LMFDB - Embedded newform 2013.1.bm.d.1829.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,1,Mod(548,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, 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("2013.548");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2013.bm (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:	$1.00461787043$
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:	$\Z[a_1, \ldots, a_{11}]$
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			1829.2
Root			$-0.891007 - 0.453990i$ of defining polynomial
Character	$\chi$	$=$	2013.1829
Dual form			2013.1.bm.d.1280.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.253116 + 0.183900i) q^{2} +(-0.309017 - 0.951057i) q^{3} +(-0.278768 + 0.857960i) q^{4} +(0.253116 + 0.183900i) q^{6} +(-0.183900 - 0.565985i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})$	\(q+(-0.253116 + 0.183900i) q^{2} +(-0.309017 - 0.951057i) q^{3} +(-0.278768 + 0.857960i) q^{4} +(0.253116 + 0.183900i) q^{6} +(-0.183900 - 0.565985i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.891007 - 0.453990i) q^{11} +0.902113 q^{12} +(-0.951057 + 0.690983i) q^{13} +(-0.579192 - 0.420808i) q^{16} +(-1.44168 - 1.04744i) q^{17} +(0.0966818 - 0.297556i) q^{18} +(-0.587785 - 1.80902i) q^{19} +(-0.142040 + 0.278768i) q^{22} +1.97538 q^{23} +(-0.481456 + 0.349798i) q^{24} +(0.309017 + 0.951057i) q^{25} +(0.113656 - 0.349798i) q^{26} +(0.809017 + 0.587785i) q^{27} +(0.610425 - 1.87869i) q^{29} +0.819101 q^{32} +(-0.707107 - 0.707107i) q^{33} +0.557537 q^{34} +(-0.278768 - 0.857960i) q^{36} +(0.481456 + 0.349798i) q^{38} +(0.951057 + 0.690983i) q^{39} +(0.141122 + 0.891007i) q^{44} +(-0.500000 + 0.363271i) q^{46} +(-0.221232 + 0.680881i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-0.253116 - 0.183900i) q^{50} +(-0.550672 + 1.69480i) q^{51} +(-0.327712 - 1.00859i) q^{52} +(0.253116 - 0.183900i) q^{53} -0.312869 q^{54} +(-1.53884 + 1.11803i) q^{57} +(0.190983 + 0.587785i) q^{58} +(0.280582 - 0.863541i) q^{59} +(0.809017 + 0.587785i) q^{61} +(0.371864 - 0.270175i) q^{64} +(0.309017 + 0.0489435i) q^{66} +(1.30056 - 0.944910i) q^{68} +(-0.610425 - 1.87869i) q^{69} +(-1.14412 - 0.831254i) q^{71} +(0.481456 + 0.349798i) q^{72} +(0.500000 - 1.53884i) q^{73} +(0.809017 - 0.587785i) q^{75} +1.71592 q^{76} -0.367799 q^{78} +(0.309017 - 0.951057i) q^{81} -1.97538 q^{87} +(-0.420808 - 0.420808i) q^{88} -1.78201 q^{89} +(-0.550672 + 1.69480i) q^{92} +(-0.253116 - 0.779012i) q^{96} +0.312869 q^{98} +(-0.453990 + 0.891007i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 16 q^{12} - 16 q^{16} + 4 q^{22} - 4 q^{25} + 4 q^{27} + 8 q^{34} - 4 q^{36} - 8 q^{46} - 4 q^{48} - 4 q^{49} - 20 q^{52} + 12 q^{58} + 4 q^{61} - 16 q^{64} - 4 q^{66} + 8 q^{73} + 4 q^{75} + 40 q^{76} - 4 q^{81}+O(q^{100}) $ 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9 - 16 * q^12 - 16 * q^16 + 4 * q^22 - 4 * q^25 + 4 * q^27 + 8 * q^34 - 4 * q^36 - 8 * q^46 - 4 * q^48 - 4 * q^49 - 20 * q^52 + 12 * q^58 + 4 * q^61 - 16 * q^64 - 4 * q^66 + 8 * q^73 + 4 * q^75 + 40 * q^76 - 4 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2013\mathbb{Z}\right)^\times$.

