LMFDB - Embedded newform 572.2.bb.a.519.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(51,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 7, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("572.51");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.bb (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.855355656503296000000000000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 12x^{14} + 95x^{12} + 552x^{10} + 1969x^{8} + 27048x^{6} + 228095x^{4} + 1411788x^{2} + 5764801 $ x^16 + 12x^14 + 95x^12 + 552x^10 + 1969x^8 + 27048x^6 + 228095x^4 + 1411788*x^2 + 5764801
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			519.1
Root			$1.64697 + 2.07063i$ of defining polynomial
Character	$\chi$	$=$	572.519
Dual form			572.2.bb.a.259.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} +(-2.59357 + 3.56974i) q^{7} +(-1.66251 - 2.28825i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})$	\(q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} +(-2.59357 + 3.56974i) q^{7} +(-1.66251 - 2.28825i) q^{8} +(0.927051 - 2.85317i) q^{9} +(0.815716 + 3.21475i) q^{11} +(-3.42908 - 1.11418i) q^{13} +(5.04837 - 3.66786i) q^{14} +(1.23607 + 3.80423i) q^{16} +(-1.49874 + 0.486971i) q^{17} +(-2.49376 + 3.43237i) q^{18} +(-5.12416 - 7.05280i) q^{19} +(0.307761 - 4.68031i) q^{22} +(-4.04508 + 2.93893i) q^{25} +(4.12519 + 2.99713i) q^{26} +(-8.39296 + 2.72704i) q^{28} +(-0.665151 + 0.915502i) q^{29} +(3.07286 - 9.45728i) q^{31} -5.65685i q^{32} +2.22862 q^{34} +(4.85410 - 3.52671i) q^{36} +(3.80979 + 11.7253i) q^{38} +(-2.45931 + 6.16050i) q^{44} +(-11.0679 + 8.04129i) q^{47} +(-3.85333 - 11.8593i) q^{49} +(6.72499 - 2.18508i) q^{50} +(-4.23858 - 5.83390i) q^{52} +(-2.11959 + 6.52342i) q^{53} +12.4803 q^{56} +(1.29472 - 0.940666i) q^{58} +(-9.33956 - 6.78558i) q^{59} +(-14.4508 + 4.69537i) q^{61} +(-8.26596 + 11.3771i) q^{62} +(7.78070 + 10.7092i) q^{63} +(-2.47214 + 7.60845i) q^{64} +5.09902 q^{67} +(-2.99749 - 0.973943i) q^{68} +(3.53887 + 10.8915i) q^{71} +(-8.06998 + 2.62210i) q^{72} -17.4355i q^{76} +(-13.5914 - 5.42577i) q^{77} +(-7.28115 - 5.29007i) q^{81} +(-0.934438 + 0.303617i) q^{83} +(6.00000 - 7.21110i) q^{88} +(12.8709 - 9.35124i) q^{91} +(18.4004 - 5.97867i) q^{94} +17.6347i q^{98} +(9.92843 + 0.652860i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) $ 16 * q + 8 * q^4 - 12 * q^9	$ 16 q + 8 q^{4} - 12 q^{9} - 8 q^{14} - 16 q^{16} - 40 q^{17} + 12 q^{22} - 20 q^{25} + 80 q^{29} + 24 q^{36} + 40 q^{38} - 20 q^{49} + 8 q^{53} + 16 q^{56} - 140 q^{62} + 32 q^{64} - 80 q^{68} - 112 q^{77} - 36 q^{81} + 96 q^{88} + 180 q^{94}+O(q^{100}) $ 16 * q + 8 * q^4 - 12 * q^9 - 8 * q^14 - 16 * q^16 - 40 * q^17 + 12 * q^22 - 20 * q^25 + 80 * q^29 + 24 * q^36 + 40 * q^38 - 20 * q^49 + 8 * q^53 + 16 * q^56 - 140 * q^62 + 32 * q^64 - 80 * q^68 - 112 * q^77 - 36 * q^81 + 96 * q^88 + 180 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{10}\right)$

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 $

$2$	−1.34500	−	0.437016i	−0.951057	−	0.309017i
$2$	−1.34500	−	0.437016i	−0.951057	−	0.309017i
$3$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$3$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$4$	1.61803	+	1.17557i	0.809017	+	0.587785i
$5$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$5$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$6$		0			0
$7$	−2.59357	+	3.56974i	−0.980276	+	1.34923i	−0.0435957	+	0.999049i	$0.513881\pi$
$7$	−2.59357	+	3.56974i	−0.980276	+	1.34923i	−0.936680	+	0.350185i	$0.886119\pi$
$8$	−1.66251	−	2.28825i	−0.587785	−	0.809017i
$9$	0.927051	−	2.85317i	0.309017	−	0.951057i
$10$		0			0
$11$	0.815716	+	3.21475i	0.245948	+	0.969283i
$11$	0.815716	+	3.21475i	0.245948	+	0.969283i
$12$		0			0
$13$	−3.42908	−	1.11418i	−0.951057	−	0.309017i
$13$	−3.42908	−	1.11418i	−0.951057	−	0.309017i
$14$	5.04837	−	3.66786i	1.34923	−	0.980276i
$15$		0			0
$16$	1.23607	+	3.80423i	0.309017	+	0.951057i
$17$	−1.49874	+	0.486971i	−0.363499	+	0.118108i	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−1.49874	+	0.486971i	−0.363499	+	0.118108i	0.121572	+	0.992583i	$0.461206\pi$
$18$	−2.49376	+	3.43237i	−0.587785	+	0.809017i
$19$	−5.12416	−	7.05280i	−1.17556	−	1.61802i	−0.590665	−	0.806917i	$-0.701135\pi$
$19$	−5.12416	−	7.05280i	−1.17556	−	1.61802i	−0.584898	−	0.811107i	$-0.698865\pi$
$20$		0			0
$21$		0			0
$22$	0.307761	−	4.68031i	0.0656149	−	0.997845i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−4.04508	+	2.93893i	−0.809017	+	0.587785i
$26$	4.12519	+	2.99713i	0.809017	+	0.587785i
$27$		0			0
$28$	−8.39296	+	2.72704i	−1.58612	+	0.515362i
$29$	−0.665151	+	0.915502i	−0.123515	+	0.170004i	−0.866297	−	0.499530i	$-0.833506\pi$
$29$	−0.665151	+	0.915502i	−0.123515	+	0.170004i	0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$	3.07286	−	9.45728i	0.551901	−	1.69858i	−0.152088	−	0.988367i	$-0.548600\pi$
$31$	3.07286	−	9.45728i	0.551901	−	1.69858i	0.703990	−	0.710210i	$-0.251400\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$	2.22862			0.382205
$35$		0			0
$36$	4.85410	−	3.52671i	0.809017	−	0.587785i
$37$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$37$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$38$	3.80979	+	11.7253i	0.618030	+	1.90210i
$39$		0			0
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−2.45931	+	6.16050i	−0.370755	+	0.928731i
$45$		0			0
$46$		0			0
