LMFDB - Embedded newform 2852.1.bn.b.735.3

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			735.3
Root			$-0.615661 + 0.788011i$ of defining polynomial
Character	$\chi$	$=$	2852.735
Dual form			2852.1.bn.b.551.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.939693 + 0.342020i) q^{2} +(0.473271 + 0.100597i) q^{3} +(0.766044 + 0.642788i) q^{4} +(0.410323 + 0.256398i) q^{6} +(0.500000 + 0.866025i) q^{8} +(-0.699680 - 0.311518i) q^{9} +(0.297884 + 0.381274i) q^{12} +(1.23219 + 1.10947i) q^{13} +(0.173648 + 0.984808i) q^{16} +(-0.550939 - 0.532036i) q^{18} +(0.809017 - 0.587785i) q^{23} +(0.149516 + 0.460163i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.778419 + 1.46399i) q^{26} +(-0.691238 - 0.502214i) q^{27} +(-1.70961 - 0.555485i) q^{29} +(0.615661 - 0.788011i) q^{31} +(-0.173648 + 0.984808i) q^{32} +(-0.335746 - 0.688382i) q^{36} +(0.471550 + 0.649033i) q^{39} +(1.88051 - 0.399715i) q^{41} +(0.961262 - 0.275637i) q^{46} +(-1.49889 + 0.487017i) q^{47} +(-0.0168859 + 0.483549i) q^{48} +(-0.669131 + 0.743145i) q^{49} +(-0.766044 + 0.642788i) q^{50} +(0.230759 + 1.64194i) q^{52} +(-0.477784 - 0.708344i) q^{54} +(-1.41652 - 1.10671i) q^{58} +(0.244415 - 1.14988i) q^{59} +(0.848048 - 0.529919i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(0.442013 - 0.196797i) q^{69} +(0.565086 - 1.26920i) q^{71} +(-0.0800578 - 0.761700i) q^{72} +(0.138749 + 0.0145831i) q^{73} +(-0.323755 + 0.359566i) q^{75} +(0.221130 + 0.771172i) q^{78} +(0.235862 + 0.261952i) q^{81} +(1.90381 + 0.267564i) q^{82} +(-0.753227 - 0.434876i) q^{87} +(0.997564 + 0.0697565i) q^{92} +(0.370646 - 0.311009i) q^{93} +(-1.57506 - 0.0550024i) q^{94} +(-0.181251 + 0.448612i) q^{96} +(-0.882948 + 0.469472i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2852\mathbb{Z}\right)^\times$.

$n$	$373$	$1427$	$2669$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{17}{30}\right)$

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 $

$2$	0.939693	+	0.342020i	0.939693	+	0.342020i
$2$	0.939693	+	0.342020i	0.939693	+	0.342020i
$3$	0.473271	+	0.100597i	0.473271	+	0.100597i	0.438371	−	0.898794i	$-0.355556\pi$
$3$	0.473271	+	0.100597i	0.473271	+	0.100597i	0.0348995	+	0.999391i	$0.488889\pi$
$4$	0.766044	+	0.642788i	0.766044	+	0.642788i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	0.410323	+	0.256398i	0.410323	+	0.256398i
$7$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$7$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$8$	0.500000	+	0.866025i	0.500000	+	0.866025i
$9$	−0.699680	−	0.311518i	−0.699680	−	0.311518i
$10$		0			0
$11$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$11$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$12$	0.297884	+	0.381274i	0.297884	+	0.381274i
$13$	1.23219	+	1.10947i	1.23219	+	1.10947i	0.990268	+	0.139173i	$0.0444444\pi$
$13$	1.23219	+	1.10947i	1.23219	+	1.10947i	0.241922	+	0.970296i	$0.422222\pi$
$14$		0			0
$15$		0			0
$16$	0.173648	+	0.984808i	0.173648	+	0.984808i
$17$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$17$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$18$	−0.550939	−	0.532036i	−0.550939	−	0.532036i
$19$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$19$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.809017	−	0.587785i	0.809017	−	0.587785i
$23$	0.809017	−	0.587785i	0.809017	−	0.587785i
$24$	0.149516	+	0.460163i	0.149516	+	0.460163i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	0.778419	+	1.46399i	0.778419	+	1.46399i
$27$	−0.691238	−	0.502214i	−0.691238	−	0.502214i
$28$		0			0
$29$	−1.70961	−	0.555485i	−1.70961	−	0.555485i	−0.719340	−	0.694658i	$-0.755556\pi$
$29$	−1.70961	−	0.555485i	−1.70961	−	0.555485i	−0.990268	+	0.139173i	$0.955556\pi$
$30$		0			0
$31$	0.615661	−	0.788011i	0.615661	−	0.788011i
$31$	0.615661	−	0.788011i	0.615661	−	0.788011i
$32$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.335746	−	0.688382i	−0.335746	−	0.688382i
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$	0.471550	+	0.649033i	0.471550	+	0.649033i
$40$		0			0
$41$	1.88051	−	0.399715i	1.88051	−	0.399715i	0.882948	−	0.469472i	$-0.155556\pi$
$41$	1.88051	−	0.399715i	1.88051	−	0.399715i	0.997564	+	0.0697565i	$0.0222222\pi$
$42$		0			0
$43$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$43$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$44$		0			0
$45$		0			0
$46$	0.961262	−	0.275637i	0.961262	−	0.275637i
$47$	−1.49889	+	0.487017i	−1.49889	+	0.487017i	−0.939693	−	0.342020i	$-0.888889\pi$
$47$	−1.49889	+	0.487017i	−1.49889	+	0.487017i	−0.559193	+	0.829038i	$0.688889\pi$
$48$	−0.0168859	+	0.483549i	−0.0168859	+	0.483549i
$49$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$50$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$51$		0			0
$52$	0.230759	+	1.64194i	0.230759	+	1.64194i
$53$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$53$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$54$	−0.477784	−	0.708344i	−0.477784	−	0.708344i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.41652	−	1.10671i	−1.41652	−	1.10671i
$59$	0.244415	−	1.14988i	0.244415	−	1.14988i	−0.669131	−	0.743145i	$-0.733333\pi$
$59$	0.244415	−	1.14988i	0.244415	−	1.14988i	0.913545	−	0.406737i	$-0.133333\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	0.848048	−	0.529919i	0.848048	−	0.529919i
$63$		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
$69$	0.442013	−	0.196797i	0.442013	−	0.196797i
$70$		0			0
$71$	0.565086	−	1.26920i	0.565086	−	1.26920i	−0.374607	−	0.927184i	$-0.622222\pi$
