1.00000i	1.00000	−	1.00000i
$470$	1.41421	−	1.41421i	1.41421	−	1.41421i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	1.41421			1.41421
$475$		0			0
$476$	0.707107	−	0.707107i	0.707107	−	0.707107i
$477$	0.707107	−	0.707107i	0.707107	−	0.707107i
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$481$	−1.00000			−1.00000
$482$	−1.41421			−1.41421
$483$		0			0
$484$		−	1.00000i		−	1.00000i
$485$		0			0
$486$	1.41421			1.41421
$487$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$487$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$491$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$492$		0			0
$493$			1.00000i			1.00000i
$494$		0			0
$495$		0			0
$496$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$497$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$498$	1.00000	+	1.00000i	1.00000	+	1.00000i
$499$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$499$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$500$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$501$		−	1.41421i		−	1.41421i
$502$		−	2.00000i		−	2.00000i
$503$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$503$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$504$		0			0
$505$	2.00000			2.00000
$506$		0			0
$507$		0			0
$508$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$			1.41421i			1.41421i
$511$		0			0
$512$			1.41421i			1.41421i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	1.00000			1.00000
$517$		0			0
$518$	−1.41421			−1.41421
$519$	1.41421			1.41421
$520$		0			0
$521$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$521$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$522$	1.00000	−	1.00000i	1.00000	−	1.00000i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$			1.41421i			1.41421i
$525$		0			0
$526$	−2.00000			−2.00000
$527$	1.41421			1.41421
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		−	1.41421i		−	1.41421i
$531$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$532$		0			0
$533$		0			0
$534$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$535$		0			0
$536$		0			0
$537$	−1.41421			−1.41421
$538$	1.00000	−	1.00000i	1.00000	−	1.00000i
$539$		0			0
$540$	0.707107	−	0.707107i	0.707107	−	0.707107i
$541$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$541$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$542$	1.41421	+	1.41421i	1.41421	+	1.41421i
$543$	1.00000			1.00000
$544$	1.00000	−	1.00000i	1.00000	−	1.00000i
$545$		0			0
$546$		−	1.41421i		−	1.41421i
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	1.41421			1.41421
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			1.00000i			1.00000i
$554$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$555$	0.707107	−	0.707107i	0.707107	−	0.707107i
$556$	1.00000	+	1.00000i	1.00000	+	1.00000i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$559$			1.00000i			1.00000i
$560$	0.707107	−	0.707107i	0.707107	−	0.707107i
$561$		0			0
$562$		0			0
$563$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$564$		−	1.41421i		−	1.41421i
$565$	1.00000	+	1.00000i	1.00000	+	1.00000i
$566$	2.82843			2.82843
$567$			1.00000i			1.00000i
$568$		0			0
$569$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$569$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$574$		0			0
$575$		0			0
$576$	−1.00000			−1.00000
$577$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$577$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$578$		0			0
$579$			1.00000i			1.00000i
$580$		−	1.00000i		−	1.00000i
$581$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.707107	+	0.707107i	0.707107	+	0.707107i
$586$	2.00000	+	2.00000i	2.00000	+	2.00000i
$587$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$587$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$588$		0			0
$589$		0			0
$590$			1.41421i			1.41421i
$591$	1.41421	+	1.41421i	1.41421	+	1.41421i
$592$	−1.00000			−1.00000
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−1.00000			−1.00000
$596$	1.41421			1.41421
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$601$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$602$			1.41421i			1.41421i
$603$	1.00000	+	1.00000i	1.00000	+	1.00000i
$604$		0			0
$605$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$606$	2.00000	−	2.00000i	2.00000	−	2.00000i
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$	0.707107	+	0.707107i	0.707107	+	0.707107i
$610$		0			0
$611$	1.41421			1.41421
$612$	0.707107	+	0.707107i	0.707107	+	0.707107i
$613$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$613$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$614$	1.41421	−	1.41421i	1.41421	−	1.41421i
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$619$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$620$	−1.41421			−1.41421
$621$		0			0
$622$		0			0
$623$	0.707107	−	0.707107i	0.707107	−	0.707107i