$47$	−11.0679	+	8.04129i	−1.61442	+	1.17294i	−0.767927	+	0.640537i	$0.778712\pi$
$47$	−11.0679	+	8.04129i	−1.61442	+	1.17294i	−0.846489	+	0.532406i	$0.821288\pi$
$48$		0			0
$49$	−3.85333	−	11.8593i	−0.550475	−	1.69419i
$50$	6.72499	−	2.18508i	0.951057	−	0.309017i
$51$		0			0
$52$	−4.23858	−	5.83390i	−0.587785	−	0.809017i
$53$	−2.11959	+	6.52342i	−0.291148	+	0.896060i	0.693341	+	0.720610i	$0.256138\pi$
$53$	−2.11959	+	6.52342i	−0.291148	+	0.896060i	−0.984488	+	0.175450i	$0.943862\pi$
$54$		0			0
$55$		0			0
$56$	12.4803			1.66775
$57$		0			0
$58$	1.29472	−	0.940666i	0.170004	−	0.123515i
$59$	−9.33956	−	6.78558i	−1.21591	−	0.883408i	−0.220153	−	0.975465i	$-0.570656\pi$
$59$	−9.33956	−	6.78558i	−1.21591	−	0.883408i	−0.995754	+	0.0920575i	$0.970656\pi$
$60$		0			0
$61$	−14.4508	+	4.69537i	−1.85024	+	0.601180i	−0.853447	+	0.521180i	$0.825492\pi$
$61$	−14.4508	+	4.69537i	−1.85024	+	0.601180i	−0.996795	+	0.0799995i	$0.974508\pi$
$62$	−8.26596	+	11.3771i	−1.04978	+	1.44490i
$63$	7.78070	+	10.7092i	0.980276	+	1.34923i
$64$	−2.47214	+	7.60845i	−0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	5.09902			0.622944			0.311472	−	0.950255i	$-0.399178\pi$
$67$	5.09902			0.622944			0.311472	+	0.950255i	$0.399178\pi$
$68$	−2.99749	−	0.973943i	−0.363499	−	0.118108i
$69$		0			0
$70$		0			0
$71$	3.53887	+	10.8915i	0.419987	+	1.29259i	0.907713	+	0.419591i	$0.137826\pi$
$71$	3.53887	+	10.8915i	0.419987	+	1.29259i	−0.487726	+	0.872997i	$0.662174\pi$
$72$	−8.06998	+	2.62210i	−0.951057	+	0.309017i
$73$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$73$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$74$		0			0
$75$		0			0
$76$		−	17.4355i		−	1.99999i
$77$	−13.5914	−	5.42577i	−1.54889	−	0.618324i
$78$		0			0
$79$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$79$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$80$		0			0
$81$	−7.28115	−	5.29007i	−0.809017	−	0.587785i
$82$		0			0
$83$	−0.934438	+	0.303617i	−0.102568	+	0.0333263i	−0.359851	−	0.933010i	$-0.617173\pi$
$83$	−0.934438	+	0.303617i	−0.102568	+	0.0333263i	0.257283	+	0.966336i	$0.417173\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	6.00000	−	7.21110i	0.639602	−	0.768706i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	12.8709	−	9.35124i	1.34923	−	0.980276i
$92$		0			0
$93$		0			0
$94$	18.4004	−	5.97867i	1.89786	−	0.616652i
$95$		0			0
$96$		0			0
$97$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$97$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$98$			17.6347i			1.78137i
$99$	9.92843	+	0.652860i	0.997845	+	0.0656149i
$100$	−10.0000			−1.00000
$101$	16.9017	+	5.49169i	1.68178	+	0.546444i	0.985256	−	0.171087i	$-0.0547281\pi$
$101$	16.9017	+	5.49169i	1.68178	+	0.546444i	0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$103$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$104$	3.15137	+	9.69891i	0.309017	+	0.951057i
$105$		0			0
$106$	5.70167	−	7.84768i	0.553796	−	0.762234i
$107$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$107$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	−16.7859	−	5.45408i	−1.58612	−	0.515362i
$113$	10.5484	−	7.66384i	0.992307	−	0.720954i	0.0318823	−	0.999492i	$-0.489850\pi$
$113$	10.5484	−	7.66384i	0.992307	−	0.720954i	0.960425	+	0.278538i	$0.0898498\pi$
$114$		0			0
$115$		0			0
$116$	−2.15247	+	0.699381i	−0.199852	+	0.0649359i
$117$	−6.35787	+	8.75086i	−0.587785	+	0.809017i
$118$	9.59627	+	13.2081i	0.883408	+	1.21591i
$119$	2.14873	−	6.61312i	0.196974	−	0.606224i
$120$		0			0
$121$	−9.66922	+	5.24464i	−0.879020	+	0.476786i
$122$	21.4883			1.94546
$123$		0			0
$124$	16.0897	−	11.6898i	1.44490	−	1.04978i
$125$		0			0
$126$	−5.78492	−	17.8042i	−0.515362	−	1.58612i
$127$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$127$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$128$	6.65003	−	9.15298i	0.587785	−	0.809017i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	38.4665			3.33547
$134$	−6.85817	−	2.22835i	−0.592455	−	0.192500i
$135$		0			0
$136$	3.60598	+	2.61990i	0.309211	+	0.224655i
$137$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$137$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$		0			0
$142$		−	16.1956i		−	1.35911i
$143$	0.784640	−	11.9325i	0.0656149	−	0.997845i
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$149$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$150$		0			0
$151$	9.67081	+	13.3107i	0.786999	+	1.08321i	0.994475	+	0.104971i	$0.0334751\pi$
$151$	9.67081	+	13.3107i	0.786999	+	1.08321i	−0.207476	+	0.978240i	$0.566525\pi$
$152$	−7.61959	+	23.4507i	−0.618030	+	1.90210i
$153$			4.72762i			0.382205i
$154$	15.9093	+	13.2373i	1.28201	+	1.06669i
$155$		0			0
$156$		0			0
$157$	−0.451627	+	0.328126i	−0.0360438	+	0.0261873i	−0.605661	−	0.795723i	$-0.707091\pi$
$157$	−0.451627	+	0.328126i	−0.0360438	+	0.0261873i	0.569618	+	0.821910i	$0.307091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	7.48128	+	10.2971i	0.587785	+	0.809017i
$163$	2.85019	−	8.77199i	0.223244	−	0.687075i	−0.775221	−	0.631690i	$-0.782361\pi$
$163$	2.85019	−	8.77199i	0.223244	−	0.687075i	0.998465	−	0.0553849i	$-0.0176386\pi$
$164$		0			0
$165$		0			0
$166$	1.38950			0.107846
$167$	17.5166	+	5.69148i	1.35547	+	0.440420i	0.894529	−	0.447010i	$-0.147511\pi$
$167$	17.5166	+	5.69148i	1.35547	+	0.440420i	0.460944	+	0.887429i	$0.347511\pi$
$168$		0			0
$169$	10.5172	+	7.64121i	0.809017	+	0.587785i
$170$		0			0
