LMFDB - Embedded newform 2775.1.bg.d.1331.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2775,1,Mod(221,2775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2775, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 6, 5]))

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

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

chi := DirichletCharacter("2775.221");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2775 = 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2775.bg (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:	$1.38490541006$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			1331.3
Root			$-0.891007 + 0.453990i$ of defining polynomial
Character	$\chi$	$=$	2775.1331
Dual form			2775.1.bg.d.221.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0966818 + 0.297556i) q^{2} +(0.809017 - 0.587785i) q^{3} +(0.729825 - 0.530249i) q^{4} +(-0.453990 - 0.891007i) q^{5} +(0.253116 + 0.183900i) q^{6} +(0.481456 + 0.349798i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})$	\(q+(0.0966818 + 0.297556i) q^{2} +(0.809017 - 0.587785i) q^{3} +(0.729825 - 0.530249i) q^{4} +(-0.453990 - 0.891007i) q^{5} +(0.253116 + 0.183900i) q^{6} +(0.481456 + 0.349798i) q^{8} +(0.309017 - 0.951057i) q^{9} +(0.221232 - 0.221232i) q^{10} +(0.278768 - 0.857960i) q^{12} +(-0.891007 - 0.453990i) q^{15} +(0.221232 - 0.680881i) q^{16} +(1.14412 + 0.831254i) q^{17} +0.312869 q^{18} +(-0.803789 - 0.409551i) q^{20} +(-0.280582 - 0.863541i) q^{23} +0.595112 q^{24} +(-0.587785 + 0.809017i) q^{25} +(-0.309017 - 0.951057i) q^{27} +(-1.59811 + 1.16110i) q^{29} +(0.0489435 - 0.309017i) q^{30} +0.819101 q^{32} +(-0.136729 + 0.420808i) q^{34} +(-0.278768 - 0.857960i) q^{36} +(-0.309017 + 0.951057i) q^{37} +(0.0930960 - 0.587785i) q^{40} +(-0.987688 + 0.156434i) q^{45} +(0.229825 - 0.166977i) q^{46} +(-0.221232 - 0.680881i) q^{48} -1.00000 q^{49} +(-0.297556 - 0.0966818i) q^{50} +1.41421 q^{51} +(0.253116 - 0.183900i) q^{54} +(-0.500000 - 0.363271i) q^{58} +(0.0966818 - 0.297556i) q^{59} +(-0.891007 + 0.141122i) q^{60} +(-0.142040 - 0.437153i) q^{64} +(1.53884 + 1.11803i) q^{67} +1.27578 q^{68} +(-0.734572 - 0.533698i) q^{69} +(0.481456 - 0.349798i) q^{72} +(0.363271 + 1.11803i) q^{73} -0.312869 q^{74} +1.00000i q^{75} +(-0.707107 + 0.111995i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(0.221232 - 1.39680i) q^{85} +(-0.610425 + 1.87869i) q^{87} +(-0.280582 - 0.863541i) q^{89} +(-0.142040 - 0.278768i) q^{90} +(-0.662667 - 0.481456i) q^{92} +(0.662667 - 0.481456i) q^{96} +(-0.0966818 - 0.297556i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) $ 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9	$ 16 q + 4 q^{3} - 4 q^{4} - 4 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{16} + 4 q^{27} + 16 q^{30} - 8 q^{34} - 4 q^{36} + 4 q^{37} - 20 q^{40} - 12 q^{46} - 4 q^{48} - 16 q^{49} - 8 q^{58} + 4 q^{64} - 4 q^{81} + 4 q^{85} + 4 q^{90}+O(q^{100}) $ 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^16 + 4 * q^27 + 16 * q^30 - 8 * q^34 - 4 * q^36 + 4 * q^37 - 20 * q^40 - 12 * q^46 - 4 * q^48 - 16 * q^49 - 8 * q^58 + 4 * q^64 - 4 * q^81 + 4 * q^85 + 4 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times$.

