LMFDB - Embedded newform 2852.1.bn.b.827.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2852,1,Mod(551,2852)] 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("2852.551"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2852, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 15, 13])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2852 = 2^{2} \cdot 23 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2852.bn (of order $30$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$1.42333341603$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$3$ over $\Q(\zeta_{30})$
Coefficient field:	$\Q(\zeta_{45})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{90}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{90} - \cdots)$

Embedding invariants

Embedding label			827.1
Root			$-0.997564 + 0.0697565i$ of defining polynomial
Character	$\chi$	$=$	2852.827
Dual form			2852.1.bn.b.2483.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.766044 + 0.642788i) q^{2} +(-0.207022 - 1.96969i) q^{3} +(0.173648 - 0.984808i) q^{4} +(1.42468 + 1.37580i) q^{6} +(0.500000 + 0.866025i) q^{8} +(-2.85866 + 0.607627i) q^{9} +(-1.97571 - 0.138155i) q^{12} +(-0.431075 + 0.968211i) q^{13} +(-0.939693 - 0.342020i) q^{16} +(1.79929 - 2.30298i) q^{18} +(-0.309017 + 0.951057i) q^{23} +(1.60229 - 1.16413i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.292131 - 1.01878i) q^{26} +(1.17662 + 3.62127i) q^{27} +(-1.17485 + 1.61705i) q^{29} +(0.997564 - 0.0697565i) q^{31} +(0.939693 - 0.342020i) q^{32} +(0.101995 + 2.92074i) q^{36} +(1.99631 + 0.648642i) q^{39} +(-0.0783141 + 0.745109i) q^{41} +(-0.374607 - 0.927184i) q^{46} +(-0.0820037 - 0.112868i) q^{47} +(-0.479135 + 1.92171i) q^{48} +(-0.913545 - 0.406737i) q^{49} +(-0.173648 - 0.984808i) q^{50} +(0.878646 + 0.592654i) q^{52} +(-3.22905 - 2.01773i) q^{54} +(-0.139428 - 1.99391i) q^{58} +(-1.89169 + 0.198825i) q^{59} +(-0.719340 + 0.694658i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(1.93726 + 0.411777i) q^{69} +(-0.327673 - 1.54158i) q^{71} +(-1.95555 - 2.17186i) q^{72} +(-0.697771 - 0.628276i) q^{73} +(1.80931 + 0.805557i) q^{75} +(-1.94620 + 0.786317i) q^{78} +(4.21932 - 1.87856i) q^{81} +(-0.418955 - 0.621126i) q^{82} +(3.42830 + 1.97933i) q^{87} +(0.882948 + 0.469472i) q^{92} +(-0.343916 - 1.95045i) q^{93} +(0.135369 + 0.0337512i) q^{94} +(-0.868210 - 1.78009i) q^{96} +(0.961262 - 0.275637i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 6 q^{6} + 12 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{18} + 6 q^{23} - 12 q^{25} + 3 q^{26} + 6 q^{27} - 6 q^{36} + 3 q^{48} - 3 q^{49} + 3 q^{52} - 3 q^{54} + 3 q^{58} - 12 q^{64} - 3 q^{72} + 6 q^{78}+ \cdots + 6 q^{96}+O(q^{100}) $ 24 * q + 6 * q^6 + 12 * q^8 + 3 * q^9 + 3 * q^12 + 6 * q^18 + 6 * q^23 - 12 * q^25 + 3 * q^26 + 6 * q^27 - 6 * q^36 + 3 * q^48 - 3 * q^49 + 3 * q^52 - 3 * q^54 + 3 * q^58 - 12 * q^64 - 3 * q^72 + 6 * q^78 + 3 * q^81 + 3 * q^82 + 9 * q^87 + 6 * q^93 - 6 * q^94 + 6 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2852\mathbb{Z}\right)^\times$.

$n$	$373$	$1427$	$2669$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{29}{30}\right)$

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 $

$2$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$2$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$3$	−0.207022	−	1.96969i	−0.207022	−	1.96969i	−0.241922	−	0.970296i	$-0.577778\pi$
$3$	−0.207022	−	1.96969i	−0.207022	−	1.96969i	0.0348995	−	0.999391i	$-0.488889\pi$
$4$	0.173648	−	0.984808i	0.173648	−	0.984808i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	1.42468	+	1.37580i	1.42468	+	1.37580i
$7$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$7$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$8$	0.500000	+	0.866025i	0.500000	+	0.866025i
$9$	−2.85866	+	0.607627i	−2.85866	+	0.607627i
$10$		0			0
$11$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$11$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$12$	−1.97571	−	0.138155i	−1.97571	−	0.138155i
$13$	−0.431075	+	0.968211i	−0.431075	+	0.968211i	0.559193	+	0.829038i	$0.311111\pi$
$13$	−0.431075	+	0.968211i	−0.431075	+	0.968211i	−0.990268	+	0.139173i	$0.955556\pi$
$14$		0			0
$15$		0			0
$16$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$17$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$17$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$18$	1.79929	−	2.30298i	1.79929	−	2.30298i
$19$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$19$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$23$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$24$	1.60229	−	1.16413i	1.60229	−	1.16413i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	−0.292131	−	1.01878i	−0.292131	−	1.01878i
$27$	1.17662	+	3.62127i	1.17662	+	3.62127i
$28$		0			0
$29$	−1.17485	+	1.61705i	−1.17485	+	1.61705i	−0.559193	+	0.829038i	$0.688889\pi$
$29$	−1.17485	+	1.61705i	−1.17485	+	1.61705i	−0.615661	+	0.788011i	$0.711111\pi$
$30$		0			0
$31$	0.997564	−	0.0697565i	0.997564	−	0.0697565i
$31$	0.997564	−	0.0697565i	0.997564	−	0.0697565i
$32$	0.939693	−	0.342020i	0.939693	−	0.342020i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.101995	+	2.92074i	0.101995	+	2.92074i
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$	1.99631	+	0.648642i	1.99631	+	0.648642i
$40$		0			0
$41$	−0.0783141	+	0.745109i	−0.0783141	+	0.745109i	0.882948	+	0.469472i	$0.155556\pi$
$41$	−0.0783141	+	0.745109i	−0.0783141	+	0.745109i	−0.961262	+	0.275637i	$0.911111\pi$
$42$		0			0
$43$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$43$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$44$		0			0