$171$	−24.8732	+	8.08179i	−1.90210	+	0.618030i
$172$		0			0
$173$	−8.47716	−	11.6678i	−0.644506	−	0.887087i	0.354339	−	0.935117i	$-0.384706\pi$
$173$	−8.47716	−	11.6678i	−0.644506	−	0.887087i	−0.998846	+	0.0480298i	$0.984706\pi$
$174$		0			0
$175$		−	22.0622i		−	1.66775i
$176$	−11.2213	+	7.07682i	−0.845841	+	0.533435i
$177$		0			0
$178$		0			0
$179$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$179$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$180$		0			0
$181$	−7.88573	−	24.2698i	−0.586142	−	1.80396i	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−7.88573	−	24.2698i	−0.586142	−	1.80396i	0.00849335	−	0.999964i	$-0.497296\pi$
$182$	−21.3979	+	6.95261i	−1.58612	+	0.515362i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.78804	−	4.42085i	−0.203882	−	0.323285i
$188$	−27.3613			−1.99553
$189$		0			0
$190$		0			0
$191$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$191$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$192$		0			0
$193$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$193$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$194$		0			0
$195$		0			0
$196$	7.70665	−	23.7186i	0.550475	−	1.69419i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−13.0684	−	5.21698i	−0.928731	−	0.370755i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	13.4500	+	4.37016i	0.951057	+	0.309017i
$201$		0			0
$202$	−20.3328	−	14.7726i	−1.43061	−	1.03940i
$203$	−1.54299	−	4.74883i	−0.108297	−	0.333303i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	14.4222i		−	1.00000i
$209$	18.4931	−	22.2260i	1.27920	−	1.53740i
$210$		0			0
$211$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$211$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$212$	−11.0983	+	8.06339i	−0.762234	+	0.553796i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	25.7904	+	35.4974i	1.75076	+	2.40972i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.68189			0.382205
$222$		0			0
$223$	−20.6260	+	14.9856i	−1.38122	+	1.00351i	−0.384452	+	0.923145i	$0.625609\pi$
$223$	−20.6260	+	14.9856i	−1.38122	+	1.00351i	−0.996765	+	0.0803677i	$0.974391\pi$
$224$	20.1935	+	14.6714i	1.34923	+	0.980276i
$225$	4.63525	+	14.2658i	0.309017	+	0.951057i
$226$	−17.5368	+	5.69804i	−1.16653	+	0.379028i
$227$	−8.24474	+	11.3479i	−0.547223	+	0.753187i	−0.989632	−	0.143626i	$-0.954124\pi$
$227$	−8.24474	+	11.3479i	−0.547223	+	0.753187i	0.442409	+	0.896813i	$0.354124\pi$
$228$		0			0
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$		0			0
$231$		0			0
$232$	3.20071			0.210137
$233$	−16.6033	−	5.39475i	−1.08772	−	0.353422i	−0.290355	−	0.956919i	$-0.593773\pi$
$233$	−16.6033	−	5.39475i	−1.08772	−	0.353422i	−0.797365	+	0.603497i	$0.793773\pi$
$234$	12.3756	−	8.99139i	0.809017	−	0.587785i
$235$		0			0
$236$	−7.13479	−	21.9586i	−0.464435	−	1.42938i
$237$		0			0
$238$	−5.78008	+	7.95559i	−0.374667	+	0.515685i
$239$	13.4319	+	18.4874i	0.868836	+	1.19585i	0.979390	+	0.201980i	$0.0647375\pi$
$239$	13.4319	+	18.4874i	0.868836	+	1.19585i	−0.110554	+	0.993870i	$0.535262\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	15.2971	−	2.82843i	0.983332	−	0.181818i
$243$		0			0
$244$	−28.9017	−	9.39073i	−1.85024	−	0.601180i
$245$		0			0
$246$		0			0
$247$	9.71310	+	29.8939i	0.618030	+	1.90210i
$248$	−26.7492	+	8.69135i	−1.69858	+	0.551901i
$249$		0			0
$250$		0			0
$251$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$			26.4746i			1.66775i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	25.6935	+	18.6674i	1.60272	+	1.16444i	0.882046	+	0.471163i	$0.156166\pi$
$257$	25.6935	+	18.6674i	1.60272	+	1.16444i	0.720670	+	0.693279i	$0.243834\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.99545	+	2.74651i	0.123515	+	0.170004i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$	−51.7373	−	16.8105i	−3.17222	−	1.03072i
$267$		0			0
$268$	8.25039	+	5.99426i	0.503973	+	0.366158i
$269$	9.30979	+	28.6526i	0.567628	+	1.74698i	0.660011	+	0.751256i	$0.270552\pi$
$269$	9.30979	+	28.6526i	0.567628	+	1.74698i	−0.0923827	+	0.995724i	$0.529448\pi$
$270$		0			0
$271$	3.05761	−	4.20844i	0.185736	−	0.255644i	−0.705987	−	0.708225i	$-0.749496\pi$
$271$	3.05761	−	4.20844i	0.185736	−	0.255644i	0.891724	+	0.452580i	$0.149496\pi$
$272$	−3.70510	−	5.09963i	−0.224655	−	0.309211i
$273$		0			0
$274$		0			0
$275$	−12.7475	−	10.6066i	−0.768706	−	0.639602i
$276$		0			0
$277$	−22.8967	−	7.43958i	−1.37573	−	0.447001i	−0.474465	−	0.880274i	$-0.657359\pi$
$277$	−22.8967	−	7.43958i	−1.37573	−	0.447001i	−0.901263	+	0.433273i	$0.857359\pi$
$278$		0			0
$279$	−24.1345	−	17.5348i	−1.44490	−	1.04978i
$280$		0			0
$281$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$281$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$282$		0			0
$283$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$283$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$284$	−7.07775	+	21.7831i	−0.419987	+	1.29259i
$285$		0			0
$286$	−6.27003	+	15.7063i	−0.370755	+	0.928731i
$287$		0			0
$288$	−16.1400	−	5.24419i	−0.951057	−	0.309017i
$289$	−11.7442	+	8.53266i	−0.690835	+	0.501921i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$293$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−7.19021	−	22.1292i	−0.413750	−	1.27339i
$303$		0			0
$304$	20.4966	−	28.2112i	1.17556	−	1.61802i
$305$		0			0
$306$	2.06605	−	6.35863i	0.118108	−	0.363499i
$307$		−	31.6766i		−	1.80788i	−0.427663	−	0.903938i	$-0.640663\pi$