$n$	$76$	$926$	$1777$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{5}\right)$

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 $

$2$	0.0966818	+	0.297556i	0.0966818	+	0.297556i	0.987688	−	0.156434i	$-0.0500000\pi$
$2$	0.0966818	+	0.297556i	0.0966818	+	0.297556i	−0.891007	+	0.453990i	$0.850000\pi$
$3$	0.809017	−	0.587785i	0.809017	−	0.587785i
$3$	0.809017	−	0.587785i	0.809017	−	0.587785i
$4$	0.729825	−	0.530249i	0.729825	−	0.530249i
$5$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$5$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$6$	0.253116	+	0.183900i	0.253116	+	0.183900i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	0.481456	+	0.349798i	0.481456	+	0.349798i
$9$	0.309017	−	0.951057i	0.309017	−	0.951057i
$10$	0.221232	−	0.221232i	0.221232	−	0.221232i
$11$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$11$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$12$	0.278768	−	0.857960i	0.278768	−	0.857960i
$13$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$13$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$14$		0			0
$15$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$16$	0.221232	−	0.680881i	0.221232	−	0.680881i
$17$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$17$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$18$	0.312869			0.312869
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$	−0.803789	−	0.409551i	−0.803789	−	0.409551i
$21$		0			0
$22$		0			0
$23$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	−0.987688	−	0.156434i	$-0.950000\pi$
$23$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$24$	0.595112			0.595112
$25$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$26$		0			0
$27$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$28$		0			0
$29$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.707107	+	0.707107i	$0.750000\pi$
$29$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$30$	0.0489435	−	0.309017i	0.0489435	−	0.309017i
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$	0.819101			0.819101
$33$		0			0
$34$	−0.136729	+	0.420808i	−0.136729	+	0.420808i
$35$		0			0
$36$	−0.278768	−	0.857960i	−0.278768	−	0.857960i
$37$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$37$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$38$		0			0
$39$		0			0
$40$	0.0930960	−	0.587785i	0.0930960	−	0.587785i
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$46$	0.229825	−	0.166977i	0.229825	−	0.166977i
$47$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$47$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$48$	−0.221232	−	0.680881i	−0.221232	−	0.680881i
$49$	−1.00000			−1.00000
$50$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i
$51$	1.41421			1.41421
$52$		0			0
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$	0.253116	−	0.183900i	0.253116	−	0.183900i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$59$	0.0966818	−	0.297556i	0.0966818	−	0.297556i	−0.891007	−	0.453990i	$-0.850000\pi$
$59$	0.0966818	−	0.297556i	0.0966818	−	0.297556i	0.987688	+	0.156434i	$0.0500000\pi$
$60$	−0.891007	+	0.141122i	−0.891007	+	0.141122i
$61$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$61$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$62$		0			0
$63$		0			0
$64$	−0.142040	−	0.437153i	−0.142040	−	0.437153i
$65$		0			0
$66$		0			0
$67$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$67$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$68$	1.27578			1.27578
$69$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$	0.481456	−	0.349798i	0.481456	−	0.349798i
$73$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$73$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$74$	−0.312869			−0.312869
$75$			1.00000i			1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$	−0.707107	+	0.111995i	−0.707107	+	0.111995i
$81$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$		0			0
$85$	0.221232	−	1.39680i	0.221232	−	1.39680i
$86$		0			0
$87$	−0.610425	+	1.87869i	−0.610425	+	1.87869i
$88$		0			0
$89$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	−0.987688	−	0.156434i	$-0.950000\pi$
$89$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$90$	−0.142040	−	0.278768i	−0.142040	−	0.278768i
$91$		0			0
$92$	−0.662667	−	0.481456i	−0.662667	−	0.481456i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.662667	−	0.481456i	0.662667	−	0.481456i
$97$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$97$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$98$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$99$		0			0
$100$			0.902113i			0.902113i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	0.136729	+	0.420808i	0.136729	+	0.420808i
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−0.729825	−	0.530249i	−0.729825	−	0.530249i
$109$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$109$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$110$		0			0
$111$	0.309017	+	0.951057i	0.309017	+	0.951057i
$112$		0			0
$113$	0.550672	−	1.69480i	0.550672	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$113$	0.550672	−	1.69480i	0.550672	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$114$		0			0
$115$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$116$	−0.550672	+	1.69480i	−0.550672	+	1.69480i
$117$		0			0
$118$	0.0978870			0.0978870
$119$		0			0