$71$	0.565086	−	1.26920i	0.565086	−	1.26920i	0.939693	−	0.342020i	$-0.111111\pi$
$72$	−0.0800578	−	0.761700i	−0.0800578	−	0.761700i
$73$	0.138749	+	0.0145831i	0.138749	+	0.0145831i	0.173648	−	0.984808i	$-0.444444\pi$
$73$	0.138749	+	0.0145831i	0.138749	+	0.0145831i	−0.0348995	+	0.999391i	$0.511111\pi$
$74$		0			0
$75$	−0.323755	+	0.359566i	−0.323755	+	0.359566i
$76$		0			0
$77$		0			0
$78$	0.221130	+	0.771172i	0.221130	+	0.771172i
$79$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$79$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$80$		0			0
$81$	0.235862	+	0.261952i	0.235862	+	0.261952i
$82$	1.90381	+	0.267564i	1.90381	+	0.267564i
$83$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$83$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.753227	−	0.434876i	−0.753227	−	0.434876i
$88$		0			0
$89$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$89$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$90$		0			0
$91$		0			0
$92$	0.997564	+	0.0697565i	0.997564	+	0.0697565i
$93$	0.370646	−	0.311009i	0.370646	−	0.311009i
$94$	−1.57506	−	0.0550024i	−1.57506	−	0.0550024i
$95$		0			0
$96$	−0.181251	+	0.448612i	−0.181251	+	0.448612i
$97$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$97$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$98$	−0.882948	+	0.469472i	−0.882948	+	0.469472i
$99$		0			0
$100$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$101$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$101$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$103$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$104$	−0.344733	+	1.62184i	−0.344733	+	1.62184i
$105$		0			0
$106$		0			0
$107$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$107$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$108$	−0.206702	−	0.829038i	−0.206702	−	0.829038i
$109$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$109$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$113$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$114$		0			0
$115$		0			0
$116$	−0.952577	−	1.52444i	−0.952577	−	1.52444i
$117$	−0.516520	−	1.16012i	−0.516520	−	1.16012i
$118$	0.622957	−	0.996940i	0.622957	−	0.996940i
$119$		0			0
$120$		0			0
$121$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$122$		0			0
$123$	0.930201			0.930201
$124$	0.978148	−	0.207912i	0.978148	−	0.207912i
$125$		0			0
$126$		0			0
$127$	−1.40724	−	0.299118i	−1.40724	−	0.299118i	−0.559193	−	0.829038i	$-0.688889\pi$
$127$	−1.40724	−	0.299118i	−1.40724	−	0.299118i	−0.848048	+	0.529919i	$0.822222\pi$
$128$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$129$		0			0
$130$		0			0
$131$	−0.754239	−	1.69405i	−0.754239	−	1.69405i	−0.719340	−	0.694658i	$-0.755556\pi$
$131$	−0.754239	−	1.69405i	−0.754239	−	1.69405i	−0.0348995	−	0.999391i	$-0.511111\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$137$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$138$	0.482665	−	0.0337512i	0.482665	−	0.0337512i
$139$	−0.473442	−	1.45710i	−0.473442	−	1.45710i	−0.848048	−	0.529919i	$-0.822222\pi$
$139$	−0.473442	−	1.45710i	−0.473442	−	1.45710i	0.374607	−	0.927184i	$-0.377778\pi$
$140$		0			0
$141$	−0.758371	+	0.0797080i	−0.758371	+	0.0797080i
$142$	0.965101	−	0.999391i	0.965101	−	0.999391i
$143$		0			0
$144$	0.185287	−	0.743145i	0.185287	−	0.743145i
$145$		0			0
$146$	0.125393	+	0.0611585i	0.125393	+	0.0611585i
$147$	−0.391438	+	0.284396i	−0.391438	+	0.284396i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.427209	+	0.227151i	−0.427209	+	0.227151i
$151$	0.606126	+	0.440376i	0.606126	+	0.440376i	0.848048	−	0.529919i	$-0.177778\pi$
$151$	0.606126	+	0.440376i	0.606126	+	0.440376i	−0.241922	+	0.970296i	$0.577778\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.0559621	+	0.800295i	−0.0559621	+	0.800295i
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.132045	+	0.326824i	0.132045	+	0.326824i
$163$	−0.622957	−	0.857427i	−0.622957	−	0.857427i	0.374607	−	0.927184i	$-0.377778\pi$
$163$	−0.622957	−	0.857427i	−0.622957	−	0.857427i	−0.997564	+	0.0697565i	$0.977778\pi$
$164$	1.69749	+	0.902570i	1.69749	+	0.902570i
$165$		0			0
$166$		0			0
$167$	−1.22256	−	1.35779i	−1.22256	−	1.35779i	−0.913545	−	0.406737i	$-0.866667\pi$
$167$	−1.22256	−	1.35779i	−1.22256	−	1.35779i	−0.309017	−	0.951057i	$-0.600000\pi$
$168$		0			0
$169$	0.182843	+	1.73963i	0.182843	+	1.73963i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.22256	+	1.35779i	−1.22256	+	1.35779i	−0.309017	+	0.951057i	$0.600000\pi$
$173$	−1.22256	+	1.35779i	−1.22256	+	1.35779i	−0.913545	+	0.406737i	$0.866667\pi$
$174$	−0.559066	−	0.666269i	−0.559066	−	0.666269i
$175$		0			0
$176$		0			0
$177$	0.231349	−	0.519618i	0.231349	−	0.519618i
$178$		0			0
$179$	1.12487	−	0.500824i	1.12487	−	0.500824i	0.241922	−	0.970296i	$-0.422222\pi$
$179$	1.12487	−	0.500824i	1.12487	−	0.500824i	0.882948	+	0.469472i	$0.155556\pi$
$180$		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
$184$	0.913545	+	0.406737i	0.913545	+	0.406737i
$185$		0			0
$186$	0.454664	−	0.165484i	0.454664	−	0.165484i
$187$		0			0
$188$	−1.46126	−	0.590388i	−1.46126	−	0.590388i
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	−0.323755	+	0.359566i	−0.323755	+	0.359566i
$193$	0.317271	−	0.141258i	0.317271	−	0.141258i	−0.241922	−	0.970296i	$-0.577778\pi$
$193$	0.317271	−	0.141258i	0.317271	−	0.141258i	0.559193	+	0.829038i	$0.311111\pi$
$194$		0			0
$195$		0			0
$196$	−0.990268	+	0.139173i	−0.990268	+	0.139173i
$197$	1.38171	+	0.145223i	1.38171	+	0.145223i	0.766044	−	0.642788i	$-0.222222\pi$