$n$	$1222$	$1343$	$1465$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{4}{5}\right)$

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 $

$2$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$3$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$3$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$4$	−0.278768	+	0.857960i	−0.278768	+	0.857960i
$5$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$5$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$6$	0.253116	+	0.183900i	0.253116	+	0.183900i
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$	−0.183900	−	0.565985i	−0.183900	−	0.565985i
$9$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$10$		0			0
$11$	0.891007	−	0.453990i	0.891007	−	0.453990i
$11$	0.891007	−	0.453990i	0.891007	−	0.453990i
$12$	0.902113			0.902113
$13$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$13$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−0.579192	−	0.420808i	−0.579192	−	0.420808i
$17$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$17$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$18$	0.0966818	−	0.297556i	0.0966818	−	0.297556i
$19$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$19$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$20$		0			0
$21$		0			0
$22$	−0.142040	+	0.278768i	−0.142040	+	0.278768i
$23$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$23$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$24$	−0.481456	+	0.349798i	−0.481456	+	0.349798i
$25$	0.309017	+	0.951057i	0.309017	+	0.951057i
$26$	0.113656	−	0.349798i	0.113656	−	0.349798i
$27$	0.809017	+	0.587785i	0.809017	+	0.587785i
$28$		0			0
$29$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.156434	−	0.987688i	$-0.450000\pi$
$29$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.453990	−	0.891007i	$-0.350000\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$	0.819101			0.819101
$33$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$34$	0.557537			0.557537
$35$		0			0
$36$	−0.278768	−	0.857960i	−0.278768	−	0.857960i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$	0.481456	+	0.349798i	0.481456	+	0.349798i
$39$	0.951057	+	0.690983i	0.951057	+	0.690983i
$40$		0			0
$41$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$41$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.141122	+	0.891007i	0.141122	+	0.891007i
$45$		0			0
$46$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	−0.221232	+	0.680881i	−0.221232	+	0.680881i
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$51$	−0.550672	+	1.69480i	−0.550672	+	1.69480i
$52$	−0.327712	−	1.00859i	−0.327712	−	1.00859i
$53$	0.253116	−	0.183900i	0.253116	−	0.183900i	−0.453990	−	0.891007i	$-0.650000\pi$
$53$	0.253116	−	0.183900i	0.253116	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$54$	−0.312869			−0.312869
$55$		0			0
$56$		0			0
$57$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$58$	0.190983	+	0.587785i	0.190983	+	0.587785i
$59$	0.280582	−	0.863541i	0.280582	−	0.863541i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	0.280582	−	0.863541i	0.280582	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$60$		0			0
$61$	0.809017	+	0.587785i	0.809017	+	0.587785i
$61$	0.809017	+	0.587785i	0.809017	+	0.587785i
$62$		0			0
$63$		0			0
$64$	0.371864	−	0.270175i	0.371864	−	0.270175i
$65$		0			0
$66$	0.309017	+	0.0489435i	0.309017	+	0.0489435i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	1.30056	−	0.944910i	1.30056	−	0.944910i
$69$	−0.610425	−	1.87869i	−0.610425	−	1.87869i
$70$		0			0
$71$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$71$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$72$	0.481456	+	0.349798i	0.481456	+	0.349798i
$73$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$73$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$74$		0			0
$75$	0.809017	−	0.587785i	0.809017	−	0.587785i
$76$	1.71592			1.71592
$77$		0			0
$78$	−0.367799			−0.367799
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$		0			0
$81$	0.309017	−	0.951057i	0.309017	−	0.951057i
$82$		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			0
$86$		0			0
$87$	−1.97538			−1.97538
$88$	−0.420808	−	0.420808i	−0.420808	−	0.420808i
$89$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$89$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$90$		0			0
$91$		0			0
$92$	−0.550672	+	1.69480i	−0.550672	+	1.69480i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.253116	−	0.779012i	−0.253116	−	0.779012i