$45$		0			0
$46$	−0.374607	−	0.927184i	−0.374607	−	0.927184i
$47$	−0.0820037	−	0.112868i	−0.0820037	−	0.112868i	0.766044	−	0.642788i	$-0.222222\pi$
$47$	−0.0820037	−	0.112868i	−0.0820037	−	0.112868i	−0.848048	+	0.529919i	$0.822222\pi$
$48$	−0.479135	+	1.92171i	−0.479135	+	1.92171i
$49$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$50$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$51$		0			0
$52$	0.878646	+	0.592654i	0.878646	+	0.592654i
$53$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$53$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$54$	−3.22905	−	2.01773i	−3.22905	−	2.01773i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.139428	−	1.99391i	−0.139428	−	1.99391i
$59$	−1.89169	+	0.198825i	−1.89169	+	0.198825i	−0.978148	−	0.207912i	$-0.933333\pi$
$59$	−1.89169	+	0.198825i	−1.89169	+	0.198825i	−0.913545	+	0.406737i	$0.866667\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	−0.719340	+	0.694658i	−0.719340	+	0.694658i
$63$		0			0
$64$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$	1.93726	+	0.411777i	1.93726	+	0.411777i
$70$		0			0
$71$	−0.327673	−	1.54158i	−0.327673	−	1.54158i	−0.766044	−	0.642788i	$-0.777778\pi$
$71$	−0.327673	−	1.54158i	−0.327673	−	1.54158i	0.438371	−	0.898794i	$-0.355556\pi$
$72$	−1.95555	−	2.17186i	−1.95555	−	2.17186i
$73$	−0.697771	−	0.628276i	−0.697771	−	0.628276i	0.241922	−	0.970296i	$-0.422222\pi$
$73$	−0.697771	−	0.628276i	−0.697771	−	0.628276i	−0.939693	+	0.342020i	$0.888889\pi$
$74$		0			0
$75$	1.80931	+	0.805557i	1.80931	+	0.805557i
$76$		0			0
$77$		0			0
$78$	−1.94620	+	0.786317i	−1.94620	+	0.786317i
$79$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$79$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$80$		0			0
$81$	4.21932	−	1.87856i	4.21932	−	1.87856i
$82$	−0.418955	−	0.621126i	−0.418955	−	0.621126i
$83$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$83$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.42830	+	1.97933i	3.42830	+	1.97933i
$88$		0			0
$89$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$89$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$90$		0			0
$91$		0			0
$92$	0.882948	+	0.469472i	0.882948	+	0.469472i
$93$	−0.343916	−	1.95045i	−0.343916	−	1.95045i
$94$	0.135369	+	0.0337512i	0.135369	+	0.0337512i
$95$		0			0
$96$	−0.868210	−	1.78009i	−0.868210	−	1.78009i
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$	0.961262	−	0.275637i	0.961262	−	0.275637i
$99$		0			0
$100$	0.766044	+	0.642788i	0.766044	+	0.642788i
$101$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$101$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$103$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$104$	−1.05403	+	0.110783i	−1.05403	+	0.110783i
$105$		0			0
$106$		0			0
$107$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$107$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$108$	3.77057	−	0.529919i	3.77057	−	0.529919i
$109$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$109$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$113$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$114$		0			0
$115$		0			0
$116$	1.38847	+	1.43780i	1.38847	+	1.43780i
$117$	0.643986	−	3.02972i	0.643986	−	3.02972i
$118$	1.32132	−	1.36827i	1.32132	−	1.36827i
$119$		0			0
$120$		0			0
$121$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$122$		0			0
$123$	1.48384			1.48384
$124$	0.104528	−	0.994522i	0.104528	−	0.994522i
$125$		0			0
$126$		0			0
$127$	−0.128708	−	1.22458i	−0.128708	−	1.22458i	−0.848048	−	0.529919i	$-0.822222\pi$
$127$	−0.128708	−	1.22458i	−0.128708	−	1.22458i	0.719340	−	0.694658i	$-0.244444\pi$
$128$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$129$		0			0
$130$		0			0
$131$	−0.373740	+	1.75831i	−0.373740	+	1.75831i	0.241922	+	0.970296i	$0.422222\pi$
$131$	−0.373740	+	1.75831i	−0.373740	+	1.75831i	−0.615661	+	0.788011i	$0.711111\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$137$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$138$	−1.74871	+	0.929805i	−1.74871	+	0.929805i
$139$	0.280969	−	0.204136i	0.280969	−	0.204136i	−0.438371	−	0.898794i	$-0.644444\pi$
$139$	0.280969	−	0.204136i	0.280969	−	0.204136i	0.719340	+	0.694658i	$0.244444\pi$
$140$		0			0
$141$	−0.205339	+	0.184888i	−0.205339	+	0.184888i
$142$	1.24192	+	0.970296i	1.24192	+	0.970296i
$143$		0			0
$144$	2.89408	+	0.406737i	2.89408	+	0.406737i
$145$		0			0
$146$	0.938371	+	0.0327686i	0.938371	+	0.0327686i
$147$	−0.612019	+	1.88360i	−0.612019	+	1.88360i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−1.90381	+	0.545910i	−1.90381	+	0.545910i
$151$	0.270928	+	0.833831i	0.270928	+	0.833831i	0.990268	+	0.139173i	$0.0444444\pi$
$151$	0.270928	+	0.833831i	0.270928	+	0.833831i	−0.719340	+	0.694658i	$0.755556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.985444	−	1.85335i	0.985444	−	1.85335i
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−2.02467	+	4.15119i	−2.02467	+	4.15119i
$163$	−1.32132	−	0.429322i	−1.32132	−	0.429322i	−0.438371	−	0.898794i	$-0.644444\pi$
$163$	−1.32132	−	0.429322i	−1.32132	−	0.429322i	−0.882948	+	0.469472i	$0.844444\pi$
$164$	0.720190	+	0.206511i	0.720190	+	0.206511i
$165$		0			0
$166$		0			0
$167$	1.78716	−	0.795697i	1.78716	−	0.795697i	0.809017	−	0.587785i	$-0.200000\pi$
$167$	1.78716	−	0.795697i	1.78716	−	0.795697i	0.978148	−	0.207912i	$-0.0666667\pi$
$168$		0			0
$169$	−0.0824755	−	0.0915983i	−0.0824755	−	0.0915983i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.78716	+	0.795697i	1.78716	+	0.795697i	0.978148	+	0.207912i	$0.0666667\pi$