$624$		−	1.00000i		−	1.00000i
$625$	1.00000			1.00000
$626$	−1.41421			−1.41421
$627$		0			0
$628$		0			0
$629$	0.707107	+	0.707107i	0.707107	+	0.707107i
$630$	1.00000	+	1.00000i	1.00000	+	1.00000i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	1.00000	−	1.00000i	1.00000	−	1.00000i
$635$	1.41421			1.41421
$636$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$637$		0			0
$638$		0			0
$639$	0.707107	−	0.707107i	0.707107	−	0.707107i
$640$		0			0
$641$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$641$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$642$		0			0
$643$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$643$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$644$		0			0
$645$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$646$		0			0
$647$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$647$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.00000	−	1.00000i	1.00000	−	1.00000i
$652$		0			0
$653$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$653$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$654$		0			0
$655$	1.00000	−	1.00000i	1.00000	−	1.00000i
$656$		0			0
$657$		0			0
$658$	2.00000			2.00000
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$662$	−1.41421			−1.41421
$663$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$664$		0			0
$665$		0			0
$666$		−	1.41421i		−	1.41421i
$667$		0			0
$668$	−1.41421			−1.41421
$669$		0			0
$670$	2.00000			2.00000
$671$		0			0
$672$		−	1.41421i		−	1.41421i
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$	−1.41421			−1.41421
$675$		0			0
$676$		0			0
$677$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$677$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$678$	2.00000			2.00000
$679$		0			0
$680$		0			0
$681$	1.41421			1.41421
$682$		0			0
$683$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$683$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$686$	−1.41421			−1.41421
$687$	−1.00000			−1.00000
$688$			1.00000i			1.00000i
$689$	0.707107	−	0.707107i	0.707107	−	0.707107i
$690$		0			0
$691$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$691$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$692$		−	1.41421i		−	1.41421i
$693$		0			0
$694$	1.00000	−	1.00000i	1.00000	−	1.00000i
$695$		−	1.41421i		−	1.41421i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	1.41421			1.41421
$703$		0			0
$704$		0			0
$705$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$706$		0			0
$707$	1.41421	+	1.41421i	1.41421	+	1.41421i
$708$	0.707107	+	0.707107i	0.707107	+	0.707107i
$709$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$709$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$710$		−	1.41421i		−	1.41421i
$711$	−1.00000			−1.00000
$712$		0			0
$713$		0			0
$714$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$715$		0			0
$716$			1.41421i			1.41421i
$717$		0			0
$718$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$719$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$719$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$720$	0.707107	+	0.707107i	0.707107	+	0.707107i
$721$		0			0
$722$	−1.41421			−1.41421
$723$	1.00000			1.00000
$724$		−	1.00000i		−	1.00000i
$725$		0			0
$726$			1.41421i			1.41421i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$	0.707107	−	0.707107i	0.707107	−	0.707107i
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$741$		0			0
$742$	1.00000	−	1.00000i	1.00000	−	1.00000i
$743$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$743$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$744$		0			0
$745$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$746$			1.41421i			1.41421i
$747$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$748$		0			0
$749$		0			0
$750$	1.00000	−	1.00000i	1.00000	−	1.00000i
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$752$	1.41421			1.41421
$753$			1.41421i			1.41421i
$754$	1.00000	−	1.00000i	1.00000	−	1.00000i
$755$		0			0
$756$	1.00000			1.00000
$757$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$757$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$758$		−	1.41421i		−	1.41421i
$759$		0			0
$760$		0			0
$761$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$761$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$762$	1.41421	−	1.41421i	1.41421	−	1.41421i
$763$		0			0
$764$	0.707107	+	0.707107i	0.707107	+	0.707107i
$765$		−	1.00000i		−	1.00000i
$766$		0			0
$767$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$768$		−	1.00000i		−	1.00000i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	1.00000			1.00000
$773$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$773$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$774$	−1.41421			−1.41421
$775$		0			0
$776$		0			0
$777$	1.00000			1.00000
$778$		0			0
$779$		0			0
$780$	0.707107	−	0.707107i	0.707107	−	0.707107i