$197$	1.38171	+	0.145223i	1.38171	+	0.145223i	0.615661	+	0.788011i	$0.288889\pi$
$198$		0			0
$199$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$199$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$200$	−1.00000			−1.00000
$201$		0			0
$202$	−0.594092	+	0.170353i	−0.594092	+	0.170353i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.749159	+	0.159239i	−0.749159	+	0.159239i
$208$	−0.878646	+	1.40613i	−0.878646	+	1.40613i
$209$		0			0
$210$		0			0
$211$	−0.704489	−	0.406737i	−0.704489	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$211$	−0.704489	−	0.406737i	−0.704489	−	0.406737i	−0.809017	+	0.587785i	$0.800000\pi$
$212$		0			0
$213$	0.395116	−	0.543831i	0.395116	−	0.543831i
$214$		0			0
$215$		0			0
$216$	0.0893109	−	0.849737i	0.0893109	−	0.849737i
$217$		0			0
$218$		0			0
$219$	0.0641987	+	0.0208594i	0.0641987	+	0.0208594i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$223$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$224$		0			0
$225$	0.619622	−	0.450182i	0.619622	−	0.450182i
$226$		0			0
$227$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$227$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$228$		0			0
$229$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$229$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$230$		0			0
$231$		0			0
$232$	−0.373740	−	1.75831i	−0.373740	−	1.75831i
$233$	−0.524123	−	1.61308i	−0.524123	−	1.61308i	−0.766044	−	0.642788i	$-0.777778\pi$
$233$	−0.524123	−	1.61308i	−0.524123	−	1.61308i	0.241922	−	0.970296i	$-0.422222\pi$
$234$	−0.0885846	−	1.26682i	−0.0885846	−	1.26682i
$235$		0			0
$236$	0.926362	−	0.723753i	0.926362	−	0.723753i
$237$		0			0
$238$		0			0
$239$	1.80931	+	0.805557i	1.80931	+	0.805557i	0.961262	+	0.275637i	$0.0888889\pi$
$239$	1.80931	+	0.805557i	1.80931	+	0.805557i	0.848048	+	0.529919i	$0.177778\pi$
$240$		0			0
$241$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$241$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$242$	−0.848048	−	0.529919i	−0.848048	−	0.529919i
$243$	0.512484	+	0.887648i	0.512484	+	0.887648i
$244$		0			0
$245$		0			0
$246$	0.874103	+	0.318147i	0.874103	+	0.318147i
$247$		0			0
$248$	0.990268	+	0.139173i	0.990268	+	0.139173i
$249$		0			0
$250$		0			0
$251$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$251$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$252$		0			0
$253$		0			0
$254$	−1.22007	−	0.762384i	−1.22007	−	0.762384i
$255$		0			0
$256$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$257$	1.39963	+	0.623157i	1.39963	+	0.623157i	0.961262	−	0.275637i	$-0.0888889\pi$
$257$	1.39963	+	0.623157i	1.39963	+	0.623157i	0.438371	+	0.898794i	$0.355556\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.02314	+	0.921235i	1.02314	+	0.921235i
$262$	−0.129354	−	1.84985i	−0.129354	−	1.84985i
$263$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$263$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.142220	−	0.669092i	−0.142220	−	0.669092i	−0.990268	−	0.139173i	$-0.955556\pi$
$269$	−0.142220	−	0.669092i	−0.142220	−	0.669092i	0.848048	−	0.529919i	$-0.177778\pi$
$270$		0			0
$271$	0.169131	−	0.122881i	0.169131	−	0.122881i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	0.169131	−	0.122881i	0.169131	−	0.122881i	0.669131	+	0.743145i	$0.266667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	0.465101	+	0.133365i	0.465101	+	0.133365i
$277$	0.264723	+	0.0860137i	0.264723	+	0.0860137i	0.438371	−	0.898794i	$-0.355556\pi$
$277$	0.264723	+	0.0860137i	0.264723	+	0.0860137i	−0.173648	+	0.984808i	$0.555556\pi$
$278$	0.0534691	−	1.53116i	0.0534691	−	1.53116i
$279$	−0.676245	+	0.359566i	−0.676245	+	0.359566i
$280$		0			0
$281$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$281$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$282$	−0.739897	−	0.184477i	−0.739897	−	0.184477i
$283$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$283$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$284$	1.24871	−	0.609036i	1.24871	−	0.609036i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.428283	−	0.634956i	0.428283	−	0.634956i
$289$	0.978148	−	0.207912i	0.978148	−	0.207912i
$290$		0			0
$291$		0			0
$292$	0.0969138	+	0.100357i	0.0969138	+	0.100357i
$293$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$293$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$294$	−0.465101	+	0.133365i	−0.465101	+	0.133365i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.64899	+	0.173316i	1.64899	+	0.173316i
$300$	−0.479135	+	0.0673380i	−0.479135	+	0.0673380i
$301$		0			0
$302$	0.418955	+	0.621126i	0.418955	+	0.621126i
$303$	−0.273179	+	0.121627i	−0.273179	+	0.121627i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.360114	+	1.69420i	−0.360114	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$307$	−0.360114	+	1.69420i	−0.360114	+	1.69420i	−0.669131	+	0.743145i	$0.733333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.79759i			1.79759i	0.438371	+	0.898794i	$0.355556\pi$
$311$			1.79759i			1.79759i	−0.438371	+	0.898794i	$0.644444\pi$
$312$	−0.326304	+	0.732891i	−0.326304	+	0.732891i
$313$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$313$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.78716	+	0.795697i	−1.78716	+	0.795697i	−0.809017	+	0.587785i	$0.800000\pi$
$317$	−1.78716	+	0.795697i	−1.78716	+	0.795697i	−0.978148	+	0.207912i	$0.933333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.0123017	+	0.352276i	0.0123017	+	0.352276i