$97$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$97$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$98$	0.312869			0.312869
$99$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$100$	−0.902113			−0.902113
$101$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$101$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$102$	−0.172288	−	0.530249i	−0.172288	−	0.530249i
$103$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$103$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$104$	0.565985	+	0.411212i	0.565985	+	0.411212i
$105$		0			0
$106$	−0.0302487	+	0.0930960i	−0.0302487	+	0.0930960i
$107$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$107$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$108$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$109$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$109$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$113$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$114$	0.183900	−	0.565985i	0.183900	−	0.565985i
$115$		0			0
$116$	1.44168	+	1.04744i	1.44168	+	1.04744i
$117$	0.363271	−	1.11803i	0.363271	−	1.11803i
$118$	0.0877853	+	0.270175i	0.0877853	+	0.270175i
$119$		0			0
$120$		0			0
$121$	0.587785	−	0.809017i	0.587785	−	0.809017i
$122$	−0.312869			−0.312869
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$127$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$128$	−0.297556	+	0.915783i	−0.297556	+	0.915783i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	0.803789	−	0.409551i	0.803789	−	0.409551i
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.327712	+	1.00859i	−0.327712	+	1.00859i
$137$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$137$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$138$	0.500000	+	0.363271i	0.500000	+	0.363271i
$139$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$139$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$140$		0			0
$141$		0			0
$142$	0.442463			0.442463
$143$	−0.533698	+	1.04744i	−0.533698	+	1.04744i
$144$	0.715921			0.715921
$145$		0			0
$146$	0.156434	+	0.481456i	0.156434	+	0.481456i
$147$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$148$		0			0
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$	−0.915783	+	0.665356i	−0.915783	+	0.665356i
$153$	1.78201			1.78201
$154$		0			0
$155$		0			0
$156$	−0.857960	+	0.623345i	−0.857960	+	0.623345i
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$160$		0			0
$161$		0			0
$162$	0.0966818	+	0.297556i	0.0966818	+	0.297556i
$163$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$163$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	0.118034	−	0.363271i	0.118034	−	0.363271i
$170$		0			0
$171$	1.53884	+	1.11803i	1.53884	+	1.11803i
$172$		0			0
$173$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$173$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$174$	0.500000	−	0.363271i	0.500000	−	0.363271i
$175$		0			0
$176$	−0.707107	−	0.111995i	−0.707107	−	0.111995i
$177$	−0.907981			−0.907981
$178$	0.451057	−	0.327712i	0.451057	−	0.327712i
$179$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$179$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$180$		0			0
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$	0.309017	−	0.951057i	0.309017	−	0.951057i
$184$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$185$		0			0
$186$		0			0
$187$	−1.76007	−	0.278768i	−1.76007	−	0.278768i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.280582	+	0.863541i	−0.280582	+	0.863541i	0.707107	+	0.707107i	$0.250000\pi$
$191$	−0.280582	+	0.863541i	−0.280582	+	0.863541i	−0.987688	+	0.156434i	$0.950000\pi$
$192$	−0.371864	−	0.270175i	−0.371864	−	0.270175i
$193$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$193$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$194$		0			0
$195$		0			0
$196$	0.729825	−	0.530249i	0.729825	−	0.530249i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i
$199$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$199$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$200$	0.481456	−	0.349798i	0.481456	−	0.349798i
$201$		0			0
$202$	−0.136729	+	0.420808i	−0.136729	+	0.420808i
$203$		0			0
$204$	−1.30056	−	0.944910i	−1.30056	−	0.944910i
$205$		0			0
$206$	−0.113656	−	0.349798i	−0.113656	−	0.349798i
$207$	−1.59811	+	1.16110i	−1.59811	+	1.16110i
$208$	0.841616			0.841616
$209$	−1.34500	−	1.34500i	−1.34500	−	1.34500i
$210$		0			0
$211$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$211$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$212$	0.0872179	+	0.268429i	0.0872179	+	0.268429i