$173$	1.78716	+	0.795697i	1.78716	+	0.795697i	0.809017	+	0.587785i	$0.200000\pi$
$174$	−3.89852	+	0.687414i	−3.89852	+	0.687414i
$175$		0			0
$176$		0			0
$177$	0.783246	+	3.68488i	0.783246	+	3.68488i
$178$		0			0
$179$	−1.95153	−	0.414810i	−1.95153	−	0.414810i	−0.990268	−	0.139173i	$-0.955556\pi$
$179$	−1.95153	−	0.414810i	−1.95153	−	0.414810i	−0.961262	−	0.275637i	$-0.911111\pi$
$180$		0			0
$181$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$181$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$182$		0			0
$183$		0			0
$184$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$185$		0			0
$186$	1.51718	+	1.27306i	1.51718	+	1.27306i
$187$		0			0
$188$	−0.125393	+	0.0611585i	−0.125393	+	0.0611585i
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	1.80931	+	0.805557i	1.80931	+	0.805557i
$193$	1.83832	+	0.390746i	1.83832	+	0.390746i	0.990268	−	0.139173i	$-0.0444444\pi$
$193$	1.83832	+	0.390746i	1.83832	+	0.390746i	0.848048	+	0.529919i	$0.177778\pi$
$194$		0			0
$195$		0			0
$196$	−0.559193	+	0.829038i	−0.559193	+	0.829038i
$197$	1.17121	+	1.05456i	1.17121	+	1.05456i	0.997564	+	0.0697565i	$0.0222222\pi$
$197$	1.17121	+	1.05456i	1.17121	+	1.05456i	0.173648	+	0.984808i	$0.444444\pi$
$198$		0			0
$199$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$199$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$200$	−1.00000			−1.00000
$201$		0			0
$202$	−0.606126	−	1.50021i	−0.606126	−	1.50021i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.305487	−	2.90651i	0.305487	−	2.90651i
$208$	0.736226	−	0.762384i	0.736226	−	0.762384i
$209$		0			0
$210$		0			0
$211$	−0.360114	−	0.207912i	−0.360114	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$211$	−0.360114	−	0.207912i	−0.360114	−	0.207912i	−0.669131	+	0.743145i	$0.733333\pi$
$212$		0			0
$213$	−2.96860	+	0.964556i	−2.96860	+	0.964556i
$214$		0			0
$215$		0			0
$216$	−2.54780	+	2.82962i	−2.54780	+	2.82962i
$217$		0			0
$218$		0			0
$219$	−1.09305	+	1.50446i	−1.09305	+	1.50446i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$223$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$224$		0			0
$225$	0.903109	−	2.77948i	0.903109	−	2.77948i
$226$		0			0
$227$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$227$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$228$		0			0
$229$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$229$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$230$		0			0
$231$		0			0
$232$	−1.98783	−	0.208930i	−1.98783	−	0.208930i
$233$	−1.16392	+	0.845635i	−1.16392	+	0.845635i	−0.990268	−	0.139173i	$-0.955556\pi$
$233$	−1.16392	+	0.845635i	−1.16392	+	0.845635i	−0.173648	+	0.984808i	$0.555556\pi$
$234$	1.45414	+	2.73484i	1.45414	+	2.73484i
$235$		0			0
$236$	−0.132685	+	1.89748i	−0.132685	+	1.89748i
$237$		0			0
$238$		0			0
$239$	−1.09395	+	0.232525i	−1.09395	+	0.232525i	−0.719340	−	0.694658i	$-0.755556\pi$
$239$	−1.09395	+	0.232525i	−1.09395	+	0.232525i	−0.374607	+	0.927184i	$0.622222\pi$
$240$		0			0
$241$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$241$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$242$	0.719340	+	0.694658i	0.719340	+	0.694658i
$243$	−2.66986	−	4.62433i	−2.66986	−	4.62433i
$244$		0			0
$245$		0			0
$246$	−1.13669	+	0.953796i	−1.13669	+	0.953796i
$247$		0			0
$248$	0.559193	+	0.829038i	0.559193	+	0.829038i
$249$		0			0
$250$		0			0
$251$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$251$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$252$		0			0
$253$		0			0
$254$	0.885740	+	0.855349i	0.885740	+	0.855349i
$255$		0			0
$256$	0.766044	+	0.642788i	0.766044	+	0.642788i
$257$	−0.339707	+	0.0722070i	−0.339707	+	0.0722070i	−0.374607	−	0.927184i	$-0.622222\pi$
$257$	−0.339707	+	0.0722070i	−0.339707	+	0.0722070i	0.0348995	+	0.999391i	$0.488889\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.37595	−	5.33646i	2.37595	−	5.33646i
$262$	−0.843916	−	1.58718i	−0.843916	−	1.58718i
$263$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$263$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.27853	−	0.134379i	−1.27853	−	0.134379i	−0.559193	−	0.829038i	$-0.688889\pi$
$269$	−1.27853	−	0.134379i	−1.27853	−	0.134379i	−0.719340	+	0.694658i	$0.755556\pi$
$270$		0			0
$271$	0.413545	−	1.27276i	0.413545	−	1.27276i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	0.413545	−	1.27276i	0.413545	−	1.27276i	0.913545	−	0.406737i	$-0.133333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	0.741922	−	1.83632i	0.741922	−	1.83632i
$277$	0.974592	−	1.34141i	0.974592	−	1.34141i	0.0348995	−	0.999391i	$-0.488889\pi$
$277$	0.974592	−	1.34141i	0.974592	−	1.34141i	0.939693	−	0.342020i	$-0.111111\pi$
$278$	−0.0840186	+	0.336980i	−0.0840186	+	0.336980i
$279$	−2.80931	+	0.805557i	−2.80931	+	0.805557i
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$	0.0384550	−	0.273621i	0.0384550	−	0.273621i
$283$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$283$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$284$	−1.57506	+	0.0550024i	−1.57506	+	0.0550024i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.47844	+	1.54870i	−2.47844	+	1.54870i
$289$	0.104528	−	0.994522i	0.104528	−	0.994522i
$290$		0			0
$291$		0			0
$292$	−0.739897	+	0.578071i	−0.739897	+	0.578071i
$293$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$293$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$294$	−0.741922	−	1.83632i	−0.741922	−	1.83632i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.787614	−	0.709170i	−0.787614	−	0.709170i
$300$	1.10750	−	1.64194i	1.10750	−	1.64194i
$301$		0			0
$302$	−0.743520	−	0.464603i	−0.743520	−	0.464603i