$307$		−	31.6766i		−	1.80788i	0.427663	−	0.903938i	$-0.359337\pi$
$308$	−15.6130	−	24.7568i	−0.889634	−	1.41065i
$309$		0			0
$310$		0			0
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$	−3.68411	−	11.3385i	−0.208238	−	0.640891i	−0.999565	−	0.0294975i	$-0.990609\pi$
$313$	−3.68411	−	11.3385i	−0.208238	−	0.640891i	0.791327	−	0.611393i	$-0.209391\pi$
$314$	0.750834	−	0.243961i	0.0423720	−	0.0137675i
$315$		0			0
$316$		0			0
$317$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$317$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$318$		0			0
$319$	−3.48568	−	1.39150i	−0.195161	−	0.0779093i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.1143	+	8.07503i	0.618417	+	0.449306i
$324$	−5.56231	−	17.1190i	−0.309017	−	0.951057i
$325$	17.1454	−	5.57088i	0.951057	−	0.309017i
$326$	−7.66700	+	10.5527i	−0.424636	+	0.584461i
$327$		0			0
$328$		0			0
$329$		−	60.3651i		−	3.32803i
$330$		0			0
$331$	−33.0326			−1.81564			−0.907819	−	0.419363i	$-0.862253\pi$
$331$	−33.0326			−1.81564			−0.907819	+	0.419363i	$0.862253\pi$
$332$	−1.86888	−	0.607234i	−0.102568	−	0.0333263i
$333$		0			0
$334$	−21.0725	−	15.3100i	−1.15303	−	0.837728i
$335$		0			0
$336$		0			0
$337$	−15.8668	+	21.8387i	−0.864318	+	1.18963i	0.116204	+	0.993225i	$0.462927\pi$
$337$	−15.8668	+	21.8387i	−0.864318	+	1.18963i	−0.980522	+	0.196407i	$0.937073\pi$
$338$	−10.8063	−	14.8736i	−0.587785	−	0.809017i
$339$		0			0
$340$		0			0
$341$	32.9093	+	2.16401i	1.78214	+	0.117188i
$342$	36.9862			1.99999
$343$	22.9532	+	7.45794i	1.23935	+	0.402691i
$344$		0			0
$345$		0			0
$346$	6.30273	+	19.3978i	0.338837	+	1.04283i
$347$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$347$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$	−9.64153	+	29.6736i	−0.515362	+	1.58612i
$351$		0			0
$352$	18.1854	−	4.61439i	0.969283	−	0.245948i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.27919	+	8.64256i	−0.331403	+	0.456137i	−0.941906	−	0.335877i	$-0.890967\pi$
$359$	−6.27919	+	8.64256i	−0.331403	+	0.456137i	0.610503	+	0.792014i	$0.290967\pi$
$360$		0			0
$361$	−17.6137	+	54.2093i	−0.927035	+	2.85312i
$362$			36.0890i			1.89680i
$363$		0			0
$364$	31.8186			1.66775
$365$		0			0
$366$		0			0
$367$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$367$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−17.7896	−	24.4853i	−0.923590	−	1.27121i
$372$		0			0
$373$		−	37.1189i		−	1.92194i	−0.276649	−	0.960971i	$-0.589224\pi$
$373$		−	37.1189i		−	1.92194i	0.276649	−	0.960971i	$-0.410776\pi$
$374$	1.81792	+	7.16446i	0.0940025	+	0.370465i
$375$		0			0
$376$	36.8009	+	11.9573i	1.89786	+	0.616652i
$377$	3.30089	−	2.39824i	0.170004	−	0.123515i
$378$		0			0
$379$	−11.0298	−	33.9462i	−0.566562	−	1.74370i	−0.663264	−	0.748385i	$-0.730829\pi$
$379$	−11.0298	−	33.9462i	−0.566562	−	1.74370i	0.0967022	−	0.995313i	$-0.469171\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.29360	+	19.3697i	−0.321588	+	0.989746i	0.651369	+	0.758761i	$0.274195\pi$
$383$	−6.29360	+	19.3697i	−0.321588	+	0.989746i	−0.972957	+	0.230985i	$0.925805\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.67328	−	4.84842i	−0.338349	−	0.245825i	0.405616	−	0.914044i	$-0.367057\pi$
$389$	−6.67328	−	4.84842i	−0.338349	−	0.245825i	−0.743965	+	0.668219i	$0.767057\pi$
$390$		0			0
$391$		0			0
$392$	−20.7308	+	28.5336i	−1.04707	+	1.44116i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	15.2971	+	12.7279i	0.768706	+	0.639602i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−16.1803	−	11.7557i	−0.809017	−	0.587785i
$401$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$401$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$402$		0			0
$403$	−21.0742	+	29.0061i	−1.04978	+	1.44490i
$404$	20.8916	+	28.7549i	1.03940	+	1.43061i
$405$		0			0
$406$			7.06147i			0.350455i
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	48.4455	−	15.7409i	2.38385	−	0.774559i
$414$		0			0
$415$		0			0
$416$	−6.30273	+	19.3978i	−0.309017	+	0.951057i
$417$		0			0
$418$	−34.5863	+	21.8121i	−1.69167	+	1.06686i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$		0			0
$423$	12.6827	+	39.0332i	0.616652	+	1.89786i
$424$	18.4510	−	5.99510i	0.896060	−	0.291148i
$425$	4.63137	−	6.37454i	0.224655	−	0.309211i
$426$		0			0
$427$	20.7180	−	63.7635i	1.00262	−	3.08573i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	38.5191	+	12.5156i	1.85540	+	0.602857i	0.995763	+	0.0919547i	$0.0293115\pi$
$431$	38.5191	+	12.5156i	1.85540	+	0.602857i	0.859639	+	0.510902i	$0.170689\pi$
$432$		0			0
$433$	14.6935	+	10.6754i	0.706124	+	0.513029i	0.881921	−	0.471397i	$-0.156250\pi$
$433$	14.6935	+	10.6754i	0.706124	+	0.513029i	−0.175797	+	0.984426i	$0.556250\pi$
$434$	−19.1750	−	59.0147i	−0.920431	−	2.83279i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−37.4089			−1.78137
$442$	−7.64213	−	2.48308i	−0.363499	−	0.118108i
$443$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$443$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$444$		0			0
$445$		0			0
$446$	34.2908	−	11.1418i	1.62372	−	0.527578i
$447$		0			0
$448$	−20.7485	−	28.5579i	−0.980276	−	1.34923i
$449$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$449$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$450$		−	21.2132i		−	1.00000i
$451$		0			0
$452$	26.0770			1.22656
$453$		0			0
$454$	16.0484	−	11.6598i	0.753187	−	0.547223i
$455$		0			0
$456$		0			0