$781$		0			0
$782$		0			0
$783$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$784$		0			0
$785$		0			0
$786$		−	2.00000i		−	2.00000i
$787$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$787$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$788$	1.41421	−	1.41421i	1.41421	−	1.41421i
$789$	1.41421			1.41421
$790$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$791$			1.41421i			1.41421i
$792$		0			0
$793$		0			0
$794$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$795$			1.00000i			1.00000i
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$800$		0			0
$801$	0.707107	+	0.707107i	0.707107	+	0.707107i
$802$		0			0
$803$		0			0
$804$	1.00000	−	1.00000i	1.00000	−	1.00000i
$805$		0			0
$806$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$807$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$808$		0			0
$809$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$809$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$810$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$811$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$811$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$812$	0.707107	−	0.707107i	0.707107	−	0.707107i
$813$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$814$		0			0
$815$		0			0
$816$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$817$		0			0
$818$		0			0
$819$			1.00000i			1.00000i
$820$		0			0
$821$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$821$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$822$	−2.00000			−2.00000
$823$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$823$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$827$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$827$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$828$		0			0
$829$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$829$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$830$	−1.41421			−1.41421
$831$	1.00000	−	1.00000i	1.00000	−	1.00000i
$832$	−1.00000			−1.00000
$833$		0			0
$834$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$835$	1.00000	+	1.00000i	1.00000	+	1.00000i
$836$		0			0
$837$	1.00000	+	1.00000i	1.00000	+	1.00000i
$838$	−2.00000			−2.00000
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$		0			0
$842$		−	1.41421i		−	1.41421i
$843$		0			0
$844$		0			0
$845$		0			0
$846$			2.00000i			2.00000i
$847$	−1.00000			−1.00000
$848$	0.707107	−	0.707107i	0.707107	−	0.707107i
$849$	−2.00000			−2.00000
$850$		0			0
$851$		0			0
$852$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$857$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$861$		0			0
$862$		0			0
$863$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$863$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$864$	1.41421			1.41421
$865$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$866$	−1.41421			−1.41421
$867$		0			0
$868$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$869$		0			0
$870$			1.41421i			1.41421i
$871$	1.00000	+	1.00000i	1.00000	+	1.00000i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.707107	+	0.707107i	0.707107	+	0.707107i
$876$		0			0
$877$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$877$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$878$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$879$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$880$		0			0
$881$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$881$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$882$		0			0
$883$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$884$	0.707107	+	0.707107i	0.707107	+	0.707107i
$885$		−	1.00000i		−	1.00000i
$886$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$	1.00000	+	1.00000i	1.00000	+	1.00000i
$890$	1.41421			1.41421
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−2.00000			−2.00000
$895$	1.00000	−	1.00000i	1.00000	−	1.00000i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.41421			1.41421
$900$		0			0
$901$	−1.00000			−1.00000
$902$		0			0
$903$		−	1.00000i		−	1.00000i
$904$		0			0
$905$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$906$		0			0
$907$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$907$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$908$		−	1.41421i		−	1.41421i
$909$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$910$	1.00000	+	1.00000i	1.00000	+	1.00000i
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$	1.41421	−	1.41421i	1.41421	−	1.41421i
$915$		0			0
$916$			1.00000i			1.00000i
$917$	1.41421			1.41421
$918$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$919$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$919$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$920$		0			0