$325$	−1.57692	+	0.512373i	−1.57692	+	0.512373i
$326$	−0.292131	−	1.01878i	−0.292131	−	1.01878i
$327$		0			0
$328$	1.28642	+	1.42871i	1.28642	+	1.42871i
$329$		0			0
$330$		0			0
$331$	1.65903	−	0.352638i	1.65903	−	0.352638i	0.719340	−	0.694658i	$-0.244444\pi$
$331$	1.65903	−	0.352638i	1.65903	−	0.352638i	0.939693	+	0.342020i	$0.111111\pi$
$332$		0			0
$333$		0			0
$334$	−0.684440	−	1.69405i	−0.684440	−	1.69405i
$335$		0			0
$336$		0			0
$337$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$337$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$338$	−0.423173	+	1.69725i	−0.423173	+	1.69725i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.61323	+	0.857767i	−1.61323	+	0.857767i
$347$	−0.104528	+	0.181049i	−0.104528	+	0.181049i	−0.913545	−	0.406737i	$-0.866667\pi$
$347$	−0.104528	+	0.181049i	−0.104528	+	0.181049i	0.809017	+	0.587785i	$0.200000\pi$
$348$	−0.297473	−	0.817299i	−0.297473	−	0.817299i
$349$	−0.709299	+	0.515336i	−0.709299	+	0.515336i	−0.882948	−	0.469472i	$-0.844444\pi$
$349$	−0.709299	+	0.515336i	−0.709299	+	0.515336i	0.173648	+	0.984808i	$0.444444\pi$
$350$		0			0
$351$	−0.294546	−	1.38573i	−0.294546	−	1.38573i
$352$		0			0
$353$	1.48538	−	1.33745i	1.48538	−	1.33745i	0.719340	−	0.694658i	$-0.244444\pi$
$353$	1.48538	−	1.33745i	1.48538	−	1.33745i	0.766044	−	0.642788i	$-0.222222\pi$
$354$	0.395116	−	0.409155i	0.395116	−	0.409155i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.22832	−	0.0858927i	1.22832	−	0.0858927i
$359$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$359$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$360$		0			0
$361$	0.104528	−	0.994522i	0.104528	−	0.994522i
$362$		0			0
$363$	−0.442013	−	0.196797i	−0.442013	−	0.196797i
$364$		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	$0.333333\pi$
$368$	0.719340	+	0.694658i	0.719340	+	0.694658i
$369$	−1.44027	−	0.306140i	−1.44027	−	0.306140i
$370$		0			0
$371$		0			0
$372$	0.483844			0.483844
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−1.17121	−	1.05456i	−1.17121	−	1.05456i
$377$	−1.49027	−	2.58122i	−1.49027	−	2.58122i
$378$		0			0
$379$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$379$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$380$		0			0
$381$	−0.635916	−	0.283128i	−0.635916	−	0.283128i
$382$		0			0
$383$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$383$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$384$	−0.427209	+	0.227151i	−0.427209	+	0.227151i
$385$		0			0
$386$	0.346450	−	0.0242262i	0.346450	−	0.0242262i
$387$		0			0
$388$		0			0
$389$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$389$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$390$		0			0
$391$		0			0
$392$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$393$	−0.186543	−	0.877618i	−0.186543	−	0.877618i
$394$	1.24871	+	0.609036i	1.24871	+	0.609036i
$395$		0			0
$396$		0			0
$397$	−0.719340	+	1.24593i	−0.719340	+	1.24593i	0.241922	+	0.970296i	$0.422222\pi$
$397$	−0.719340	+	1.24593i	−0.719340	+	1.24593i	−0.961262	+	0.275637i	$0.911111\pi$
$398$		0			0
$399$		0			0
$400$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$401$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$401$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$402$		0			0
$403$	1.63289	−	0.287922i	1.63289	−	0.287922i
$404$	−0.616528	−	0.0431119i	−0.616528	−	0.0431119i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$409$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	−0.766044	+	0.642788i	$0.777778\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.758442	−	0.106592i	−0.758442	−	0.106592i
$415$		0			0
$416$	−1.30658	+	1.02081i	−1.30658	+	1.02081i
$417$	−0.0774861	−	0.737231i	−0.0774861	−	0.737231i
$418$		0			0
$419$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$419$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$420$		0			0
$421$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$421$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$422$	−0.522891	−	0.623157i	−0.522891	−	0.623157i
$423$	1.20045	+	0.126173i	1.20045	+	0.126173i
$424$		0			0
$425$		0			0
$426$	0.557289	−	0.375896i	0.557289	−	0.375896i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$431$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$432$	0.374552	−	0.767945i	0.374552	−	0.767945i
$433$		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
$438$	0.0531927	+	0.0415587i	0.0531927	+	0.0415587i
$439$	−1.43594	+	0.829038i	−1.43594	+	0.829038i	−0.997564	−	0.0697565i	$-0.977778\pi$
$439$	−1.43594	+	0.829038i	−1.43594	+	0.829038i	−0.438371	+	0.898794i	$0.644444\pi$
$440$		0			0
$441$	0.699680	−	0.311518i	0.699680	−	0.311518i
$442$		0			0
$443$	−0.789310	+	1.77282i	−0.789310	+	1.77282i	−0.173648	+	0.984808i	$0.555556\pi$
$443$	−0.789310	+	1.77282i	−0.789310	+	1.77282i	−0.615661	+	0.788011i	$0.711111\pi$
$444$		0			0
$445$		0			0
$446$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$447$		0			0
$448$		0			0
$449$	1.64728	−	0.535233i	1.64728	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$449$	1.64728	−	0.535233i	1.64728	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$450$	0.736226	−	0.211109i	0.736226	−	0.211109i
$451$		0			0
$452$		0			0
$453$	0.242561	+	0.269392i	0.242561	+	0.269392i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$457$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.15771	+	1.59345i	−1.15771	+	1.59345i	−0.438371	+	0.898794i	$0.644444\pi$
$461$	−1.15771	+	1.59345i	−1.15771	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$462$		0			0