$457$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$457$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−14.7450			−0.685260			−0.342630	−	0.939470i	$-0.611318\pi$
$463$	−14.7450			−0.685260			−0.342630	+	0.939470i	$0.611318\pi$
$464$	−4.30495	−	1.39876i	−0.199852	−	0.0649359i
$465$		0			0
$466$	19.9738	+	14.5118i	0.925270	+	0.672248i
$467$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$467$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$468$	−20.5745	+	6.68506i	−0.951057	+	0.309017i
$469$	−13.2246	+	18.2022i	−0.610658	+	0.840498i
$470$		0			0
$471$		0			0
$472$			32.6523i			1.50294i
$473$		0			0
$474$		0			0
$475$	41.4553	+	13.4697i	1.90210	+	0.618030i
$476$	11.2509	−	8.17426i	0.515685	−	0.374667i
$477$	16.6474	+	12.0951i	0.762234	+	0.553796i
$478$	−9.98655	−	30.7354i	−0.456774	−	1.40581i
$479$	23.0043	−	7.47456i	1.05109	−	0.341521i	0.267997	−	0.963420i	$-0.413638\pi$
$479$	23.0043	−	7.47456i	1.05109	−	0.341521i	0.783098	+	0.621899i	$0.213638\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−21.8106	−	2.88083i	−0.991389	−	0.130947i
$485$		0			0
$486$		0			0
$487$	−35.0066	+	25.4338i	−1.58630	+	1.15252i	−0.677315	+	0.735693i	$0.736856\pi$
$487$	−35.0066	+	25.4338i	−1.58630	+	1.15252i	−0.908988	+	0.416823i	$0.863144\pi$
$488$	34.7688	+	25.2610i	1.57391	+	1.14351i
$489$		0			0
$490$		0			0
$491$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$491$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$492$		0			0
$493$	0.551068	−	1.69601i	0.0248188	−	0.0763846i
$494$		−	44.4519i		−	1.99999i
$495$		0			0
$496$	39.7759			1.78599
$497$	−48.0582	−	15.6151i	−2.15571	−	0.700431i
$498$		0			0
$499$	29.5977	+	21.5040i	1.32497	+	0.962650i	0.999856	+	0.0169761i	$0.00540392\pi$
$499$	29.5977	+	21.5040i	1.32497	+	0.962650i	0.325118	+	0.945674i	$0.394596\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$503$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$504$	11.5698	−	35.6083i	0.515362	−	1.58612i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$	21.5200	−	6.99226i	0.951057	−	0.309017i
$513$		0			0
$514$	−26.3997	−	36.3361i	−1.16444	−	1.60272i
$515$		0			0
$516$		0			0
$517$	−34.8790	−	29.0211i	−1.53398	−	1.27634i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−35.4717	−	25.7717i	−1.55404	−	1.12908i	−0.940690	−	0.339267i	$-0.889821\pi$
$521$	−35.4717	−	25.7717i	−1.55404	−	1.12908i	−0.613351	−	0.789810i	$-0.710179\pi$
$522$	−1.48361	−	4.56609i	−0.0649359	−	0.199852i
$523$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$523$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			15.6704i			0.682615i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−28.0187	+	20.3568i	−1.21591	+	0.883408i
$532$	62.2401	+	45.2201i	2.69845	+	1.96054i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−8.47716	−	11.6678i	−0.366158	−	0.503973i
$537$		0			0
$538$		−	42.6062i		−	1.83688i
$539$	34.9815	−	22.0613i	1.50676	−	0.950248i
$540$		0			0
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$	−5.95163	+	4.32411i	−0.255644	+	0.185736i
$543$		0			0
$544$	2.75473	+	8.47818i	0.118108	+	0.363499i
$545$		0			0
$546$		0			0
$547$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$547$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$548$		0			0
$549$			45.5836i			1.94546i
$550$	12.5102	+	19.8367i	0.533435	+	0.845841i
$551$	9.86519			0.420271
$552$		0			0
$553$		0			0
$554$	27.5447	+	20.0124i	1.17026	+	0.850247i
$555$		0			0
$556$		0			0
$557$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$557$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$558$	24.7979	+	34.1314i	1.04978	+	1.44490i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$563$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	37.7683	−	12.2717i	1.58612	−	0.515362i
$568$	19.0391	−	26.2051i	0.798863	−	1.09954i
$569$	−16.0035	−	22.0269i	−0.670900	−	0.923415i	0.328880	−	0.944372i	$-0.393329\pi$
$569$	−16.0035	−	22.0269i	−0.670900	−	0.923415i	−0.999780	+	0.0209564i	$0.993329\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$	15.2971	−	18.3848i	0.639602	−	0.768706i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	19.4164	+	14.1068i	0.809017	+	0.587785i
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$	19.5248	−	6.34400i	0.812125	−	0.263876i
$579$		0			0
$580$		0			0
$581$	1.33969	−	4.12315i	0.0555798	−	0.171057i
$582$		0			0
$583$	−22.7001	−	1.49268i	−0.940143	−	0.0618206i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−37.1267	−	26.9742i	−1.53238	−	1.11334i	−0.954895	−	0.296945i	$-0.904032\pi$
$587$	−37.1267	−	26.9742i	−1.53238	−	1.11334i	−0.577490	−	0.816398i	$-0.695968\pi$
$588$		0			0
$589$	−82.4461	+	26.7884i	−3.39713	+	1.10380i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$599$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$600$		0			0
$601$	−12.7157	+	17.5017i	−0.518686	+	0.713910i	−0.985354	−	0.170523i	$-0.945454\pi$
$601$	−12.7157	+	17.5017i	−0.518686	+	0.713910i	0.466668	+	0.884433i	$0.345454\pi$
$602$		0			0
$603$	4.72705	−	14.5484i	0.192500	−	0.592455i
$604$			32.9059i			1.33892i
$605$		0			0
$606$		0			0
$607$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$607$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$608$	−39.8967	+	28.9866i	−1.61802	+	1.17556i
$609$		0			0
$610$		0			0
$611$	46.9121	−	15.2427i	1.89786	−	0.616652i
$612$	−5.55765	+	7.64945i	−0.224655	+	0.309211i