$303$	3.13455	+	0.666269i	3.13455	+	0.666269i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.72256	+	0.181049i	−1.72256	+	0.181049i	−0.913545	−	0.406737i	$-0.866667\pi$
$307$	−1.72256	+	0.181049i	−1.72256	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.99878i			1.99878i	0.0348995	+	0.999391i	$0.488889\pi$
$311$			1.99878i			1.99878i	−0.0348995	+	0.999391i	$0.511111\pi$
$312$	0.436417	+	2.05318i	0.436417	+	2.05318i
$313$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$313$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.204489	+	0.0434654i	0.204489	+	0.0434654i	0.309017	−	0.951057i	$-0.400000\pi$
$317$	0.204489	+	0.0434654i	0.204489	+	0.0434654i	−0.104528	+	0.994522i	$0.533333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.11735	−	4.48143i	−1.11735	−	4.48143i
$325$	−0.622957	−	0.857427i	−0.622957	−	0.857427i
$326$	1.28815	−	0.520447i	1.28815	−	0.520447i
$327$		0			0
$328$	−0.684440	+	0.304732i	−0.684440	+	0.304732i
$329$		0			0
$330$		0			0
$331$	−0.150383	+	1.43080i	−0.150383	+	1.43080i	0.615661	+	0.788011i	$0.288889\pi$
$331$	−0.150383	+	1.43080i	−0.150383	+	1.43080i	−0.766044	+	0.642788i	$0.777778\pi$
$332$		0			0
$333$		0			0
$334$	−0.857583	+	1.75831i	−0.857583	+	1.75831i
$335$		0			0
$336$		0			0
$337$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$337$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$338$	0.122058	+	0.0171542i	0.122058	+	0.0171542i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.88051	+	0.539228i	−1.88051	+	0.539228i
$347$	0.669131	−	1.15897i	0.669131	−	1.15897i	−0.309017	−	0.951057i	$-0.600000\pi$
$347$	0.669131	−	1.15897i	0.669131	−	1.15897i	0.978148	−	0.207912i	$-0.0666667\pi$
$348$	2.54458	−	3.03251i	2.54458	−	3.03251i
$349$	0.0215691	−	0.0663828i	0.0215691	−	0.0663828i	−0.939693	−	0.342020i	$-0.888889\pi$
$349$	0.0215691	−	0.0663828i	0.0215691	−	0.0663828i	0.961262	+	0.275637i	$0.0888889\pi$
$350$		0			0
$351$	−4.01336	−	0.421821i	−4.01336	−	0.421821i
$352$		0			0
$353$	0.789310	+	1.77282i	0.789310	+	1.77282i	0.615661	+	0.788011i	$0.288889\pi$
$353$	0.789310	+	1.77282i	0.789310	+	1.77282i	0.173648	+	0.984808i	$0.444444\pi$
$354$	−2.96860	−	2.31932i	−2.96860	−	2.31932i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.76159	−	0.936656i	1.76159	−	0.936656i
$359$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$359$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$360$		0			0
$361$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$362$		0			0
$363$	−1.93726	+	0.411777i	−1.93726	+	0.411777i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	0.615661	−	0.788011i	0.615661	−	0.788011i
$369$	−0.228875	−	2.17760i	−0.228875	−	2.17760i
$370$		0			0
$371$		0			0
$372$	−1.98054			−1.98054
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	0.0567450	−	0.127451i	0.0567450	−	0.127451i
$377$	−1.05919	−	1.83458i	−1.05919	−	1.83458i
$378$		0			0
$379$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$379$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$380$		0			0
$381$	−2.38539	+	0.507030i	−2.38539	+	0.507030i
$382$		0			0
$383$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$383$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$384$	−1.90381	+	0.545910i	−1.90381	+	0.545910i
$385$		0			0
$386$	−1.65940	+	0.882318i	−1.65940	+	0.882318i
$387$		0			0
$388$		0			0
$389$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$389$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$390$		0			0
$391$		0			0
$392$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$393$	3.54069	+	0.372141i	3.54069	+	0.372141i
$394$	−1.57506	−	0.0550024i	−1.57506	−	0.0550024i
$395$		0			0
$396$		0			0
$397$	−0.615661	+	1.06636i	−0.615661	+	1.06636i	0.374607	+	0.927184i	$0.377778\pi$
$397$	−0.615661	+	1.06636i	−0.615661	+	1.06636i	−0.990268	+	0.139173i	$0.955556\pi$
$398$		0			0
$399$		0			0
$400$	0.766044	−	0.642788i	0.766044	−	0.642788i
$401$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$401$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$402$		0			0
$403$	−0.362486	+	0.995922i	−0.362486	+	0.995922i
$404$	1.42864	+	0.759621i	1.42864	+	0.759621i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$409$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.63425	+	2.42288i	1.63425	+	2.42288i
$415$		0			0
$416$	−0.0739306	+	1.05726i	−0.0739306	+	1.05726i
$417$	−0.460250	−	0.511160i	−0.460250	−	0.511160i
$418$		0			0
$419$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$419$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$421$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$422$	0.409506	−	0.0722070i	0.409506	−	0.0722070i
$423$	0.303002	+	0.272825i	0.303002	+	0.272825i
$424$		0			0
$425$		0			0
$426$	1.65407	−	2.64707i	1.65407	−	2.64707i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$431$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$432$	0.132884	−	3.80531i	0.132884	−	3.80531i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.129720	−	1.85508i	−0.129720	−	1.85508i
$439$	−0.917847	+	0.529919i	−0.917847	+	0.529919i	−0.882948	−	0.469472i	$-0.844444\pi$
$439$	−0.917847	+	0.529919i	−0.917847	+	0.529919i	−0.0348995	+	0.999391i	$0.511111\pi$
$440$		0			0
$441$	2.85866	+	0.607627i	2.85866	+	0.607627i
$442$		0			0
$443$	−0.0578714	−	0.272264i	−0.0578714	−	0.272264i	0.939693	−	0.342020i	$-0.111111\pi$
$443$	−0.0578714	−	0.272264i	−0.0578714	−	0.272264i	−0.997564	+	0.0697565i	$0.977778\pi$
$444$		0			0
$445$		0			0
$446$	0.107320	+	0.608645i	0.107320	+	0.608645i
$447$		0			0
$448$		0			0