$213$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$214$		0			0
$215$		0			0
$216$	0.183900	−	0.565985i	0.183900	−	0.565985i
$217$		0			0
$218$	0.156434	−	0.113656i	0.156434	−	0.113656i
$219$	−1.61803			−1.61803
$220$		0			0
$221$	2.09488			2.09488
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$226$		0			0
$227$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$227$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$228$	−0.530249	−	1.63194i	−0.530249	−	1.63194i
$229$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$229$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$230$		0			0
$231$		0			0
$232$	−1.17557			−1.17557
$233$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	−0.891007	−	0.453990i	$-0.850000\pi$
$233$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	0.156434	+	0.987688i	$0.450000\pi$
$234$	0.113656	+	0.349798i	0.113656	+	0.349798i
$235$		0			0
$236$	0.662667	+	0.481456i	0.662667	+	0.481456i
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$			0.312869i			0.312869i
$243$	−1.00000			−1.00000
$244$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$245$		0			0
$246$		0			0
$247$	1.80902	+	1.31433i	1.80902	+	1.31433i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.453990	−	0.891007i	$-0.350000\pi$
$251$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$252$		0			0
$253$	1.76007	−	0.896802i	1.76007	−	0.896802i
$254$		0			0
$255$		0			0
$256$	0.0489435	+	0.150633i	0.0489435	+	0.150633i
$257$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$257$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.610425	+	1.87869i	0.610425	+	1.87869i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−0.270175	+	0.530249i	−0.270175	+	0.530249i
$265$		0			0
$266$		0			0
$267$	0.550672	+	1.69480i	0.550672	+	1.69480i
$268$		0			0
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$271$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$272$	0.394238	+	1.21334i	0.394238	+	1.21334i
$273$		0			0
$274$		0			0
$275$	0.707107	+	0.707107i	0.707107	+	0.707107i
$276$	1.78201			1.78201
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$		0			0
$279$		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
$283$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$283$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$284$	1.03213	−	0.749885i	1.03213	−	0.749885i
$285$		0			0
$286$	−0.0575365	−	0.363271i	−0.0575365	−	0.363271i
$287$		0			0
$288$	−0.662667	+	0.481456i	−0.662667	+	0.481456i
$289$	0.672288	+	2.06909i	0.672288	+	2.06909i
$290$		0			0
$291$		0			0
$292$	1.18088	+	0.857960i	1.18088	+	0.857960i
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$295$		0			0
$296$		0			0
$297$	0.987688	+	0.156434i	0.987688	+	0.156434i
$298$		0			0
$299$	−1.87869	+	1.36495i	−1.87869	+	1.36495i
$300$	0.278768	+	0.857960i	0.278768	+	0.857960i
$301$		0			0
$302$		0			0
$303$	−1.14412	−	0.831254i	−1.14412	−	0.831254i
$304$	−0.420808	+	1.29511i	−0.420808	+	1.29511i
$305$		0			0
$306$	−0.451057	+	0.327712i	−0.451057	+	0.327712i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	1.17557			1.17557
$310$		0			0
$311$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$311$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$312$	0.216187	−	0.665356i	0.216187	−	0.665356i
$313$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$313$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$317$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$318$	0.0978870			0.0978870
$319$	−0.309017	−	1.95106i	−0.309017	−	1.95106i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.04744	+	3.22369i	−1.04744	+	3.22369i
$324$	0.729825	+	0.530249i	0.729825	+	0.530249i
$325$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$326$	0.113656	−	0.349798i	0.113656	−	0.349798i
$327$	0.190983	+	0.587785i	0.190983	+	0.587785i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$337$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$338$	0.0369292	+	0.113656i	0.0369292	+	0.113656i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−0.595112			−0.595112
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.357960	−	0.260074i	−0.357960	−	0.260074i
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$	0.550672	−	1.69480i	0.550672	−	1.69480i
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$	−1.17557			−1.17557
$352$	0.729825	−	0.371864i	0.729825	−	0.371864i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$	0.229825	−	0.166977i	0.229825	−	0.166977i
$355$		0			0
$356$	0.496769	−	1.52890i	0.496769	−	1.52890i
$357$		0			0