$613$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$613$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$614$	−13.8432	+	42.6049i	−0.558664	+	1.71939i
$615$		0			0
$616$	10.1804	+	40.1209i	0.410178	+	1.61652i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	36.8096	−	26.7438i	1.47950	−	1.07492i	0.501788	−	0.864991i	$-0.332676\pi$
$619$	36.8096	−	26.7438i	1.47950	−	1.07492i	0.977716	−	0.209932i	$-0.0673243\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.72542	−	23.7764i	0.309017	−	0.951057i
$626$			16.8603i			0.673873i
$627$		0			0
$628$	−1.11648			−0.0445525
$629$		0			0
$630$		0			0
$631$	−18.2799	−	13.2811i	−0.727709	−	0.528712i	0.161129	−	0.986933i	$-0.448487\pi$
$631$	−18.2799	−	13.2811i	−0.727709	−	0.528712i	−0.888838	+	0.458222i	$0.848487\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			44.9599i			1.78137i
$638$	4.08012	+	3.39487i	0.161534	+	0.134404i
$639$	34.3561			1.35911
$640$		0			0
$641$	26.6216	−	19.3417i	1.05149	−	0.763953i	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	26.6216	−	19.3417i	1.05149	−	0.763953i	0.972495	+	0.232922i	$0.0748288\pi$
$642$		0			0
$643$	−1.58924	−	4.89119i	−0.0626737	−	0.192890i	0.914817	−	0.403869i	$-0.132335\pi$
$643$	−1.58924	−	4.89119i	−0.0626737	−	0.192890i	−0.977491	+	0.210979i	$0.932335\pi$
$644$		0			0
$645$		0			0
$646$	−11.4198	−	15.7180i	−0.449306	−	0.618417i
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$			25.4558i			1.00000i
$649$	14.1955	−	35.5594i	0.557223	−	1.39583i
$650$	−25.4951			−1.00000
$651$		0			0
$652$	14.9238	−	10.8428i	0.584461	−	0.424636i
$653$	−40.8386	−	29.6710i	−1.59814	−	1.16112i	−0.890962	−	0.454079i	$-0.849969\pi$
$653$	−40.8386	−	29.6710i	−1.59814	−	1.16112i	−0.707177	−	0.707037i	$-0.750031\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−26.3805	+	81.1908i	−1.02842	+	3.16515i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	44.4288	+	14.4358i	1.72677	+	0.561063i
$663$		0			0
$664$	2.24826	+	1.63346i	0.0872494	+	0.0633904i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	21.6517	+	29.8010i	0.837728	+	1.15303i
$669$		0			0
$670$		0			0
$671$	−26.8822	−	42.6258i	−1.03778	−	1.64555i
$672$		0			0
$673$	32.1304	+	10.4398i	1.23854	+	0.402425i	0.853800	−	0.520602i	$-0.174292\pi$
$673$	32.1304	+	10.4398i	1.23854	+	0.402425i	0.384736	+	0.923026i	$0.374292\pi$
$674$	30.8846	−	22.4390i	1.18963	−	0.864318i
$675$		0			0
$676$	8.03444	+	24.7275i	0.309017	+	0.951057i
$677$	−39.0476	+	12.6873i	−1.50072	+	0.487614i	−0.940228	−	0.340545i	$-0.889388\pi$
$677$	−39.0476	+	12.6873i	−1.50072	+	0.487614i	−0.560493	+	0.828159i	$0.689388\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−43.3173	−	17.2925i	−1.65870	−	0.662164i
$683$	−51.3202			−1.96371			−0.981857	−	0.189625i	$-0.939273\pi$
$683$	−51.3202			−1.96371			−0.981857	+	0.189625i	$0.939273\pi$
$684$	−49.7464	−	16.1636i	−1.90210	−	0.618030i
$685$		0			0
$686$	−27.6127	−	20.0618i	−1.05426	−	0.765963i
$687$		0			0
$688$		0			0
$689$	14.5365	−	20.0077i	0.553796	−	0.762234i
$690$		0			0
$691$	−8.22949	+	25.3278i	−0.313065	+	0.963514i	0.663479	+	0.748195i	$0.269079\pi$
$691$	−8.22949	+	25.3278i	−0.313065	+	0.963514i	−0.976544	+	0.215319i	$0.930921\pi$
$692$		−	28.8444i		−	1.09650i
$693$	−28.0806	+	33.7487i	−1.06669	+	1.28201i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	25.9357	−	35.6974i	0.980276	−	1.34923i
$701$	−18.1128	−	24.9302i	−0.684112	−	0.941599i	0.315862	−	0.948805i	$-0.397706\pi$
$701$	−18.1128	−	24.9302i	−0.684112	−	0.941599i	−0.999974	+	0.00720582i	$0.997706\pi$
$702$		0			0
$703$		0			0
$704$	−26.4758	−	1.74096i	−0.997845	−	0.0656149i
$705$		0			0
$706$		0			0
$707$	−63.4396	+	46.0916i	−2.38589	+	1.73345i
$708$		0			0
$709$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$709$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	12.2224	−	8.88011i	0.456137	−	0.331403i
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$		0			0
$722$	47.3807	−	65.2139i	1.76333	−	2.42701i
$723$		0			0
$724$	15.7715	−	48.5396i	0.586142	−	1.80396i
$725$		−	5.65811i		−	0.210137i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	−42.7959	−	13.9052i	−1.58612	−	0.515362i
$729$	−21.8435	+	15.8702i	−0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$733$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.15935	+	16.3921i	0.153212	+	0.603809i
$738$		0			0
$739$	20.2316	+	6.57363i	0.744230	+	0.241815i	0.656497	−	0.754329i	$-0.272038\pi$
$739$	20.2316	+	6.57363i	0.744230	+	0.241815i	0.0877335	+	0.996144i	$0.472038\pi$
$740$		0			0
$741$		0			0
$742$	13.2265	+	40.7070i	0.485560	+	1.49440i
$743$	−48.8120	+	15.8600i	−1.79074	+	0.581846i	−0.999558	−	0.0297385i	$-0.990533\pi$
$743$	−48.8120	+	15.8600i	−1.79074	+	0.581846i	−0.791179	+	0.611584i	$0.790533\pi$
$744$		0			0
$745$		0			0
$746$	−16.2215	+	49.9248i	−0.593913	+	1.82788i
$747$			2.94758i			0.107846i
$748$	0.685883	−	10.4306i	0.0250784	−	0.381382i
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$	−44.2715	−	32.1652i	−1.61442	−	1.17294i
$753$		0			0
$754$	−5.48775	+	1.78308i	−0.199852	+	0.0649359i
$755$		0			0
$756$		0			0
$757$	9.46903	−	29.1427i	0.344158	−	1.05921i	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	9.46903	−	29.1427i	0.344158	−	1.05921i	0.962033	−	0.272932i	$-0.0879936\pi$
$758$			50.4777i			1.83343i
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	16.9297	−	23.3018i	0.611697	−	0.841928i