$449$	1.01807	+	1.40126i	1.01807	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$449$	1.01807	+	1.40126i	1.01807	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$450$	1.09480	+	2.70972i	1.09480	+	2.70972i
$451$		0			0
$452$		0			0
$453$	1.58630	−	0.706266i	1.58630	−	0.706266i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$457$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.650561	+	0.211380i	−0.650561	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$461$	−0.650561	+	0.211380i	−0.650561	+	0.211380i	−0.0348995	+	0.999391i	$0.511111\pi$
$462$		0			0
$463$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$463$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$464$	1.65707	−	1.11770i	1.65707	−	1.11770i
$465$		0			0
$466$	0.348048	−	1.39594i	0.348048	−	1.39594i
$467$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$467$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$468$	−2.87186	−	1.16031i	−2.87186	−	1.16031i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.688547	−	0.881300i	0.688547	−	0.881300i
$479$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$479$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.997564	−	0.0697565i	−0.997564	−	0.0697565i
$485$		0			0
$486$	5.01769	+	1.82629i	5.01769	+	1.82629i
$487$	1.49861	−	0.318539i	1.49861	−	0.318539i	0.615661	−	0.788011i	$-0.288889\pi$
$487$	1.49861	−	0.318539i	1.49861	−	0.318539i	0.882948	+	0.469472i	$0.155556\pi$
$488$		0			0
$489$	−0.572088	+	2.69146i	−0.572088	+	2.69146i
$490$		0			0
$491$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$491$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$492$	0.257667	−	1.46130i	0.257667	−	1.46130i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.961262	−	0.275637i	−0.961262	−	0.275637i
$497$		0			0
$498$		0			0
$499$	0.196449	+	1.86909i	0.196449	+	1.86909i	0.438371	+	0.898794i	$0.355556\pi$
$499$	0.196449	+	1.86909i	0.196449	+	1.86909i	−0.241922	+	0.970296i	$0.577778\pi$
$500$		0			0
$501$	−1.93726	−	3.35543i	−1.93726	−	3.35543i
$502$		0			0
$503$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$503$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.163346	+	0.181414i	−0.163346	+	0.181414i
$508$	−1.22832	−	0.0858927i	−1.22832	−	0.0858927i
$509$	0.731145	−	1.64218i	0.731145	−	1.64218i	−0.0348995	−	0.999391i	$-0.511111\pi$
$509$	0.731145	−	1.64218i	0.731145	−	1.64218i	0.766044	−	0.642788i	$-0.222222\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	0.213817	−	0.273673i	0.213817	−	0.273673i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.19729	−	3.68488i	1.19729	−	3.68488i
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	1.61013	+	5.61520i	1.61013	+	5.61520i
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$	1.66669	+	0.673388i	1.66669	+	0.673388i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$530$		0			0
$531$	5.28689	−	1.71782i	5.28689	−	1.71782i
$532$		0			0
$533$	−0.687663	−	0.397023i	−0.687663	−	0.397023i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.413036	+	3.92978i	−0.413036	+	3.92978i
$538$	1.06579	−	0.718885i	1.06579	−	0.718885i
$539$		0			0
$540$		0			0
$541$	−0.232387	−	0.258091i	−0.232387	−	0.258091i	0.615661	−	0.788011i	$-0.288889\pi$
$541$	−0.232387	−	0.258091i	−0.232387	−	0.258091i	−0.848048	+	0.529919i	$0.822222\pi$
$542$	0.501321	+	1.24081i	0.501321	+	1.24081i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.37806	+	1.24081i	1.37806	+	1.24081i	0.939693	+	0.342020i	$0.111111\pi$
$547$	1.37806	+	1.24081i	1.37806	+	1.24081i	0.438371	+	0.898794i	$0.355556\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	0.612019	+	1.88360i	0.612019	+	1.88360i
$553$		0			0
$554$	0.115661	+	1.65404i	0.115661	+	1.65404i
$555$		0			0
$556$	−0.152245	−	0.312148i	−0.152245	−	0.312148i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	1.63425	−	2.42288i	1.63425	−	2.42288i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$	0.146422	+	0.234325i	0.146422	+	0.234325i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.17121	−	1.05456i	1.17121	−	1.05456i
$569$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$569$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$570$		0			0
$571$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$571$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$576$	0.903109	−	2.77948i	0.903109	−	2.77948i
$577$	−1.39963	+	0.623157i	−1.39963	+	0.623157i	−0.961262	−	0.275637i	$-0.911111\pi$
$577$	−1.39963	+	0.623157i	−1.39963	+	0.623157i	−0.438371	+	0.898794i	$0.644444\pi$
$578$	0.559193	+	0.829038i	0.559193	+	0.829038i
$579$	0.389075	−	3.70180i	0.389075	−	3.70180i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.195217	−	0.918425i	0.195217	−	0.918425i
$585$		0			0
$586$		0			0
$587$	0.709299	+	0.515336i	0.709299	+	0.515336i	0.882948	−	0.469472i	$-0.155556\pi$
$587$	0.709299	+	0.515336i	0.709299	+	0.515336i	−0.173648	+	0.984808i	$0.555556\pi$
$588$	1.74871	+	0.929805i	1.74871	+	0.929805i
$589$		0			0
$590$		0			0
$591$	1.83469	−	2.52524i	1.83469	−	2.52524i
$592$		0			0
$593$	0.564602	+	1.73767i	0.564602	+	1.73767i	0.669131	+	0.743145i	$0.266667\pi$
$593$	0.564602	+	1.73767i	0.564602	+	1.73767i	−0.104528	+	0.994522i	$0.533333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.05919	+	0.0369878i	1.05919	+	0.0369878i
$599$	−1.72256	−	0.181049i	−1.72256	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$599$	−1.72256	−	0.181049i	−1.72256	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$600$	0.207022	+	1.96969i	0.207022	+	1.96969i
$601$	0.565086	+	1.26920i	0.565086	+	1.26920i	0.939693	+	0.342020i	$0.111111\pi$
$601$	0.565086	+	1.26920i	0.565086	+	1.26920i	−0.374607	+	0.927184i	$0.622222\pi$
$602$		0			0
$603$		0			0