$921$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$922$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$923$	0.707107	−	0.707107i	0.707107	−	0.707107i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	1.00000	−	1.00000i	1.00000	−	1.00000i
$929$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$930$	2.00000			2.00000
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$937$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$938$	1.41421	+	1.41421i	1.41421	+	1.41421i
$939$	1.00000			1.00000
$940$	1.00000	+	1.00000i	1.00000	+	1.00000i
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$945$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$946$		0			0
$947$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$948$			1.00000i			1.00000i
$949$		0			0
$950$		0			0
$951$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$952$		0			0
$953$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$953$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$954$	1.00000	+	1.00000i	1.00000	+	1.00000i
$955$		−	1.00000i		−	1.00000i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1.41421i		−	1.41421i
$960$	0.707107	−	0.707107i	0.707107	−	0.707107i
$961$		−	1.00000i		−	1.00000i
$962$		−	1.41421i		−	1.41421i
$963$		0			0
$964$		−	1.00000i		−	1.00000i
$965$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$966$		0			0
$967$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$967$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$971$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$972$			1.00000i			1.00000i
$973$	1.00000	−	1.00000i	1.00000	−	1.00000i
$974$		−	1.41421i		−	1.41421i
$975$		0			0
$976$		0			0
$977$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$977$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.00000	−	1.00000i	1.00000	−	1.00000i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−2.00000			−2.00000
$986$	−1.41421			−1.41421
$987$	−1.41421			−1.41421
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$993$	1.00000			1.00000
$994$	1.00000	−	1.00000i	1.00000	−	1.00000i
$995$		0			0
$996$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$997$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$997$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$998$			1.41421i			1.41421i
$999$			1.00000i			1.00000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1011.1.f.b.863.2	yes	4
3.2	odd	2	inner	1011.1.f.b.863.1	yes	4
337.148	even	4	inner	1011.1.f.b.485.2	yes	4
1011.485	odd	4	inner	1011.1.f.b.485.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1011.1.f.b.485.1	✓	4	1011.485	odd	4	inner
1011.1.f.b.485.2	yes	4	337.148	even	4	inner
1011.1.f.b.863.1	yes	4	3.2	odd	2	inner
1011.1.f.b.863.2	yes	4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1011 = 3 \cdot 337 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1011.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.707107 + 0.707107i) q^{5} +1.41421 q^{6} +1.00000i q^{7} -1.00000 q^{9} +(-1.00000 + 1.00000i) q^{10} +1.00000i q^{12} -1.00000 q^{13} -1.41421 q^{14} +(0.707107 - 0.707107i) q^{15} -1.00000 q^{16} +(0.707107 + 0.707107i) q^{17} -1.41421i q^{18} +(-0.707107 - 0.707107i) q^{20} +1.00000 q^{21} -1.41421i q^{26} +1.00000i q^{27} -1.00000i q^{28} +(0.707107 + 0.707107i) q^{29} +(1.00000 + 1.00000i) q^{30} +(1.00000 - 1.00000i) q^{31} -1.41421i q^{32} +(-1.00000 + 1.00000i) q^{34} +(-0.707107 + 0.707107i) q^{35} +1.00000 q^{36} +1.00000 q^{37} +1.00000i q^{39} +1.41421i q^{42} -1.00000i q^{43} +(-0.707107 - 0.707107i) q^{45} -1.41421 q^{47} +1.00000i q^{48} +(0.707107 - 0.707107i) q^{51} +1.00000 q^{52} +(-0.707107 + 0.707107i) q^{53} -1.41421 q^{54} +(-1.00000 + 1.00000i) q^{58} +(0.707107 - 0.707107i) q^{59} +(-0.707107 + 0.707107i) q^{60} +(1.41421 + 1.41421i) q^{62} -1.00000i q^{63} +1.00000 q^{64} +(-0.707107 - 0.707107i) q^{65} +(-1.00000 - 1.00000i) q^{67} +(-0.707107 - 0.707107i) q^{68} +(-1.00000 - 1.00000i) q^{70} +(-0.707107 + 0.707107i) q^{71} +1.41421i q^{74} -1.41421 q^{78} +1.00000 q^{79} +(-0.707107 - 0.707107i) q^{80} +1.00000 q^{81} +(0.707107 + 0.707107i) q^{83} -1.00000 q^{84} +1.00000i q^{85} +1.41421 q^{86} +(0.707107 - 0.707107i) q^{87} +(-0.707107 - 0.707107i) q^{89} +(1.00000 - 1.00000i) q^{90} -1.00000i q^{91} +(-1.00000 - 1.00000i) q^{93} -2.00000i q^{94} -1.41421 q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9} - 4 q^{10} - 4 q^{13} - 4 q^{16} + 4 q^{21} + 4 q^{30} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 4 q^{37} + 4 q^{52} - 4 q^{58} + 4 q^{64} - 4 q^{67} - 4 q^{70} + 4 q^{79} + 4 q^{81} - 4 q^{84}+ \cdots - 4 q^{93}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^10 - 4 * q^13 - 4 * q^16 + 4 * q^21 + 4 * q^30 + 4 * q^31 - 4 * q^34 + 4 * q^36 + 4 * q^37 + 4 * q^52 - 4 * q^58 + 4 * q^64 - 4 * q^67 - 4 * q^70 + 4 * q^79 + 4 * q^81 - 4 * q^84 + 4 * q^90 - 4 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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