$463$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$463$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$464$	0.250176	−	1.78009i	0.250176	−	1.78009i
$465$		0			0
$466$	0.0591929	−	1.69506i	0.0591929	−	1.69506i
$467$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$467$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$468$	0.350035	−	1.22072i	0.350035	−	1.22072i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.11803	−	0.363271i	1.11803	−	0.363271i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.42468	+	1.37580i	1.42468	+	1.37580i
$479$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$479$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.615661	−	0.788011i	−0.615661	−	0.788011i
$485$		0			0
$486$	0.177984	+	1.00940i	0.177984	+	1.00940i
$487$	1.71690	+	0.764415i	1.71690	+	0.764415i	0.997564	+	0.0697565i	$0.0222222\pi$
$487$	1.71690	+	0.764415i	1.71690	+	0.764415i	0.719340	+	0.694658i	$0.244444\pi$
$488$		0			0
$489$	−0.208573	−	0.468463i	−0.208573	−	0.468463i
$490$		0			0
$491$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$491$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.0348995	+	0.999391i	$0.488889\pi$
$492$	0.712575	+	0.597922i	0.712575	+	0.597922i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.882948	+	0.469472i	0.882948	+	0.469472i
$497$		0			0
$498$		0			0
$499$	−0.339707	−	0.0722070i	−0.339707	−	0.0722070i	0.0348995	−	0.999391i	$-0.488889\pi$
$499$	−0.339707	−	0.0722070i	−0.339707	−	0.0722070i	−0.374607	+	0.927184i	$0.622222\pi$
$500$		0			0
$501$	−0.442013	−	0.765589i	−0.442013	−	0.765589i
$502$		0			0
$503$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$503$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.0884672	+	0.841710i	−0.0884672	+	0.841710i
$508$	−0.885740	−	1.13369i	−0.885740	−	1.13369i
$509$	−1.37806	−	1.24081i	−1.37806	−	1.24081i	−0.939693	−	0.342020i	$-0.888889\pi$
$509$	−1.37806	−	1.24081i	−1.37806	−	1.24081i	−0.438371	−	0.898794i	$-0.644444\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	1.10209	+	1.06428i	1.10209	+	1.06428i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.715193	+	0.519618i	−0.715193	+	0.519618i
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	0.646352	+	1.21561i	0.646352	+	1.21561i
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$	0.511133	−	1.78253i	0.511133	−	1.78253i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.309017	−	0.951057i	0.309017	−	0.951057i
$530$		0			0
$531$	−0.529221	+	0.728410i	−0.529221	+	0.728410i
$532$		0			0
$533$	2.76062	+	1.59384i	2.76062	+	1.59384i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.582749	−	0.123867i	0.582749	−	0.123867i
$538$	0.0952000	−	0.677383i	0.0952000	−	0.677383i
$539$		0			0
$540$		0			0
$541$	0.160147	+	1.52370i	0.160147	+	1.52370i	0.719340	+	0.694658i	$0.244444\pi$
$541$	0.160147	+	1.52370i	0.160147	+	1.52370i	−0.559193	+	0.829038i	$0.688889\pi$
$542$	0.200958	−	0.0576239i	0.200958	−	0.0576239i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.548255	−	0.0576239i	−0.548255	−	0.0576239i	−0.173648	−	0.984808i	$-0.555556\pi$
$547$	−0.548255	−	0.0576239i	−0.548255	−	0.0576239i	−0.374607	+	0.927184i	$0.622222\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	0.391438	+	0.284396i	0.391438	+	0.284396i
$553$		0			0
$554$	0.219340	+	0.171367i	0.219340	+	0.171367i
$555$		0			0
$556$	0.573931	−	1.42053i	0.573931	−	1.42053i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−0.758442	+	0.106592i	−0.758442	+	0.106592i
$559$		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	$0.833333\pi$
$564$	−0.632181	−	0.426412i	−0.632181	−	0.426412i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.38171	−	0.145223i	1.38171	−	0.145223i
$569$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$569$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$570$		0			0
$571$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$571$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.104528	+	0.994522i	0.104528	+	0.994522i
$576$	0.619622	−	0.450182i	0.619622	−	0.450182i
$577$	1.25755	+	1.39666i	1.25755	+	1.39666i	0.882948	+	0.469472i	$0.155556\pi$
$577$	1.25755	+	1.39666i	1.25755	+	1.39666i	0.374607	+	0.927184i	$0.377778\pi$
$578$	0.990268	+	0.139173i	0.990268	+	0.139173i
$579$	0.164365	−	0.0349369i	0.164365	−	0.0349369i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.0567450	+	0.127451i	0.0567450	+	0.127451i
$585$		0			0
$586$		0			0
$587$	0.231520	−	0.712544i	0.231520	−	0.712544i	−0.766044	−	0.642788i	$-0.777778\pi$
$587$	0.231520	−	0.712544i	0.231520	−	0.712544i	0.997564	−	0.0697565i	$-0.0222222\pi$
$588$	−0.482665	−	0.0337512i	−0.482665	−	0.0337512i
$589$		0			0
$590$		0			0
$591$	0.639312	+	0.207725i	0.639312	+	0.207725i
$592$		0			0
$593$	−1.08268	−	0.786610i	−1.08268	−	0.786610i	−0.104528	−	0.994522i	$-0.533333\pi$
$593$	−1.08268	−	0.786610i	−1.08268	−	0.786610i	−0.978148	+	0.207912i	$0.933333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.49027	+	0.726852i	1.49027	+	0.726852i
$599$	−0.360114	−	1.69420i	−0.360114	−	1.69420i	−0.669131	−	0.743145i	$-0.733333\pi$
$599$	−0.360114	−	1.69420i	−0.360114	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$600$	−0.473271	−	0.100597i	−0.473271	−	0.100597i
$601$	0.787614	−	0.709170i	0.787614	−	0.709170i	−0.173648	−	0.984808i	$-0.555556\pi$
$601$	0.787614	−	0.709170i	0.787614	−	0.709170i	0.961262	+	0.275637i	$0.0888889\pi$
$602$		0			0
$603$		0			0
$604$	0.181251	+	0.726958i	0.181251	+	0.726958i
$605$		0			0