$120$	−0.270175	−	0.530249i	−0.270175	−	0.530249i
$121$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.987688	+	0.156434i	0.987688	+	0.156434i
$126$		0			0
$127$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$127$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$128$	0.779012	−	0.565985i	0.779012	−	0.565985i
$129$		0			0
$130$		0			0
$131$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$131$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$	−0.183900	+	0.565985i	−0.183900	+	0.565985i
$135$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$136$	0.260074	+	0.800424i	0.260074	+	0.800424i
$137$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$137$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$138$	0.0877853	−	0.270175i	0.0877853	−	0.270175i
$139$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$139$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.579192	−	0.420808i	−0.579192	−	0.420808i
$145$	1.76007	+	0.896802i	1.76007	+	0.896802i
$146$	−0.297556	+	0.216187i	−0.297556	+	0.216187i
$147$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$148$	0.278768	+	0.857960i	0.278768	+	0.857960i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i
$151$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$151$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$152$		0			0
$153$	1.14412	−	0.831254i	1.14412	−	0.831254i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$157$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$158$		0			0
$159$		0			0
$160$	−0.371864	−	0.729825i	−0.371864	−	0.729825i
$161$		0			0
$162$	0.0966818	−	0.297556i	0.0966818	−	0.297556i
$163$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$163$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.44168	+	1.04744i	1.44168	+	1.04744i	0.987688	+	0.156434i	$0.0500000\pi$
$167$	1.44168	+	1.04744i	1.44168	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$168$		0			0
$169$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$170$	0.437016	−	0.0692165i	0.437016	−	0.0692165i
$171$		0			0
$172$		0			0
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$	−0.618034			−0.618034
$175$		0			0
$176$		0			0
$177$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$178$	0.229825	−	0.166977i	0.229825	−	0.166977i
$179$	0.734572	−	0.533698i	0.734572	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$179$	0.734572	−	0.533698i	0.734572	−	0.533698i	0.891007	+	0.453990i	$0.150000\pi$
$180$	−0.637890	+	0.637890i	−0.637890	+	0.637890i
$181$	−0.951057	−	0.690983i	−0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$181$	−0.951057	−	0.690983i	−0.951057	−	0.690983i	−0.951057	+	0.309017i	$0.900000\pi$
$182$		0			0
$183$		0			0
$184$	0.166977	−	0.513904i	0.166977	−	0.513904i
$185$	0.987688	−	0.156434i	0.987688	−	0.156434i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$191$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$192$	−0.371864	−	0.270175i	−0.371864	−	0.270175i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$197$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$197$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.565985	+	0.183900i	−0.565985	+	0.183900i
$201$	1.90211			1.90211
$202$		0			0
$203$		0			0
$204$	1.03213	−	0.749885i	1.03213	−	0.749885i
$205$		0			0
$206$		0			0
$207$	−0.907981			−0.907981
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$211$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$	0.183900	−	0.565985i	0.183900	−	0.565985i
$217$		0			0
$218$		0			0
$219$	0.951057	+	0.690983i	0.951057	+	0.690983i
$220$		0			0
$221$		0			0
$222$	−0.253116	+	0.183900i	−0.253116	+	0.183900i
$223$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$223$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$224$		0			0
$225$	0.587785	+	0.809017i	0.587785	+	0.809017i
$226$	0.557537			0.557537
$227$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$227$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$228$		0			0
$229$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$229$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$230$	−0.253116	−	0.128969i	−0.253116	−	0.128969i
$231$		0			0
$232$	−1.17557			−1.17557
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$		0			0
$235$		0			0
$236$	−0.0872179	−	0.268429i	−0.0872179	−	0.268429i
$237$		0			0
$238$		0			0
$239$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i	0.891007	−	0.453990i	$-0.150000\pi$
$239$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i	−0.987688	+	0.156434i	$0.950000\pi$
$240$	−0.506233	+	0.506233i	−0.506233	+	0.506233i
$241$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$241$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$242$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$243$	−1.00000			−1.00000
$244$		0			0
$245$	0.453990	+	0.891007i	0.453990	+	0.891007i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.0489435	+	0.309017i	0.0489435	+	0.309017i
$251$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$251$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$252$		0			0
$253$		0			0
$254$	0.409551	−	0.297556i	0.409551	−	0.297556i
$255$	−0.642040	−	1.26007i	−0.642040	−	1.26007i
$256$	−0.128136	−	0.0930960i	−0.128136	−	0.0930960i
$257$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$257$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.610425	+	1.87869i	0.610425	+	1.87869i