$767$	24.4658	+	33.6742i	0.883408	+	1.21591i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$773$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$774$		0			0
$775$	15.3643	+	47.2864i	0.551901	+	1.69858i
$776$		0			0
$777$		0			0
$778$	6.85670	+	9.43744i	0.245825	+	0.338349i
$779$		0			0
$780$		0			0
$781$	−32.1268	+	20.2610i	−1.14959	+	0.724995i
$782$		0			0
$783$		0			0
$784$	40.3526	−	29.3178i	1.44116	−	1.04707i
$785$		0			0
$786$		0			0
$787$	−1.34500	+	0.437016i	−0.0479440	+	0.0155779i	−0.332891	−	0.942965i	$-0.608024\pi$
$787$	−1.34500	+	0.437016i	−0.0479440	+	0.0155779i	0.284947	+	0.958543i	$0.408024\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			57.5316i			2.04559i
$792$	−15.0122	−	23.8041i	−0.533435	−	0.845841i
$793$	54.7846			1.94546
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.6902	−	39.0564i	−0.449510	−	1.38345i	−0.877461	−	0.479648i	$-0.840764\pi$
$797$	−12.6902	−	39.0564i	−0.449510	−	1.38345i	0.427951	−	0.903802i	$-0.359236\pi$
$798$		0			0
$799$	12.6720	−	17.4416i	0.448305	−	0.617039i
$800$	16.6251	+	22.8825i	0.587785	+	0.809017i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	41.0208	−	29.8034i	1.44490	−	1.04978i
$807$		0			0
$808$	−15.5329	−	47.8052i	−0.546444	−	1.68178i
$809$	34.2908	−	11.1418i	1.20560	−	0.391724i	0.363783	−	0.931484i	$-0.381485\pi$
$809$	34.2908	−	11.1418i	1.20560	−	0.391724i	0.841819	+	0.539760i	$0.181485\pi$
$810$		0			0
$811$	5.81878	+	8.00886i	0.204325	+	0.281229i	0.898866	−	0.438224i	$-0.144392\pi$
$811$	5.81878	+	8.00886i	0.204325	+	0.281229i	−0.694541	+	0.719453i	$0.744392\pi$
$812$	3.08598	−	9.49766i	0.108297	−	0.333303i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−14.7487	−	45.3919i	−0.515362	−	1.58612i
$820$		0			0
$821$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$821$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$822$		0			0
$823$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$823$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$824$		0			0
$825$		0			0
$826$	−72.0381			−2.50653
$827$	12.1050	+	3.93314i	0.420931	+	0.136769i	0.511821	−	0.859092i	$-0.328971\pi$
$827$	12.1050	+	3.93314i	0.420931	+	0.136769i	−0.0908898	+	0.995861i	$0.528971\pi$
$828$		0			0
$829$	45.4458	+	33.0183i	1.57840	+	1.14677i	0.918501	+	0.395418i	$0.129400\pi$
$829$	45.4458	+	33.0183i	1.57840	+	1.14677i	0.659897	+	0.751356i	$0.270600\pi$
$830$		0			0
$831$		0			0
$832$	16.9543	−	23.3356i	0.587785	−	0.809017i
$833$	11.5503	+	15.8976i	0.400194	+	0.550820i
$834$		0			0
$835$		0			0
$836$	56.0507	−	14.2224i	1.93855	−	0.491892i
$837$		0			0
$838$		0			0
$839$	−13.2237	+	9.60758i	−0.456533	+	0.331691i	−0.792170	−	0.610301i	$-0.791049\pi$
$839$	−13.2237	+	9.60758i	−0.456533	+	0.331691i	0.335637	+	0.941991i	$0.391049\pi$
$840$		0			0
$841$	8.56578	+	26.3627i	0.295372	+	0.909060i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		−	58.0421i		−	1.99553i
$847$	6.35575	−	48.1189i	0.218386	−	1.65339i
$848$	−27.4365			−0.942173
$849$		0			0
$850$	−9.01496	+	6.54975i	−0.309211	+	0.224655i
$851$		0			0
$852$		0			0
$853$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$853$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$854$	−55.7313	+	76.7076i	−1.90709	+	2.62488i
$855$		0			0
$856$		0			0
$857$			8.31626i			0.284078i	0.989861	+	0.142039i	$0.0453659\pi$
$857$			8.31626i			0.284078i	−0.989861	+	0.142039i	$0.954634\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−46.3386	−	33.6670i	−1.57830	−	1.14670i
$863$	−11.0298	−	33.9462i	−0.375458	−	1.15554i	−0.943169	−	0.332314i	$-0.892171\pi$
$863$	−11.0298	−	33.9462i	−0.375458	−	1.15554i	0.567711	−	0.823228i	$-0.307829\pi$
$864$		0			0
$865$		0			0
$866$	−15.0974	−	20.7797i	−0.513029	−	0.706124i
$867$		0			0
$868$			87.7543i			2.97858i
$869$		0			0
$870$		0			0
$871$	−17.4850	−	5.68121i	−0.592455	−	0.192500i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$877$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−45.2349			−1.52400			−0.762001	−	0.647576i	$-0.775783\pi$
$881$	−45.2349			−1.52400			−0.762001	+	0.647576i	$0.775783\pi$
$882$	50.3148	+	16.3483i	1.69419	+	0.550475i
$883$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$883$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$884$	9.19349	+	6.67946i	0.309211	+	0.224655i
$885$		0			0
$886$		0			0
$887$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$887$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	11.0669	−	27.7223i	0.370755	−	0.928731i
$892$	−50.9902			−1.70728
$893$	113.427	+	36.8547i	3.79570	+	1.23330i
$894$		0			0
$895$		0			0
$896$	15.4265	+	47.4777i	0.515362	+	1.58612i
$897$		0			0
$898$		0			0
$899$	6.61424	+	9.10372i	0.220597	+	0.303626i
$900$	−9.27051	+	28.5317i	−0.309017	+	0.951057i
$901$		−	10.8091i		−	0.360104i
$902$		0			0
$903$		0			0
$904$	−35.0735	−	11.3961i	−1.16653	−	0.379028i
$905$		0			0
$906$		0			0
$907$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$907$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$908$	−26.6805	+	8.66903i	−0.885425	+	0.287692i
$909$	31.3375	−	43.1323i	1.03940	−	1.43061i
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$	−1.73829	−	2.75632i	−0.0575289	−	0.0912208i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	41.2909i		−	1.35911i
$924$		0			0
$925$		0			0
$926$	19.8320	+	6.44382i	0.651721	+	0.211757i
$927$		0			0
$928$	5.17886	+	3.76266i	0.170004	+	0.123515i