$606$	−0.298303	+	0.0208594i	−0.298303	+	0.0208594i
$607$	1.41355	+	1.27276i	1.41355	+	1.27276i	0.913545	+	0.406737i	$0.133333\pi$
$607$	1.41355	+	1.27276i	1.41355	+	1.27276i	0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.38724	−	1.06287i	−2.38724	−	1.06287i
$612$		0			0
$613$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$613$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$614$	−0.917847	+	1.46886i	−0.917847	+	1.46886i
$615$		0			0
$616$		0			0
$617$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$617$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−0.854417			−0.854417
$622$	−0.614811	+	1.68918i	−0.614811	+	1.68918i
$623$		0			0
$624$	−0.557289	+	0.577090i	−0.557289	+	0.577090i
$625$	−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		0.104528	−	0.994522i	$-0.466667\pi$
$631$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$632$		0			0
$633$	−0.292497	−	0.263366i	−0.292497	−	0.263366i
$634$	−1.95153	+	0.136464i	−1.95153	+	0.136464i
$635$		0			0
$636$		0			0
$637$	−1.64899	+	0.173316i	−1.64899	+	0.173316i
$638$		0			0
$639$	−0.790759	+	0.712002i	−0.790759	+	0.712002i
$640$		0			0
$641$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$641$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$642$		0			0
$643$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$643$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.23949	+	0.900539i	1.23949	+	0.900539i	0.997564	−	0.0697565i	$-0.0222222\pi$
$647$	1.23949	+	0.900539i	1.23949	+	0.900539i	0.241922	+	0.970296i	$0.422222\pi$
$648$	−0.108926	+	0.335239i	−0.108926	+	0.335239i
$649$		0			0
$650$	−1.65707	−	0.0578660i	−1.65707	−	0.0578660i
$651$		0			0
$652$	0.0739306	−	1.05726i	0.0739306	−	1.05726i
$653$	−0.580762	+	1.78740i	−0.580762	+	1.78740i	0.0348995	+	0.999391i	$0.488889\pi$
$653$	−0.580762	+	1.78740i	−0.580762	+	1.78740i	−0.615661	+	0.788011i	$0.711111\pi$
$654$		0			0
$655$		0			0
$656$	0.720190	+	1.78253i	0.720190	+	1.78253i
$657$	−0.0925368	−	0.0534261i	−0.0925368	−	0.0534261i
$658$		0			0
$659$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$659$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$660$		0			0
$661$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$661$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$662$	1.67959	+	0.236051i	1.67959	+	0.236051i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.70961	+	0.555485i	−1.70961	+	0.555485i
$668$	−0.0637646	−	1.82598i	−0.0637646	−	1.82598i
$669$	−0.523846	+	0.581790i	−0.523846	+	0.581790i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.801115	−	1.79933i	0.801115	−	1.79933i	0.241922	−	0.970296i	$-0.422222\pi$
$673$	0.801115	−	1.79933i	0.801115	−	1.79933i	0.559193	−	0.829038i	$-0.311111\pi$
$674$		0			0
$675$	0.780549	−	0.347523i	0.780549	−	0.347523i
$676$	−0.978148	+	1.45016i	−0.978148	+	1.45016i
$677$		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
$683$		−	0.278346i		−	0.278346i	−0.990268	−	0.139173i	$-0.955556\pi$
$683$		−	0.278346i		−	0.278346i	0.990268	−	0.139173i	$-0.0444444\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.809017	+	1.81708i	−0.809017	+	1.81708i	−0.309017	+	0.951057i	$0.600000\pi$
$691$	−0.809017	+	1.81708i	−0.809017	+	1.81708i	−0.500000	+	0.866025i	$0.666667\pi$
$692$	−1.80931	+	0.254282i	−1.80931	+	0.254282i
$693$		0			0
$694$	−0.160147	+	0.134379i	−0.160147	+	0.134379i
$695$		0			0
$696$		−	0.869752i		−	0.869752i
$697$		0			0
$698$	−0.842779	+	0.241663i	−0.842779	+	0.241663i
$699$	−0.0857808	−	0.816150i	−0.0857808	−	0.816150i
$700$		0			0
$701$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$701$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$702$	0.197165	−	1.40290i	0.197165	−	1.40290i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.85324	−	0.748757i	1.85324	−	0.748757i
$707$		0			0
$708$	0.511227	−	0.249342i	0.511227	−	0.249342i
$709$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$709$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.0348995	−	0.999391i	0.0348995	−	0.999391i
$714$		0			0
$715$		0			0
$716$	1.18362	+	0.339399i	1.18362	+	0.339399i
$717$	0.775257	+	0.563257i	0.775257	+	0.563257i
$718$		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
$722$	0.438371	−	0.898794i	0.438371	−	0.898794i
$723$		0			0
$724$		0			0
$725$	1.33587	−	1.20282i	1.33587	−	1.20282i
$726$	−0.348048	−	0.336106i	−0.348048	−	0.336106i
$727$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$727$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$728$		0			0
$729$	0.0443234	+	0.136414i	0.0443234	+	0.136414i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$733$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$734$		0			0
$735$		0			0
$736$	0.438371	+	0.898794i	0.438371	+	0.898794i
$737$		0			0
$738$	−1.24871	−	0.780280i	−1.24871	−	0.780280i
$739$	0.990268	+	1.71519i	0.990268	+	1.71519i	0.615661	+	0.788011i	$0.288889\pi$
$739$	0.990268	+	1.71519i	0.990268	+	1.71519i	0.374607	+	0.927184i	$0.377778\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$	0.454664	+	0.165484i	0.454664	+	0.165484i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$751$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$752$	−0.739897	−	1.39154i	−0.739897	−	1.39154i
$753$		0			0
$754$	−0.517565	−	2.93526i	−0.517565	−	2.93526i
$755$		0			0
$756$		0			0
$757$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$757$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.933799	+	0.0981463i	−0.933799	+	0.0981463i	−0.559193	−	0.829038i	$-0.688889\pi$