$262$	0.0302487	−	0.0930960i	0.0302487	−	0.0930960i
$263$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$263$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$268$	1.71592			1.71592
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$	−0.278768	−	0.142040i	−0.278768	−	0.142040i
$271$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$271$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$272$	0.819101	−	0.595112i	0.819101	−	0.595112i
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−0.819101			−0.819101
$277$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$277$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$278$	−0.481456	+	0.349798i	−0.481456	+	0.349798i
$279$		0			0
$280$		0			0
$281$	1.59811	+	1.16110i	1.59811	+	1.16110i	0.891007	+	0.453990i	$0.150000\pi$
$281$	1.59811	+	1.16110i	1.59811	+	1.16110i	0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$283$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.253116	−	0.779012i	0.253116	−	0.779012i
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i
$291$		0			0
$292$	0.857960	+	0.623345i	0.857960	+	0.623345i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$295$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i
$296$	−0.481456	+	0.349798i	−0.481456	+	0.349798i
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.530249	+	0.729825i	0.530249	+	0.729825i
$301$		0			0
$302$	−0.0597526	−	0.183900i	−0.0597526	−	0.183900i
$303$		0			0
$304$		0			0
$305$		0			0
$306$	0.357960	+	0.260074i	0.357960	+	0.260074i
$307$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.550672	+	1.69480i	0.550672	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$311$	0.550672	+	1.69480i	0.550672	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$	−0.113656	−	0.349798i	−0.113656	−	0.349798i
$315$		0			0
$316$		0			0
$317$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$317$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$318$		0			0
$319$		0			0
$320$	−0.325021	+	0.325021i	−0.325021	+	0.325021i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.902113			−0.902113
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$331$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$332$		0			0
$333$	0.809017	+	0.587785i	0.809017	+	0.587785i
$334$	−0.172288	+	0.530249i	−0.172288	+	0.530249i
$335$	0.297556	−	1.87869i	0.297556	−	1.87869i
$336$		0			0
$337$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$337$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$338$	0.0966818	−	0.297556i	0.0966818	−	0.297556i
$339$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$340$	−0.579192	−	1.13673i	−0.579192	−	1.13673i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.142040	+	0.896802i	−0.142040	+	0.896802i
$346$		0			0
$347$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	−0.891007	−	0.453990i	$-0.850000\pi$
$347$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	0.156434	+	0.987688i	$0.450000\pi$
$348$	0.550672	+	1.69480i	0.550672	+	1.69480i
$349$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$349$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.453990	−	0.891007i	$-0.350000\pi$
$353$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$354$	0.0791922	−	0.0575365i	0.0791922	−	0.0575365i
$355$		0			0
$356$	−0.662667	−	0.481456i	−0.662667	−	0.481456i
$357$		0			0
$358$	0.229825	+	0.166977i	0.229825	+	0.166977i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$	−0.530249	−	0.270175i	−0.530249	−	0.270175i
$361$	0.309017	+	0.951057i	0.309017	+	0.951057i
$362$	0.113656	−	0.349798i	0.113656	−	0.349798i
$363$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$364$		0			0
$365$	0.831254	−	0.831254i	0.831254	−	0.831254i
$366$		0			0
$367$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$367$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$368$	−0.650043			−0.650043
$369$		0			0
$370$	0.142040	+	0.278768i	0.142040	+	0.278768i
$371$		0			0
$372$		0			0
$373$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$373$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$374$		0			0
$375$	0.891007	−	0.453990i	0.891007	−	0.453990i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$379$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$380$		0			0
$381$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$382$	−0.442463			−0.442463
$383$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$383$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$384$	0.297556	−	0.915783i	0.297556	−	0.915783i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	−0.987688	−	0.156434i	$-0.950000\pi$
$389$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$390$		0			0
$391$	0.396802	−	1.22123i	0.396802	−	1.22123i
$392$	−0.481456	−	0.349798i	−0.481456	−	0.349798i
$393$	−0.312869			−0.312869
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$397$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$398$		0			0
$399$		0			0
$400$	0.420808	+	0.579192i	0.420808	+	0.579192i
$401$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$401$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$402$	0.183900	+	0.565985i	0.183900	+	0.565985i
$403$		0			0
$404$		0			0
$405$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$406$		0			0
$407$		0			0
$408$	0.680881	+	0.494689i	0.680881	+	0.494689i