$604$	0.868210	−	0.122019i	0.868210	−	0.122019i
$605$		0			0
$606$	−2.82947	+	1.50446i	−2.82947	+	1.50446i
$607$	−0.478148	+	1.07394i	−0.478148	+	1.07394i	0.500000	+	0.866025i	$0.333333\pi$
$607$	−0.478148	+	1.07394i	−0.478148	+	1.07394i	−0.978148	+	0.207912i	$0.933333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.144630	−	0.0307421i	0.144630	−	0.0307421i
$612$		0			0
$613$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$613$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$614$	1.20318	−	1.24593i	1.20318	−	1.24593i
$615$		0			0
$616$		0			0
$617$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$617$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−3.80763			−3.80763
$622$	−1.28479	−	1.53116i	−1.28479	−	1.53116i
$623$		0			0
$624$	−1.65407	−	1.29230i	−1.65407	−	1.29230i
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$631$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$632$		0			0
$633$	−0.334969	+	0.752353i	−0.334969	+	0.752353i
$634$	−0.184586	+	0.0981463i	−0.184586	+	0.0981463i
$635$		0			0
$636$		0			0
$637$	0.787614	−	0.709170i	0.787614	−	0.709170i
$638$		0			0
$639$	1.87341	+	4.20775i	1.87341	+	4.20775i
$640$		0			0
$641$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$641$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.107320	−	0.330298i	−0.107320	−	0.330298i	0.882948	−	0.469472i	$-0.155556\pi$
$647$	−0.107320	−	0.330298i	−0.107320	−	0.330298i	−0.990268	+	0.139173i	$0.955556\pi$
$648$	3.73654	+	2.71476i	3.73654	+	2.71476i
$649$		0			0
$650$	1.02836	+	0.256398i	1.02836	+	0.256398i
$651$		0			0
$652$	−0.652245	+	1.22669i	−0.652245	+	1.22669i
$653$	−1.23949	−	0.900539i	−1.23949	−	0.900539i	−0.241922	−	0.970296i	$-0.577778\pi$
$653$	−1.23949	−	0.900539i	−1.23949	−	0.900539i	−0.997564	+	0.0697565i	$0.977778\pi$
$654$		0			0
$655$		0			0
$656$	0.328433	−	0.673388i	0.328433	−	0.673388i
$657$	2.37645	+	1.37204i	2.37645	+	1.37204i
$658$		0			0
$659$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$659$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$660$		0			0
$661$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$661$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$662$	−0.804499	−	1.19272i	−0.804499	−	1.19272i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.17485	−	1.61705i	−1.17485	−	1.61705i
$668$	−0.473271	−	1.89818i	−0.473271	−	1.89818i
$669$	−1.11821	−	0.497861i	−1.11821	−	0.497861i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.142220	−	0.669092i	−0.142220	−	0.669092i	−0.990268	−	0.139173i	$-0.955556\pi$
$673$	−0.142220	−	0.669092i	−0.142220	−	0.669092i	0.848048	−	0.529919i	$-0.177778\pi$
$674$		0			0
$675$	−3.72442	−	0.791650i	−3.72442	−	0.791650i
$676$	−0.104528	+	0.0653166i	−0.104528	+	0.0653166i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.65808i		−	1.65808i	−0.559193	−	0.829038i	$-0.688889\pi$
$683$		−	1.65808i		−	1.65808i	0.559193	−	0.829038i	$-0.311111\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.309017	+	1.45381i	0.309017	+	1.45381i	0.809017	+	0.587785i	$0.200000\pi$
$691$	0.309017	+	1.45381i	0.309017	+	1.45381i	−0.500000	+	0.866025i	$0.666667\pi$
$692$	1.09395	−	1.62184i	1.09395	−	1.62184i
$693$		0			0
$694$	0.232387	+	1.31793i	0.232387	+	1.31793i
$695$		0			0
$696$			3.95866i			3.95866i
$697$		0			0
$698$	0.0261472	+	0.0647165i	0.0261472	+	0.0647165i
$699$	1.90659	+	2.11748i	1.90659	+	2.11748i
$700$		0			0
$701$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$701$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$702$	3.34556	−	2.25661i	3.34556	−	2.25661i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.74419	−	0.850699i	−1.74419	−	0.850699i
$707$		0			0
$708$	3.76491	−	0.131474i	3.76491	−	0.131474i
$709$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$709$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.241922	+	0.970296i	−0.241922	+	0.970296i
$714$		0			0
$715$		0			0
$716$	−0.747388	+	1.84985i	−0.747388	+	1.84985i
$717$	0.684474	+	2.10659i	0.684474	+	2.10659i
$718$		0			0
$719$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$720$		0			0
$721$		0			0
$722$	0.0348995	−	0.999391i	0.0348995	−	0.999391i
$723$		0			0
$724$		0			0
$725$	−0.812978	−	1.82598i	−0.812978	−	1.82598i
$726$	1.21934	−	1.56068i	1.21934	−	1.56068i
$727$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$727$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$728$		0			0
$729$	−4.81922	+	3.50137i	−4.81922	+	3.50137i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$733$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$734$		0			0
$735$		0			0
$736$	0.0348995	+	0.999391i	0.0348995	+	0.999391i
$737$		0			0
$738$	1.57506	+	1.52102i	1.57506	+	1.52102i
$739$	0.559193	+	0.968551i	0.559193	+	0.968551i	0.997564	+	0.0697565i	$0.0222222\pi$
$739$	0.559193	+	0.968551i	0.559193	+	0.968551i	−0.438371	+	0.898794i	$0.644444\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$	1.51718	−	1.27306i	1.51718	−	1.27306i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$751$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$752$	0.0384550	+	0.134108i	0.0384550	+	0.134108i
$753$		0			0
$754$	1.99063	+	0.724531i	1.99063	+	0.724531i
$755$		0			0
$756$		0			0
$757$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$757$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.409677	+	0.368875i	−0.409677	+	0.368875i	−0.848048	−	0.529919i	$-0.822222\pi$
$761$	−0.409677	+	0.368875i	−0.409677	+	0.368875i	0.438371	+	0.898794i	$0.355556\pi$
$762$	1.50140	−	1.92171i	1.50140	−	1.92171i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.622957	−	1.91727i	0.622957	−	1.91727i