$761$	−0.933799	+	0.0981463i	−0.933799	+	0.0981463i	−0.374607	+	0.927184i	$0.622222\pi$
$762$	−0.500730	−	0.483549i	−0.500730	−	0.483549i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.57692	−	1.14570i	1.57692	−	1.14570i
$768$	−0.479135	+	0.0673380i	−0.479135	+	0.0673380i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	0.599718	+	0.435720i	0.599718	+	0.435720i
$772$	0.333843	+	0.0957278i	0.333843	+	0.0957278i
$773$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$773$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$774$		0			0
$775$	0.374607	+	0.927184i	0.374607	+	0.927184i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.902774	+	1.24256i	0.902774	+	1.24256i
$784$	−0.848048	−	0.529919i	−0.848048	−	0.529919i
$785$		0			0
$786$	0.124869	−	0.888493i	0.124869	−	0.888493i
$787$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$787$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$788$	0.965101	+	0.999391i	0.965101	+	0.999391i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−1.10209	+	0.924765i	−1.10209	+	0.924765i
$795$		0			0
$796$		0			0
$797$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$797$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$798$		0			0
$799$		0			0
$800$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	1.63289	+	0.287922i	1.63289	+	0.287922i
$807$		−	0.330969i		−	0.330969i
$808$	−0.564602	−	0.251377i	−0.564602	−	0.251377i
$809$	−0.413545	+	1.94558i	−0.413545	+	1.94558i	−0.104528	+	0.994522i	$0.533333\pi$
$809$	−0.413545	+	1.94558i	−0.413545	+	1.94558i	−0.309017	+	0.951057i	$0.600000\pi$
$810$		0			0
$811$	−1.68060	+	0.970296i	−1.68060	+	0.970296i	−0.719340	+	0.694658i	$0.755556\pi$
$811$	−1.68060	+	0.970296i	−1.68060	+	0.970296i	−0.961262	+	0.275637i	$0.911111\pi$
$812$		0			0
$813$	0.0924059	−	0.0411418i	0.0924059	−	0.0411418i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$819$		0			0
$820$		0			0
$821$	1.41355	−	0.459289i	1.41355	−	0.459289i	0.500000	−	0.866025i	$-0.333333\pi$
$821$	1.41355	−	0.459289i	1.41355	−	0.459289i	0.913545	+	0.406737i	$0.133333\pi$
$822$		0			0
$823$	0.0916445	+	0.871939i	0.0916445	+	0.871939i	0.939693	+	0.342020i	$0.111111\pi$
$823$	0.0916445	+	0.871939i	0.0916445	+	0.871939i	−0.848048	+	0.529919i	$0.822222\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$827$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$828$	−0.676245	−	0.359566i	−0.676245	−	0.359566i
$829$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$829$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$830$		0			0
$831$	0.116633	+	0.0673380i	0.116633	+	0.0673380i
$832$	−1.57692	+	0.512373i	−1.57692	+	0.512373i
$833$		0			0
$834$	0.179335	−	0.719272i	0.179335	−	0.719272i
$835$		0			0
$836$		0			0
$837$	−0.821319	+	0.235509i	−0.821319	+	0.235509i
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	1.80518	+	1.31154i	1.80518	+	1.31154i
$842$		0			0
$843$		0			0
$844$	−0.278224	−	0.764415i	−0.278224	−	0.764415i
$845$		0			0
$846$	1.08490	+	0.529143i	1.08490	+	0.529143i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	0.652245	−	0.162623i	0.652245	−	0.162623i
$853$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$853$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.0916445	+	0.871939i	−0.0916445	+	0.871939i	0.848048	+	0.529919i	$0.177778\pi$
$857$	−0.0916445	+	0.871939i	−0.0916445	+	0.871939i	−0.939693	+	0.342020i	$0.888889\pi$
$858$		0			0
$859$	1.61323	+	0.718254i	1.61323	+	0.718254i	0.997564	−	0.0697565i	$-0.0222222\pi$
$859$	1.61323	+	0.718254i	1.61323	+	0.718254i	0.615661	+	0.788011i	$0.288889\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$863$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$864$	0.614616	−	0.593528i	0.614616	−	0.593528i
$865$		0			0
$866$		0			0
$867$	0.483844			0.483844
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.0357709	+	0.0572453i	0.0357709	+	0.0572453i
$877$	1.22256	+	0.544320i	1.22256	+	0.544320i	0.913545	−	0.406737i	$-0.133333\pi$
$877$	1.22256	+	0.544320i	1.22256	+	0.544320i	0.309017	+	0.951057i	$0.400000\pi$
$878$	−1.63289	+	0.287922i	−1.63289	+	0.287922i
$879$		0			0
$880$		0			0
$881$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$881$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$882$	0.764030	−	0.0534261i	0.764030	−	0.0534261i
$883$	0.564602	+	1.73767i	0.564602	+	1.73767i	0.669131	+	0.743145i	$0.266667\pi$
$883$	0.564602	+	1.73767i	0.564602	+	1.73767i	−0.104528	+	0.994522i	$0.533333\pi$
$884$		0			0
$885$		0			0
$886$	−1.34805	+	1.39594i	−1.34805	+	1.39594i
$887$	−0.787614	+	0.709170i	−0.787614	+	0.709170i	−0.961262	−	0.275637i	$-0.911111\pi$
$887$	−0.787614	+	0.709170i	−0.787614	+	0.709170i	0.173648	+	0.984808i	$0.444444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1.52045	+	0.553400i	−1.52045	+	0.553400i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.762984	+	0.247909i	0.762984	+	0.247909i
$898$	1.73100	+	0.0604477i	1.73100	+	0.0604477i
$899$	−1.49027	+	1.00520i	−1.49027	+	1.00520i
$900$	0.764030	+	0.0534261i	0.764030	+	0.0534261i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	0.135796	+	0.336106i	0.135796	+	0.336106i
$907$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$907$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$908$		0			0
$909$	0.463005	−	0.0984148i	0.463005	−	0.0984148i
$910$		0			0
$911$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$911$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$919$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$920$		0			0