$358$		0			0
$359$	0.280582	−	0.863541i	0.280582	−	0.863541i	−0.707107	−	0.707107i	$-0.750000\pi$
$359$	0.280582	−	0.863541i	0.280582	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$360$		0			0
$361$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$362$		0			0
$363$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$364$		0			0
$365$		0			0
$366$	0.0966818	+	0.297556i	0.0966818	+	0.297556i
$367$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$367$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$368$	−1.14412	−	0.831254i	−1.14412	−	0.831254i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	0.496769	−	0.253116i	0.496769	−	0.253116i
$375$		0			0
$376$		0			0
$377$	0.717598	+	2.20854i	0.717598	+	2.20854i
$378$		0			0
$379$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$379$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$380$		0			0
$381$		0			0
$382$	−0.0877853	−	0.270175i	−0.0877853	−	0.270175i
$383$	−1.44168	+	1.04744i	−1.44168	+	1.04744i	−0.453990	+	0.891007i	$0.650000\pi$
$383$	−1.44168	+	1.04744i	−1.44168	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$384$	0.962912			0.962912
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.437016	−	1.34500i	0.437016	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$389$	0.437016	−	1.34500i	0.437016	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$390$		0			0
$391$	−2.84786	−	2.06909i	−2.84786	−	2.06909i
$392$	−0.183900	+	0.565985i	−0.183900	+	0.565985i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.637890	−	0.637890i	−0.637890	−	0.637890i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−0.156434	+	0.113656i	−0.156434	+	0.113656i
$399$		0			0
$400$	0.221232	−	0.680881i	0.221232	−	0.680881i
$401$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$401$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$		0			0
$404$	0.394238	+	1.21334i	0.394238	+	1.21334i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.06050			1.06050
$409$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$409$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$		0			0
$412$	−0.857960	−	0.623345i	−0.857960	−	0.623345i
$413$		0			0
$414$	0.190983	−	0.587785i	0.190983	−	0.587785i
$415$		0			0
$416$	−0.779012	+	0.565985i	−0.779012	+	0.565985i
$417$		0			0
$418$	0.587785	+	0.0930960i	0.587785	+	0.0930960i
$419$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$419$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$		0			0
$424$	−0.150633	−	0.109441i	−0.150633	−	0.109441i
$425$	0.550672	−	1.69480i	0.550672	−	1.69480i
$426$	−0.136729	−	0.420808i	−0.136729	−	0.420808i
$427$		0			0
$428$		0			0
$429$	1.16110	+	0.183900i	1.16110	+	0.183900i
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	−0.221232	−	0.680881i	−0.221232	−	0.680881i
$433$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$433$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$434$		0			0
$435$		0			0
$436$	0.172288	−	0.530249i	0.172288	−	0.530249i
$437$	−1.16110	−	3.57349i	−1.16110	−	3.57349i
$438$	0.409551	−	0.297556i	0.409551	−	0.297556i
$439$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$439$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$	−0.530249	+	0.385248i	−0.530249	+	0.385248i
$443$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$443$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$	0.312869			0.312869
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.136729	−	0.420808i	−0.136729	−	0.420808i
$455$		0			0
$456$	0.915783	+	0.665356i	0.915783	+	0.665356i
$457$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$457$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$458$	−0.113656	+	0.349798i	−0.113656	+	0.349798i
$459$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$463$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$464$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$465$		0			0
$466$	0.0877853	−	0.270175i	0.0877853	−	0.270175i
$467$	0.734572	+	0.533698i	0.734572	+	0.533698i	0.891007	−	0.453990i	$-0.150000\pi$
$467$	0.734572	+	0.533698i	0.734572	+	0.533698i	−0.156434	+	0.987688i	$0.550000\pi$
$468$	0.857960	+	0.623345i	0.857960	+	0.623345i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.540350			−0.540350
$473$		0			0
$474$		0			0
$475$	1.53884	−	1.11803i	1.53884	−	1.11803i
$476$		0			0
$477$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.530249	+	0.729825i	0.530249	+	0.729825i
$485$		0			0
$486$	0.253116	−	0.183900i	0.253116	−	0.183900i