$409$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$409$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.0877853	−	0.270175i	−0.0877853	−	0.270175i
$415$		0			0
$416$		0			0
$417$	1.53884	+	1.11803i	1.53884	+	1.11803i
$418$		0			0
$419$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$419$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$	−0.156434	+	0.113656i	−0.156434	+	0.113656i
$423$		0			0
$424$		0			0
$425$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$431$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$432$	−0.715921			−0.715921
$433$	−0.951057	−	0.690983i	−0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$433$	−0.951057	−	0.690983i	−0.951057	−	0.690983i	−0.951057	+	0.309017i	$0.900000\pi$
$434$		0			0
$435$	1.95106	−	0.309017i	1.95106	−	0.309017i
$436$		0			0
$437$		0			0
$438$	−0.113656	+	0.349798i	−0.113656	+	0.349798i
$439$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$439$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$440$		0			0
$441$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	0.729825	+	0.530249i	0.729825	+	0.530249i
$445$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$446$	−0.409551	+	0.297556i	−0.409551	+	0.297556i
$447$		0			0
$448$		0			0
$449$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$449$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$450$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$451$		0			0
$452$	−0.496769	−	1.52890i	−0.496769	−	1.52890i
$453$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$454$	−0.357960	+	0.260074i	−0.357960	+	0.260074i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	0.156434	+	0.113656i	0.156434	+	0.113656i
$459$	0.437016	−	1.34500i	0.437016	−	1.34500i
$460$	−0.128136	+	0.809017i	−0.128136	+	0.809017i
$461$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i	0.891007	−	0.453990i	$-0.150000\pi$
$461$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i	−0.987688	+	0.156434i	$0.950000\pi$
$462$		0			0
$463$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$463$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$464$	0.437016	+	1.34500i	0.437016	+	1.34500i
$465$		0			0
$466$		0			0
$467$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$467$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$472$	0.150633	−	0.109441i	0.150633	−	0.109441i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.0791922	−	0.0575365i	0.0791922	−	0.0575365i
$479$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.453990	−	0.891007i	$-0.350000\pi$
$479$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$480$	−0.729825	−	0.371864i	−0.729825	−	0.371864i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.278768	+	0.857960i	−0.278768	+	0.857960i
$485$		0			0
$486$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$487$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$487$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$488$		0			0
$489$		0			0
$490$	−0.221232	+	0.221232i	−0.221232	+	0.221232i
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$	−2.79360			−2.79360
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	0.803789	−	0.409551i	0.803789	−	0.409551i
$501$	1.78201			1.78201
$502$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$503$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.707107	+	0.707107i	$0.750000\pi$
$503$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.00000			−1.00000
$508$	−1.18088	−	0.857960i	−1.18088	−	0.857960i
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$	0.312869	−	0.312869i	0.312869	−	0.312869i
$511$		0			0
$512$	0.312869	−	0.962912i	0.312869	−	0.962912i
$513$		0			0
$514$	0.172288	+	0.530249i	0.172288	+	0.530249i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$	−0.282243			−0.282243
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.142040	−	0.103198i	0.142040	−	0.103198i
$530$		0			0
$531$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$532$		0			0
$533$		0			0
$534$	0.0877853	−	0.270175i	0.0877853	−	0.270175i
$535$		0			0
$536$	0.349798	+	1.07657i	0.349798	+	1.07657i
$537$	0.280582	−	0.863541i	0.280582	−	0.863541i
$538$		0			0
$539$		0			0
$540$	−0.141122	+	0.891007i	−0.141122	+	0.891007i
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$	0.297556	+	0.216187i	0.297556	+	0.216187i
$543$	−1.17557			−1.17557
$544$	0.937153	+	0.680881i	0.937153	+	0.680881i
$545$		0			0
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−0.166977	−	0.513904i	−0.166977	−	0.513904i
$553$		0			0
$554$		0			0
$555$	0.707107	−	0.707107i	0.707107	−	0.707107i
$556$	1.38821	+	1.00859i	1.38821	+	1.00859i
$557$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$557$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$563$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	−0.987688	−	0.156434i	$-0.950000\pi$
$563$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	0.891007	+	0.453990i	$0.150000\pi$
$564$		0			0
$565$	−1.76007	+	0.278768i	−1.76007	+	0.278768i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$569$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$570$		0			0
$571$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$571$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$572$		0			0
$573$	0.437016	+	1.34500i	0.437016	+	1.34500i
$574$		0			0
$575$	0.863541	+	0.280582i	0.863541	+	0.280582i
$576$	−0.459650			−0.459650