$929$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$929$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$	−63.8964	+	87.9458i	−2.09412	+	2.88231i
$932$	−20.5228	−	28.2473i	−0.672248	−	0.925270i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	30.5941			1.00000
$937$	7.99343	+	2.59722i	0.261134	+	0.0848475i	0.436658	−	0.899628i	$-0.356162\pi$
$937$	7.99343	+	2.59722i	0.261134	+	0.0848475i	−0.175524	+	0.984475i	$0.556162\pi$
$938$	25.7418	−	18.7025i	0.840498	−	0.610658i
$939$		0			0
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$	14.2696	−	43.9172i	0.464435	−	1.42938i
$945$		0			0
$946$		0			0
$947$	−24.8238			−0.806664			−0.403332	−	0.915054i	$-0.632148\pi$
$947$	−24.8238			−0.806664			−0.403332	+	0.915054i	$0.632148\pi$
$948$		0			0
$949$		0			0
$950$	−49.8708	−	36.2333i	−1.61802	−	1.17556i
$951$		0			0
$952$	−18.7047	+	6.07753i	−0.606224	+	0.196974i
$953$	33.9086	−	46.6712i	1.09841	−	1.51183i	0.260931	−	0.965357i	$-0.415970\pi$
$953$	33.9086	−	46.6712i	1.09841	−	1.51183i	0.837477	−	0.546472i	$-0.184030\pi$
$954$	−17.1050	−	23.5430i	−0.553796	−	0.762234i
$955$		0			0
$956$			45.7033i			1.47815i
$957$		0			0
$958$	−34.2072			−1.10519
$959$		0			0
$960$		0			0
$961$	−54.9181	−	39.9004i	−1.77155	−	1.28711i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	26.8701i		−	0.864083i	−0.901854	−	0.432041i	$-0.857793\pi$
$967$		−	26.8701i		−	0.864083i	0.901854	−	0.432041i	$-0.142207\pi$
$968$	28.0762	+	13.4063i	0.902402	+	0.430894i
$969$		0			0
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$		0			0
$973$		0			0
$974$	58.1988	−	18.9099i	1.86481	−	0.605914i
$975$		0			0
$976$	−35.7245	−	49.1705i	−1.14351	−	1.57391i
$977$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$977$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−38.9295	−	28.2839i	−1.24166	−	0.902117i	−0.243950	−	0.969788i	$-0.578443\pi$
$983$	−38.9295	−	28.2839i	−1.24166	−	0.902117i	−0.997708	+	0.0676705i	$0.978443\pi$
$984$		0			0
$985$		0			0
$986$	−1.48237	+	2.04031i	−0.0472083	+	0.0649766i
$987$		0			0
$988$	−19.4262	+	59.7877i	−0.618030	+	1.90210i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−53.4984	−	17.3827i	−1.69858	−	0.551901i
$993$		0			0
$994$	57.8141	+	42.0044i	1.83375	+	1.33230i
$995$		0			0
$996$		0			0
$997$	−17.8560	+	24.5767i	−0.565506	+	0.778352i	−0.992013	−	0.126132i	$-0.959744\pi$
$997$	−17.8560	+	24.5767i	−0.565506	+	0.778352i	0.426508	+	0.904484i	$0.359744\pi$
$998$	−30.4112	−	41.8574i	−0.962650	−	1.32497i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bb.a.519.1	yes	16
4.3	odd	2	inner	572.2.bb.a.519.4	yes	16
11.6	odd	10	inner	572.2.bb.a.259.1	✓	16
13.12	even	2	inner	572.2.bb.a.519.4	yes	16
44.39	even	10	inner	572.2.bb.a.259.4	yes	16
52.51	odd	2	CM	572.2.bb.a.519.1	yes	16
143.116	odd	10	inner	572.2.bb.a.259.4	yes	16
572.259	even	10	inner	572.2.bb.a.259.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bb.a.259.1	✓	16	11.6	odd	10	inner
572.2.bb.a.259.1	✓	16	572.259	even	10	inner
572.2.bb.a.259.4	yes	16	44.39	even	10	inner
572.2.bb.a.259.4	yes	16	143.116	odd	10	inner
572.2.bb.a.519.1	yes	16	1.1	even	1	trivial
572.2.bb.a.519.1	yes	16	52.51	odd	2	CM
572.2.bb.a.519.4	yes	16	4.3	odd	2	inner
572.2.bb.a.519.4	yes	16	13.12	even	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bb (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} +(-2.59357 + 3.56974i) q^{7} +(-1.66251 - 2.28825i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\)	\(q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} +(-2.59357 + 3.56974i) q^{7} +(-1.66251 - 2.28825i) q^{8} +(0.927051 - 2.85317i) q^{9} +(0.815716 + 3.21475i) q^{11} +(-3.42908 - 1.11418i) q^{13} +(5.04837 - 3.66786i) q^{14} +(1.23607 + 3.80423i) q^{16} +(-1.49874 + 0.486971i) q^{17} +(-2.49376 + 3.43237i) q^{18} +(-5.12416 - 7.05280i) q^{19} +(0.307761 - 4.68031i) q^{22} +(-4.04508 + 2.93893i) q^{25} +(4.12519 + 2.99713i) q^{26} +(-8.39296 + 2.72704i) q^{28} +(-0.665151 + 0.915502i) q^{29} +(3.07286 - 9.45728i) q^{31} -5.65685i q^{32} +2.22862 q^{34} +(4.85410 - 3.52671i) q^{36} +(3.80979 + 11.7253i) q^{38} +(-2.45931 + 6.16050i) q^{44} +(-11.0679 + 8.04129i) q^{47} +(-3.85333 - 11.8593i) q^{49} +(6.72499 - 2.18508i) q^{50} +(-4.23858 - 5.83390i) q^{52} +(-2.11959 + 6.52342i) q^{53} +12.4803 q^{56} +(1.29472 - 0.940666i) q^{58} +(-9.33956 - 6.78558i) q^{59} +(-14.4508 + 4.69537i) q^{61} +(-8.26596 + 11.3771i) q^{62} +(7.78070 + 10.7092i) q^{63} +(-2.47214 + 7.60845i) q^{64} +5.09902 q^{67} +(-2.99749 - 0.973943i) q^{68} +(3.53887 + 10.8915i) q^{71} +(-8.06998 + 2.62210i) q^{72} -17.4355i q^{76} +(-13.5914 - 5.42577i) q^{77} +(-7.28115 - 5.29007i) q^{81} +(-0.934438 + 0.303617i) q^{83} +(6.00000 - 7.21110i) q^{88} +(12.8709 - 9.35124i) q^{91} +(18.4004 - 5.97867i) q^{94} +17.6347i q^{98} +(9.92843 + 0.652860i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 16 * q + 8 * q^4 - 12 * q^9	\( 16 q + 8 q^{4} - 12 q^{9} - 8 q^{14} - 16 q^{16} - 40 q^{17} + 12 q^{22} - 20 q^{25} + 80 q^{29} + 24 q^{36} + 40 q^{38} - 20 q^{49} + 8 q^{53} + 16 q^{56} - 140 q^{62} + 32 q^{64} - 80 q^{68} - 112 q^{77} - 36 q^{81} + 96 q^{88} + 180 q^{94}+O(q^{100}) \) 16 * q + 8 * q^4 - 12 * q^9 - 8 * q^14 - 16 * q^16 - 40 * q^17 + 12 * q^22 - 20 * q^25 + 80 * q^29 + 24 * q^36 + 40 * q^38 - 20 * q^49 + 8 * q^53 + 16 * q^56 - 140 * q^62 + 32 * q^64 - 80 * q^68 - 112 * q^77 - 36 * q^81 + 96 * q^88 + 180 * q^94

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