$487$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$487$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$488$	0.183900	−	0.565985i	0.183900	−	0.565985i
$489$	0.951057	+	0.690983i	0.951057	+	0.690983i
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$	−2.84786	+	2.06909i	−2.84786	+	2.06909i
$494$	−0.699596			−0.699596
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$		0			0
$501$		0			0
$502$	−0.172288	+	0.530249i	−0.172288	+	0.530249i
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$		0			0
$506$	−0.280582	+	0.550672i	−0.280582	+	0.550672i
$507$	−0.381966			−0.381966
$508$		0			0
$509$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	−0.987688	−	0.156434i	$-0.950000\pi$
$509$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$510$		0			0
$511$		0			0
$512$	−0.819101	−	0.595112i	−0.819101	−	0.595112i
$513$	0.587785	−	1.80902i	0.587785	−	1.80902i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.14412	−	0.831254i	1.14412	−	0.831254i
$520$		0			0
$521$	0.550672	−	1.69480i	0.550672	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$521$	0.550672	−	1.69480i	0.550672	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$522$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.111995	+	0.707107i	0.111995	+	0.707107i
$529$	2.90211			2.90211
$530$		0			0
$531$	0.280582	+	0.863541i	0.280582	+	0.863541i
$532$		0			0
$533$		0			0
$534$	−0.451057	−	0.327712i	−0.451057	−	0.327712i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$	−0.0597526	−	0.183900i	−0.0597526	−	0.183900i
$543$		0			0
$544$	−1.18088	−	0.857960i	−1.18088	−	0.857960i
$545$		0			0
$546$		0			0
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$		0			0
$549$	−1.00000			−1.00000
$550$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i
$551$	−3.75739			−3.75739
$552$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.280582	−	0.863541i	0.280582	−	0.863541i	−0.707107	−	0.707107i	$-0.750000\pi$
$557$	0.280582	−	0.863541i	0.280582	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.278768	+	1.76007i	0.278768	+	1.76007i
$562$	−0.618034			−0.618034
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$		0			0
$565$		0			0
$566$	−0.156434	−	0.113656i	−0.156434	−	0.113656i
$567$		0			0
$568$	−0.260074	+	0.800424i	−0.260074	+	0.800424i
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$571$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$572$	−0.749885	−	0.749885i	−0.749885	−	0.749885i
$573$	0.907981			0.907981
$574$		0			0
$575$	0.610425	+	1.87869i	0.610425	+	1.87869i
$576$	−0.142040	+	0.437153i	−0.142040	+	0.437153i
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	−0.550672	−	0.400087i	−0.550672	−	0.400087i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.142040	−	0.278768i	0.142040	−	0.278768i
$584$	−0.962912			−0.962912
$585$		0			0
$586$		0			0
$587$	0.0966818	−	0.297556i	0.0966818	−	0.297556i	−0.891007	−	0.453990i	$-0.850000\pi$
$587$	0.0966818	−	0.297556i	0.0966818	−	0.297556i	0.987688	+	0.156434i	$0.0500000\pi$
$588$	−0.729825	−	0.530249i	−0.729825	−	0.530249i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$593$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$594$	−0.278768	+	0.142040i	−0.278768	+	0.142040i
$595$		0			0
$596$		0			0
$597$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$598$	0.224514	−	0.690983i	0.224514	−	0.690983i
$599$	0.253116	+	0.183900i	0.253116	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$599$	0.253116	+	0.183900i	0.253116	+	0.183900i	−0.453990	+	0.891007i	$0.650000\pi$
$600$	−0.481456	−	0.349798i	−0.481456	−	0.349798i
$601$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$601$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.442463			0.442463
$607$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$607$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$608$	−0.481456	−	1.48177i	−0.481456	−	1.48177i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.496769	+	1.52890i	−0.496769	+	1.52890i
$613$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$613$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$617$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$618$	−0.297556	+	0.216187i	−0.297556	+	0.216187i
$619$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$619$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$620$		0			0
$621$	1.59811	+	1.16110i	1.59811	+	1.16110i
$622$	0.357960	+	0.260074i	0.357960	+	0.260074i
$623$		0			0
$624$	−0.260074	−	0.800424i	−0.260074	−	0.800424i
$625$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$626$		0			0