$921$	−0.340862	+	0.765589i	−0.340862	+	0.765589i
$922$	−1.63289	+	1.10140i	−1.63289	+	1.10140i
$923$	2.10444	−	0.936955i	2.10444	−	0.936955i
$924$		0			0
$925$		0			0
$926$	0.380500	−	0.487017i	0.380500	−	0.487017i
$927$		0			0
$928$	0.843916	−	1.58718i	0.843916	−	1.58718i
$929$		−	0.278346i		−	0.278346i	−0.990268	−	0.139173i	$-0.955556\pi$
$929$		−	0.278346i		−	0.278346i	0.990268	−	0.139173i	$-0.0444444\pi$
$930$		0			0
$931$		0			0
$932$	0.635369	−	1.57259i	0.635369	−	1.57259i
$933$	−0.180832	+	0.850746i	−0.180832	+	0.850746i
$934$		0			0
$935$		0			0
$936$	0.746435	−	1.02738i	0.746435	−	1.02738i
$937$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$937$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$941$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$942$		0			0
$943$	1.28642	−	1.42871i	1.28642	−	1.42871i
$944$	1.17485	+	0.0410268i	1.17485	+	0.0410268i
$945$		0			0
$946$		0			0
$947$	−0.200958	−	1.91199i	−0.200958	−	1.91199i	−0.374607	−	0.927184i	$-0.622222\pi$
$947$	−0.200958	−	1.91199i	−0.200958	−	1.91199i	0.173648	−	0.984808i	$-0.444444\pi$
$948$		0			0
$949$	0.154785	+	0.171906i	0.154785	+	0.171906i
$950$		0			0
$951$	−0.925857	+	0.196797i	−0.925857	+	0.196797i
$952$		0			0
$953$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$953$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$954$		0			0
$955$		0			0
$956$	0.868210	+	1.78009i	0.868210	+	1.78009i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.241922	−	0.970296i	−0.241922	−	0.970296i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$967$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$968$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$969$		0			0
$970$		0			0
$971$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$971$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$972$	−0.177984	+	1.00940i	−0.177984	+	1.00940i
$973$		0			0
$974$	1.35192	+	1.30553i	1.35192	+	1.30553i
$975$	−0.797855	+	0.0838579i	−0.797855	+	0.0838579i
$976$		0			0
$977$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$977$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$978$	−0.0357709	−	0.511547i	−0.0357709	−	0.511547i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	0.306644	+	1.73907i	0.306644	+	1.73907i
$983$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$983$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$984$	0.465101	+	0.805578i	0.465101	+	0.805578i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$991$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$992$	0.669131	+	0.743145i	0.669131	+	0.743145i
$993$	0.820646			0.820646
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	0.809017	−	0.587785i	$-0.200000\pi$
$997$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$998$	−0.294524	−	0.184039i	−0.294524	−	0.184039i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2852.1.bn.b.735.3	yes	24
4.3	odd	2		2852.1.bn.a.735.1	yes	24
23.22	odd	2	CM	2852.1.bn.b.735.3	yes	24
31.24	odd	30		2852.1.bn.a.551.1	✓	24
92.91	even	2		2852.1.bn.a.735.1	yes	24
124.55	even	30	inner	2852.1.bn.b.551.3	yes	24
713.551	even	30		2852.1.bn.a.551.1	✓	24
2852.551	odd	30	inner	2852.1.bn.b.551.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2852.1.bn.a.551.1	✓	24	31.24	odd	30
2852.1.bn.a.551.1	✓	24	713.551	even	30
2852.1.bn.a.735.1	yes	24	4.3	odd	2
2852.1.bn.a.735.1	yes	24	92.91	even	2
2852.1.bn.b.551.3	yes	24	124.55	even	30	inner
2852.1.bn.b.551.3	yes	24	2852.551	odd	30	inner
2852.1.bn.b.735.3	yes	24	1.1	even	1	trivial
2852.1.bn.b.735.3	yes	24	23.22	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.939693 + 0.342020i) q^{2} +(0.473271 + 0.100597i) q^{3} +(0.766044 + 0.642788i) q^{4} +(0.410323 + 0.256398i) q^{6} +(0.500000 + 0.866025i) q^{8} +(-0.699680 - 0.311518i) q^{9} +(0.297884 + 0.381274i) q^{12} +(1.23219 + 1.10947i) q^{13} +(0.173648 + 0.984808i) q^{16} +(-0.550939 - 0.532036i) q^{18} +(0.809017 - 0.587785i) q^{23} +(0.149516 + 0.460163i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.778419 + 1.46399i) q^{26} +(-0.691238 - 0.502214i) q^{27} +(-1.70961 - 0.555485i) q^{29} +(0.615661 - 0.788011i) q^{31} +(-0.173648 + 0.984808i) q^{32} +(-0.335746 - 0.688382i) q^{36} +(0.471550 + 0.649033i) q^{39} +(1.88051 - 0.399715i) q^{41} +(0.961262 - 0.275637i) q^{46} +(-1.49889 + 0.487017i) q^{47} +(-0.0168859 + 0.483549i) q^{48} +(-0.669131 + 0.743145i) q^{49} +(-0.766044 + 0.642788i) q^{50} +(0.230759 + 1.64194i) q^{52} +(-0.477784 - 0.708344i) q^{54} +(-1.41652 - 1.10671i) q^{58} +(0.244415 - 1.14988i) q^{59} +(0.848048 - 0.529919i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(0.442013 - 0.196797i) q^{69} +(0.565086 - 1.26920i) q^{71} +(-0.0800578 - 0.761700i) q^{72} +(0.138749 + 0.0145831i) q^{73} +(-0.323755 + 0.359566i) q^{75} +(0.221130 + 0.771172i) q^{78} +(0.235862 + 0.261952i) q^{81} +(1.90381 + 0.267564i) q^{82} +(-0.753227 - 0.434876i) q^{87} +(0.997564 + 0.0697565i) q^{92} +(0.370646 - 0.311009i) q^{93} +(-1.57506 - 0.0550024i) q^{94} +(-0.181251 + 0.448612i) q^{96} +(-0.882948 + 0.469472i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{6} + 12 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{18} + 6 q^{23} - 12 q^{25} + 3 q^{26} + 6 q^{27} - 6 q^{36} + 3 q^{48} - 3 q^{49} + 3 q^{52} - 3 q^{54} + 3 q^{58} - 12 q^{64} - 3 q^{72} + 6 q^{78}+ \cdots + 6 q^{96}+O(q^{100}) \) 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

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