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$	−0.253116	+	0.183900i	−0.253116	+	0.183900i
$579$		0			0
$580$	1.76007	−	0.278768i	1.76007	−	0.278768i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.216187	+	0.665356i	−0.216187	+	0.665356i
$585$		0			0
$586$		0			0
$587$	−0.610425	+	1.87869i	−0.610425	+	1.87869i	−0.156434	+	0.987688i	$0.550000\pi$
$587$	−0.610425	+	1.87869i	−0.610425	+	1.87869i	−0.453990	+	0.891007i	$0.650000\pi$
$588$	−0.278768	+	0.857960i	−0.278768	+	0.857960i
$589$		0			0
$590$	−0.0444398	−	0.0872179i	−0.0444398	−	0.0872179i
$591$		0			0
$592$	0.579192	+	0.420808i	0.579192	+	0.420808i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−0.349798	+	0.481456i	−0.349798	+	0.481456i
$601$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$601$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$	1.53884	−	1.11803i	1.53884	−	1.11803i
$604$	−0.451057	+	0.327712i	−0.451057	+	0.327712i
$605$	0.891007	+	0.453990i	0.891007	+	0.453990i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.394238	−	1.21334i	0.394238	−	1.21334i
$613$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$613$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$614$	0.0597526	+	0.183900i	0.0597526	+	0.183900i
$615$		0			0
$616$		0			0
$617$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$617$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$618$		0			0
$619$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$619$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$620$		0			0
$621$	−0.734572	+	0.533698i	−0.734572	+	0.533698i
$622$	−0.451057	+	0.327712i	−0.451057	+	0.327712i
$623$		0			0
$624$		0			0
$625$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$626$		0			0
$627$		0			0
$628$	−0.857960	+	0.623345i	−0.857960	+	0.623345i
$629$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$		0			0
$633$	0.500000	+	0.363271i	0.500000	+	0.363271i
$634$		0			0
$635$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.857960	−	0.437153i	−0.857960	−	0.437153i
$641$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$641$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$647$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$648$	−0.183900	−	0.565985i	−0.183900	−	0.565985i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	−0.891007	−	0.453990i	$-0.850000\pi$
$653$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	0.156434	+	0.987688i	$0.450000\pi$
$654$		0			0
$655$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i
$656$		0			0
$657$	1.17557			1.17557
$658$		0			0
$659$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$659$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$660$		0			0
$661$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$661$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i
$667$	1.45106	+	1.05425i	1.45106	+	1.05425i
$668$	1.60758			1.60758
$669$	1.30902	+	0.951057i	1.30902	+	0.951057i
$670$	0.587785	−	0.0930960i	0.587785	−	0.0930960i
$671$		0			0
$672$		0			0
$673$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$673$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$674$	0.506233			0.506233
$675$	0.951057	+	0.309017i	0.951057	+	0.309017i
$676$	−0.902113			−0.902113
$677$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$677$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$678$	0.451057	−	0.327712i	0.451057	−	0.327712i
$679$		0			0
$680$	0.595112	−	0.595112i	0.595112	−	0.595112i
$681$	1.14412	+	0.831254i	1.14412	+	0.831254i
$682$		0			0
$683$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$683$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.190983	−	0.587785i	0.190983	−	0.587785i
$688$		0			0
$689$		0			0
$690$	−0.280582	+	0.0444398i	−0.280582	+	0.0444398i
$691$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$691$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$692$		0			0
$693$		0			0
$694$	−0.229825	−	0.166977i	−0.229825	−	0.166977i
$695$	1.34500	−	1.34500i	1.34500	−	1.34500i
$696$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$697$		0			0
$698$	−0.113656	−	0.349798i	−0.113656	−	0.349798i
$699$		0			0
$700$		0			0
$701$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$701$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.451057	+	0.327712i	0.451057	+	0.327712i
$707$		0			0
$708$	−0.228339	−	0.165898i	−0.228339	−	0.165898i
$709$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$709$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$710$		0			0
$711$		0			0
$712$	0.166977	−	0.513904i	0.166977	−	0.513904i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.253116	−	0.779012i	0.253116	−	0.779012i
$717$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$718$		0			0
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$	−0.111995	+	0.707107i	−0.111995	+	0.707107i
$721$		0			0
$722$	−0.253116	+	0.183900i	−0.253116	+	0.183900i
$723$		0			0
$724$	−1.06050			−1.06050
$725$		−	1.97538i		−	1.97538i
$726$	−0.312869			−0.312869
$727$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$727$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$	0.327712	+	0.166977i	0.327712	+	0.166977i
$731$		0			0
$732$		0			0
$733$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$733$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$734$	−0.156434	+	0.481456i	−0.156434	+	0.481456i