$768$	1.10750	−	1.64194i	1.10750	−	1.64194i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	0.212552	+	0.654168i	0.212552	+	0.654168i
$772$	0.704030	−	1.74254i	0.704030	−	1.74254i
$773$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$773$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$774$		0			0
$775$	−0.438371	+	0.898794i	−0.438371	+	0.898794i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−7.23813	−	2.35181i	−7.23813	−	2.35181i
$784$	0.719340	+	0.694658i	0.719340	+	0.694658i
$785$		0			0
$786$	−2.95153	+	1.99083i	−2.95153	+	1.99083i
$787$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$787$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$788$	1.24192	−	0.970296i	1.24192	−	0.970296i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.213817	−	1.21262i	−0.213817	−	1.21262i
$795$		0			0
$796$		0			0
$797$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$797$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$798$		0			0
$799$		0			0
$800$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.362486	−	0.995922i	−0.362486	−	0.995922i
$807$			2.54613i			2.54613i
$808$	−1.58268	+	0.336408i	−1.58268	+	0.336408i
$809$	1.47815	−	0.155360i	1.47815	−	0.155360i	0.669131	−	0.743145i	$-0.266667\pi$
$809$	1.47815	−	0.155360i	1.47815	−	0.155360i	0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$	−0.241055	+	0.139173i	−0.241055	+	0.139173i	−0.615661	−	0.788011i	$-0.711111\pi$
$811$	−0.241055	+	0.139173i	−0.241055	+	0.139173i	0.374607	+	0.927184i	$0.377778\pi$
$812$		0			0
$813$	−2.59256	−	0.551065i	−2.59256	−	0.551065i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.26604	−	0.223238i	1.26604	−	0.223238i
$819$		0			0
$820$		0			0
$821$	−0.478148	−	0.658114i	−0.478148	−	0.658114i	0.500000	−	0.866025i	$-0.333333\pi$
$821$	−0.478148	−	0.658114i	−0.478148	−	0.658114i	−0.978148	+	0.207912i	$0.933333\pi$
$822$		0			0
$823$	−0.0467046	−	0.0518708i	−0.0467046	−	0.0518708i	0.719340	−	0.694658i	$-0.244444\pi$
$823$	−0.0467046	−	0.0518708i	−0.0467046	−	0.0518708i	−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$827$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$828$	−2.80931	−	0.805557i	−2.80931	−	0.805557i
$829$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$829$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$	−2.84392	−	1.64194i	−2.84392	−	1.64194i
$832$	−0.622957	−	0.857427i	−0.622957	−	0.857427i
$833$		0			0
$834$	0.681139	+	0.0957278i	0.681139	+	0.0957278i
$835$		0			0
$836$		0			0
$837$	1.42636	+	3.53037i	1.42636	+	3.53037i
$838$		0			0
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$		0			0
$841$	−0.925545	−	2.84854i	−0.925545	−	2.84854i
$842$		0			0
$843$		0			0
$844$	−0.267286	+	0.318539i	−0.267286	+	0.318539i
$845$		0			0
$846$	−0.407481	−	0.0142296i	−0.407481	−	0.0142296i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	0.434410	+	3.09099i	0.434410	+	3.09099i
$853$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$853$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.0467046	−	0.0518708i	0.0467046	−	0.0518708i	−0.719340	−	0.694658i	$-0.755556\pi$
$857$	0.0467046	−	0.0518708i	0.0467046	−	0.0518708i	0.766044	+	0.642788i	$0.222222\pi$
$858$		0			0
$859$	1.88051	−	0.399715i	1.88051	−	0.399715i	0.882948	−	0.469472i	$-0.155556\pi$
$859$	1.88051	−	0.399715i	1.88051	−	0.399715i	0.997564	+	0.0697565i	$0.0222222\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$863$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$864$	2.34421	+	3.00045i	2.34421	+	3.00045i
$865$		0			0
$866$		0			0
$867$	−1.98054			−1.98054
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	1.29179	+	1.33769i	1.29179	+	1.33769i
$877$	−1.78716	+	0.379874i	−1.78716	+	0.379874i	−0.978148	−	0.207912i	$-0.933333\pi$
$877$	−1.78716	+	0.379874i	−1.78716	+	0.379874i	−0.809017	+	0.587785i	$0.800000\pi$
$878$	0.362486	−	0.995922i	0.362486	−	0.995922i
$879$		0			0
$880$		0			0
$881$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$881$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$882$	−2.58043	+	1.37204i	−2.58043	+	1.37204i
$883$	1.58268	−	1.14988i	1.58268	−	1.14988i	0.669131	−	0.743145i	$-0.266667\pi$
$883$	1.58268	−	1.14988i	1.58268	−	1.14988i	0.913545	−	0.406737i	$-0.133333\pi$
$884$		0			0
$885$		0			0
$886$	0.219340	+	0.171367i	0.219340	+	0.171367i
$887$	−0.565086	−	1.26920i	−0.565086	−	1.26920i	−0.939693	−	0.342020i	$-0.888889\pi$
$887$	−0.565086	−	1.26920i	−0.565086	−	1.26920i	0.374607	−	0.927184i	$-0.377778\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−0.473442	−	0.397265i	−0.473442	−	0.397265i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.23379	+	1.69817i	−1.23379	+	1.69817i
$898$	−1.68060	−	0.419021i	−1.68060	−	0.419021i
$899$	−1.05919	+	1.69506i	−1.05919	+	1.69506i
$900$	−2.58043	−	1.37204i	−2.58043	−	1.37204i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.761196	+	1.56068i	−0.761196	+	1.56068i
$907$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$907$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$908$		0			0
$909$	0.494288	−	4.70284i	0.494288	−	4.70284i
$910$		0			0
$911$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$911$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$919$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$920$		0			0
$921$	0.713218	+	3.35543i	0.713218	+	3.35543i
$922$	0.362486	−	0.580099i	0.362486	−	0.580099i
$923$	1.63383	+	0.347281i	1.63383	+	0.347281i
$924$		0			0
$925$		0			0
$926$	−1.61409	+	0.112868i	−1.61409	+	0.112868i
$927$		0			0
$928$	−0.550939	+	1.92135i	−0.550939	+	1.92135i
$929$		−	1.65808i		−	1.65808i	−0.559193	−	0.829038i	$-0.688889\pi$
$929$		−	1.65808i		−	1.65808i	0.559193	−	0.829038i	$-0.311111\pi$
$930$		0			0
$931$		0			0
$932$	0.630676	+	1.29308i	0.630676	+	1.29308i
$933$	3.93697	−	0.413793i	3.93697	−	0.413793i
$934$		0			0
$935$		0			0
$936$	2.94581	−	0.957150i	2.94581	−	0.957150i
$937$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$937$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$941$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$942$		0			0
$943$	−0.684440	−	0.304732i	−0.684440	−	0.304732i
$944$	1.84561	+	0.460163i	1.84561	+	0.460163i
$945$		0			0
$946$		0			0
$947$	−0.501321	−	0.556774i	−0.501321	−	0.556774i	0.438371	−	0.898794i	$-0.355556\pi$
$947$	−0.501321	−	0.556774i	−0.501321	−	0.556774i	−0.939693	+	0.342020i	$0.888889\pi$
$948$		0			0
$949$	0.909095	−	0.404755i	0.909095	−	0.404755i
$950$		0			0
$951$	0.0432795	−	0.411777i	0.0432795	−	0.411777i
$952$		0			0
$953$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$953$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$954$		0			0
$955$		0			0
$956$	0.0390311	+	1.11770i	0.0390311	+	1.11770i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.990268	−	0.139173i	0.990268	−	0.139173i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$967$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$968$	0.809017	−	0.587785i	0.809017	−	0.587785i
$969$		0			0
$970$		0			0
$971$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$971$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$972$	−5.01769	+	1.82629i	−5.01769	+	1.82629i
$973$		0			0
$974$	−0.943248	+	1.20730i	−0.943248	+	1.20730i
$975$	−1.55990	+	1.40454i	−1.55990	+	1.40454i
$976$		0			0
$977$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$977$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$978$	−1.29179	−	2.42951i	−1.29179	−	2.42951i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.80658	+	0.657542i	1.80658	+	0.657542i
$983$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$983$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$984$	0.741922	+	1.28505i	0.741922	+	1.28505i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$991$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$992$	0.913545	−	0.406737i	0.913545	−	0.406737i
$993$	2.84936			2.84936
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.669131	+	1.15897i	0.669131	+	1.15897i	0.978148	+	0.207912i	$0.0666667\pi$
$997$	0.669131	+	1.15897i	0.669131	+	1.15897i	−0.309017	+	0.951057i	$0.600000\pi$
$998$	−1.35192	−	1.30553i	−1.35192	−	1.30553i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2852.1.bn.b.827.1	yes	24
4.3	odd	2		2852.1.bn.a.827.2	✓	24
23.22	odd	2	CM	2852.1.bn.b.827.1	yes	24
31.3	odd	30		2852.1.bn.a.2483.2	yes	24
92.91	even	2		2852.1.bn.a.827.2	✓	24
124.3	even	30	inner	2852.1.bn.b.2483.1	yes	24
713.344	even	30		2852.1.bn.a.2483.2	yes	24
2852.2483	odd	30	inner	2852.1.bn.b.2483.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2852.1.bn.a.827.2	✓	24	4.3	odd	2
2852.1.bn.a.827.2	✓	24	92.91	even	2
2852.1.bn.a.2483.2	yes	24	31.3	odd	30
2852.1.bn.a.2483.2	yes	24	713.344	even	30
2852.1.bn.b.827.1	yes	24	1.1	even	1	trivial
2852.1.bn.b.827.1	yes	24	23.22	odd	2	CM
2852.1.bn.b.2483.1	yes	24	124.3	even	30	inner
2852.1.bn.b.2483.1	yes	24	2852.2483	odd	30	inner

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:	\( N \)	\(=\)	\( 2852 = 2^{2} \cdot 23 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2852.bn (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.766044 + 0.642788i) q^{2} +(-0.207022 - 1.96969i) q^{3} +(0.173648 - 0.984808i) q^{4} +(1.42468 + 1.37580i) q^{6} +(0.500000 + 0.866025i) q^{8} +(-2.85866 + 0.607627i) q^{9} +(-1.97571 - 0.138155i) q^{12} +(-0.431075 + 0.968211i) q^{13} +(-0.939693 - 0.342020i) q^{16} +(1.79929 - 2.30298i) q^{18} +(-0.309017 + 0.951057i) q^{23} +(1.60229 - 1.16413i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.292131 - 1.01878i) q^{26} +(1.17662 + 3.62127i) q^{27} +(-1.17485 + 1.61705i) q^{29} +(0.997564 - 0.0697565i) q^{31} +(0.939693 - 0.342020i) q^{32} +(0.101995 + 2.92074i) q^{36} +(1.99631 + 0.648642i) q^{39} +(-0.0783141 + 0.745109i) q^{41} +(-0.374607 - 0.927184i) q^{46} +(-0.0820037 - 0.112868i) q^{47} +(-0.479135 + 1.92171i) q^{48} +(-0.913545 - 0.406737i) q^{49} +(-0.173648 - 0.984808i) q^{50} +(0.878646 + 0.592654i) q^{52} +(-3.22905 - 2.01773i) q^{54} +(-0.139428 - 1.99391i) q^{58} +(-1.89169 + 0.198825i) q^{59} +(-0.719340 + 0.694658i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(1.93726 + 0.411777i) q^{69} +(-0.327673 - 1.54158i) q^{71} +(-1.95555 - 2.17186i) q^{72} +(-0.697771 - 0.628276i) q^{73} +(1.80931 + 0.805557i) q^{75} +(-1.94620 + 0.786317i) q^{78} +(4.21932 - 1.87856i) q^{81} +(-0.418955 - 0.621126i) q^{82} +(3.42830 + 1.97933i) q^{87} +(0.882948 + 0.469472i) q^{92} +(-0.343916 - 1.95045i) q^{93} +(0.135369 + 0.0337512i) q^{94} +(-0.868210 - 1.78009i) q^{96} +(0.961262 - 0.275637i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{6} + 12 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{18} + 6 q^{23} - 12 q^{25} + 3 q^{26} + 6 q^{27} - 6 q^{36} + 3 q^{48} - 3 q^{49} + 3 q^{52} - 3 q^{54} + 3 q^{58} - 12 q^{64} - 3 q^{72} + 6 q^{78}+ \cdots + 6 q^{96}+O(q^{100}) \) 24 * q + 6 * q^6 + 12 * q^8 + 3 * q^9 + 3 * q^12 + 6 * q^18 + 6 * q^23 - 12 * q^25 + 3 * q^26 + 6 * q^27 - 6 * q^36 + 3 * q^48 - 3 * q^49 + 3 * q^52 - 3 * q^54 + 3 * q^58 - 12 * q^64 - 3 * q^72 + 6 * q^78 + 3 * q^81 + 3 * q^82 + 9 * q^87 + 6 * q^93 - 6 * q^94 + 6 * q^96

Introduction

L-functions

Modular forms

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