$735$	0.891007	+	0.453990i	0.891007	+	0.453990i
$736$	−0.229825	−	0.707328i	−0.229825	−	0.707328i
$737$		0			0
$738$		0			0
$739$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$739$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$740$	0.637890	−	0.637890i	0.637890	−	0.637890i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$	0.221232	+	0.221232i	0.221232	+	0.221232i
$751$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$751$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$752$		0			0
$753$	−1.59811	+	1.16110i	−1.59811	+	1.16110i
$754$		0			0
$755$	0.280582	+	0.550672i	0.280582	+	0.550672i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	0.481456	+	0.349798i	0.481456	+	0.349798i
$759$		0			0
$760$		0			0
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$	0.156434	−	0.481456i	0.156434	−	0.481456i
$763$		0			0
$764$	0.394238	+	1.21334i	0.394238	+	1.21334i
$765$	−1.26007	−	0.642040i	−1.26007	−	0.642040i
$766$	0.190983	−	0.587785i	0.190983	−	0.587785i
$767$		0			0
$768$	−0.158384			−0.158384
$769$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$769$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$770$		0			0
$771$	1.44168	−	1.04744i	1.44168	−	1.04744i
$772$		0			0
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.229825	−	0.166977i	0.229825	−	0.166977i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.401748			0.401748
$783$	1.59811	+	1.16110i	1.59811	+	1.16110i
$784$	−0.221232	+	0.680881i	−0.221232	+	0.680881i
$785$	0.533698	+	1.04744i	0.533698	+	1.04744i
$786$	−0.0302487	−	0.0930960i	−0.0302487	−	0.0930960i
$787$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$787$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.481456	−	0.349798i	−0.481456	−	0.349798i
$795$		0			0
$796$		0			0
$797$	0.253116	−	0.183900i	0.253116	−	0.183900i	−0.453990	−	0.891007i	$-0.650000\pi$
$797$	0.253116	−	0.183900i	0.253116	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$	−0.481456	+	0.662667i	−0.481456	+	0.662667i
$801$	−0.907981			−0.907981
$802$	−0.172288	−	0.530249i	−0.172288	−	0.530249i
$803$		0			0
$804$	1.38821	−	1.00859i	1.38821	−	1.00859i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.437016	−	1.34500i	0.437016	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$809$	0.437016	−	1.34500i	0.437016	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$810$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i
$811$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$811$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$812$		0			0
$813$	0.363271	−	1.11803i	0.363271	−	1.11803i
$814$		0			0
$815$		0			0
$816$	0.312869	−	0.962912i	0.312869	−	0.962912i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$823$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$827$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$828$	−0.662667	+	0.481456i	−0.662667	+	0.481456i
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.14412	−	0.831254i	−1.14412	−	0.831254i
$834$	−0.183900	+	0.565985i	−0.183900	+	0.565985i
$835$	0.278768	−	1.76007i	0.278768	−	1.76007i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	0.896802	−	2.76007i	0.896802	−	2.76007i
$842$		0			0
$843$	1.97538			1.97538
$844$	0.451057	+	0.327712i	0.451057	+	0.327712i
$845$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	−0.260074	−	0.357960i	−0.260074	−	0.357960i
$851$	0.907981			0.907981
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$857$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$858$		0			0
$859$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$859$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$860$		0			0
$861$		0			0
$862$	0.172288	−	0.530249i	0.172288	−	0.530249i
$863$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$863$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$864$	−0.253116	−	0.779012i	−0.253116	−	0.779012i
$865$		0			0
$866$	0.113656	−	0.349798i	0.113656	−	0.349798i
$867$	0.809017	+	0.587785i	0.809017	+	0.587785i
$868$		0			0
$869$		0			0
$870$	0.280582	+	0.550672i	0.280582	+	0.550672i
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	1.06050			1.06050
$877$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$877$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$881$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$882$	−0.312869			−0.312869
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$	−0.221232	+	0.221232i	−0.221232	+	0.221232i
$886$		0			0
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$	−0.183900	+	0.565985i	−0.183900	+	0.565985i
$889$		0			0
$890$	−0.253116	−	0.128969i	−0.253116	−	0.128969i
$891$		0			0
$892$	1.18088	+	0.857960i	1.18088	+	0.857960i
$893$		0			0
$894$		0			0
$895$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$896$		0			0
$897$		0			0
$898$	−0.136729	−	0.420808i	−0.136729	−	0.420808i
$899$		0			0
$900$	0.857960	+	0.278768i	0.857960	+	0.278768i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.857960	−	0.623345i	0.857960	−	0.623345i
$905$	−0.183900	+	1.16110i	−0.183900	+	1.16110i
$906$	−0.156434	−	0.113656i	−0.156434	−	0.113656i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	1.03213	+	0.749885i	1.03213	+	0.749885i
$909$		0			0
$910$		0			0
$911$	−0.550672	−	1.69480i	−0.550672	−	1.69480i	−0.707107	−	0.707107i	$-0.750000\pi$
$911$	−0.550672	−	1.69480i	−0.550672	−	1.69480i	0.156434	−	0.987688i	$-0.450000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.172288	−	0.530249i	0.172288	−	0.530249i
$917$		0			0
$918$	0.442463			0.442463
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$	−0.533698	+	0.0845294i	−0.533698	+	0.0845294i
$921$	0.500000	−	0.363271i	0.500000	−	0.363271i
$922$	0.0791922	−	0.0575365i	0.0791922	−	0.0575365i
$923$		0			0
$924$		0			0
$925$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$926$		0			0
$927$		0			0
$928$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	1.44168	+	1.04744i	1.44168	+	1.04744i
$934$	−0.136729	+	0.420808i	−0.136729	+	0.420808i
$935$		0			0
$936$		0			0
$937$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$937$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$	−0.297556	−	0.216187i	−0.297556	−	0.216187i
$943$		0			0
$944$	−0.181211	−	0.131658i	−0.181211	−	0.131658i
$945$		0			0
$946$		0			0
$947$	−1.44168	+	1.04744i	−1.44168	+	1.04744i	−0.453990	+	0.891007i	$0.650000\pi$
$947$	−1.44168	+	1.04744i	−1.44168	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$953$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$954$		0			0
$955$	1.39680	−	0.221232i	1.39680	−	0.221232i
$956$	−0.228339	−	0.165898i	−0.228339	−	0.165898i
$957$		0			0
$958$	0.451057	+	0.327712i	0.451057	+	0.327712i
$959$		0			0
$960$	−0.0719050	+	0.453990i	−0.0719050	+	0.453990i
$961$	0.309017	+	0.951057i	0.309017	+	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$967$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$968$	−0.595112			−0.595112
$969$		0			0
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$977$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$978$		0			0
$979$		0			0
$980$	0.803789	+	0.409551i	0.803789	+	0.409551i
$981$		0			0
$982$		0			0
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$		0			0
$985$		0			0
$986$	−0.270091	−	0.831254i	−0.270091	−	0.831254i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$991$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$997$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$998$		0			0
$999$	1.00000			1.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2775.1.bg.d.1331.3	yes	16
3.2	odd	2	inner	2775.1.bg.d.1331.2	yes	16
25.21	even	5	inner	2775.1.bg.d.221.3	yes	16
37.36	even	2	inner	2775.1.bg.d.1331.2	yes	16
75.71	odd	10	inner	2775.1.bg.d.221.2	✓	16
111.110	odd	2	CM	2775.1.bg.d.1331.3	yes	16
925.221	even	10	inner	2775.1.bg.d.221.2	✓	16
2775.221	odd	10	inner	2775.1.bg.d.221.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2775.1.bg.d.221.2	✓	16	75.71	odd	10	inner
2775.1.bg.d.221.2	✓	16	925.221	even	10	inner
2775.1.bg.d.221.3	yes	16	25.21	even	5	inner
2775.1.bg.d.221.3	yes	16	2775.221	odd	10	inner
2775.1.bg.d.1331.2	yes	16	3.2	odd	2	inner
2775.1.bg.d.1331.2	yes	16	37.36	even	2	inner
2775.1.bg.d.1331.3	yes	16	1.1	even	1	trivial
2775.1.bg.d.1331.3	yes	16	111.110	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2775.bg (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0966818 + 0.297556i) q^{2} +(0.809017 - 0.587785i) q^{3} +(0.729825 - 0.530249i) q^{4} +(-0.453990 - 0.891007i) q^{5} +(0.253116 + 0.183900i) q^{6} +(0.481456 + 0.349798i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})\)	\(q+(0.0966818 + 0.297556i) q^{2} +(0.809017 - 0.587785i) q^{3} +(0.729825 - 0.530249i) q^{4} +(-0.453990 - 0.891007i) q^{5} +(0.253116 + 0.183900i) q^{6} +(0.481456 + 0.349798i) q^{8} +(0.309017 - 0.951057i) q^{9} +(0.221232 - 0.221232i) q^{10} +(0.278768 - 0.857960i) q^{12} +(-0.891007 - 0.453990i) q^{15} +(0.221232 - 0.680881i) q^{16} +(1.14412 + 0.831254i) q^{17} +0.312869 q^{18} +(-0.803789 - 0.409551i) q^{20} +(-0.280582 - 0.863541i) q^{23} +0.595112 q^{24} +(-0.587785 + 0.809017i) q^{25} +(-0.309017 - 0.951057i) q^{27} +(-1.59811 + 1.16110i) q^{29} +(0.0489435 - 0.309017i) q^{30} +0.819101 q^{32} +(-0.136729 + 0.420808i) q^{34} +(-0.278768 - 0.857960i) q^{36} +(-0.309017 + 0.951057i) q^{37} +(0.0930960 - 0.587785i) q^{40} +(-0.987688 + 0.156434i) q^{45} +(0.229825 - 0.166977i) q^{46} +(-0.221232 - 0.680881i) q^{48} -1.00000 q^{49} +(-0.297556 - 0.0966818i) q^{50} +1.41421 q^{51} +(0.253116 - 0.183900i) q^{54} +(-0.500000 - 0.363271i) q^{58} +(0.0966818 - 0.297556i) q^{59} +(-0.891007 + 0.141122i) q^{60} +(-0.142040 - 0.437153i) q^{64} +(1.53884 + 1.11803i) q^{67} +1.27578 q^{68} +(-0.734572 - 0.533698i) q^{69} +(0.481456 - 0.349798i) q^{72} +(0.363271 + 1.11803i) q^{73} -0.312869 q^{74} +1.00000i q^{75} +(-0.707107 + 0.111995i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(0.221232 - 1.39680i) q^{85} +(-0.610425 + 1.87869i) q^{87} +(-0.280582 - 0.863541i) q^{89} +(-0.142040 - 0.278768i) q^{90} +(-0.662667 - 0.481456i) q^{92} +(0.662667 - 0.481456i) q^{96} +(-0.0966818 - 0.297556i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9	\( 16 q + 4 q^{3} - 4 q^{4} - 4 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{16} + 4 q^{27} + 16 q^{30} - 8 q^{34} - 4 q^{36} + 4 q^{37} - 20 q^{40} - 12 q^{46} - 4 q^{48} - 16 q^{49} - 8 q^{58} + 4 q^{64} - 4 q^{81} + 4 q^{85} + 4 q^{90}+O(q^{100}) \) 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^16 + 4 * q^27 + 16 * q^30 - 8 * q^34 - 4 * q^36 + 4 * q^37 - 20 * q^40 - 12 * q^46 - 4 * q^48 - 16 * q^49 - 8 * q^58 + 4 * q^64 - 4 * q^81 + 4 * q^85 + 4 * q^90

Introduction

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