$627$	−0.863541	+	1.69480i	−0.863541	+	1.69480i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.228339	−	0.165898i	0.228339	−	0.165898i
$637$	1.17557			1.17557
$638$	0.437016	+	0.437016i	0.437016	+	0.437016i
$639$	1.41421			1.41421
$640$		0			0
$641$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$641$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$642$		0			0
$643$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$643$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$644$		0			0
$645$		0			0
$646$	−0.327712	−	1.00859i	−0.327712	−	1.00859i
$647$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.707107	−	0.707107i	$-0.250000\pi$
$647$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$648$	−0.595112			−0.595112
$649$	−0.142040	−	0.896802i	−0.142040	−	0.896802i
$650$	0.367799			0.367799
$651$		0			0
$652$	−0.327712	−	1.00859i	−0.327712	−	1.00859i
$653$	0.280582	−	0.863541i	0.280582	−	0.863541i	−0.707107	−	0.707107i	$-0.750000\pi$
$653$	0.280582	−	0.863541i	0.280582	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$654$	−0.156434	−	0.113656i	−0.156434	−	0.113656i
$655$		0			0
$656$		0			0
$657$	0.500000	+	1.53884i	0.500000	+	1.53884i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	−0.647354	−	1.99235i	−0.647354	−	1.99235i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.20582	−	3.71113i	1.20582	−	3.71113i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.987688	+	0.156434i	0.987688	+	0.156434i
$672$		0			0
$673$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$673$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$674$		0			0
$675$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$676$	0.278768	+	0.202537i	0.278768	+	0.202537i
$677$	−0.734572	−	0.533698i	−0.734572	−	0.533698i	0.156434	−	0.987688i	$-0.450000\pi$
$677$	−0.734572	−	0.533698i	−0.734572	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.41421			1.41421
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−1.38821	+	1.00859i	−1.38821	+	1.00859i
$685$		0			0
$686$		0			0
$687$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$688$		0			0
$689$	−0.113656	+	0.349798i	−0.113656	+	0.349798i
$690$		0			0
$691$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$691$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$692$	−1.27578			−1.27578
$693$		0			0
$694$		0			0
$695$		0			0
$696$	0.363271	+	1.11803i	0.363271	+	1.11803i
$697$		0			0
$698$		0			0
$699$	0.734572	+	0.533698i	0.734572	+	0.533698i
$700$		0			0
$701$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	−0.987688	−	0.156434i	$-0.950000\pi$
$701$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$702$	0.297556	−	0.216187i	0.297556	−	0.216187i
$703$		0			0
$704$	0.208677	−	0.409551i	0.208677	−	0.409551i
$705$		0			0
$706$		0			0
$707$		0			0
$708$	0.253116	−	0.779012i	0.253116	−	0.779012i
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$	0.327712	+	1.00859i	0.327712	+	1.00859i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.0877853	+	0.270175i	0.0877853	+	0.270175i
$719$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$719$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$720$		0			0
$721$		0			0
$722$	0.253116	−	0.779012i	0.253116	−	0.779012i
$723$		0			0
$724$		0			0
$725$	1.97538			1.97538
$726$	0.297556	−	0.0966818i	0.297556	−	0.0966818i
$727$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$727$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$728$		0			0
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$	0.729825	+	0.530249i	0.729825	+	0.530249i
$733$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$733$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$734$		0			0
$735$		0			0
$736$	1.61803			1.61803
$737$		0			0
$738$		0			0
$739$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$739$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$	0.690983	−	2.12663i	0.690983	−	2.12663i
$742$		0			0
$743$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$743$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	0.729825	−	1.43236i	0.729825	−	1.43236i
$749$		0			0
$750$		0			0
$751$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$751$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$752$		0			0
$753$	−1.44168	−	1.04744i	−1.44168	−	1.04744i
$754$	−0.587785	−	0.427051i	−0.587785	−	0.427051i
$755$		0			0
$756$		0			0
$757$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$757$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$758$	−0.506233			−0.506233
$759$	−1.39680	−	1.39680i	−1.39680	−	1.39680i
$760$		0			0
$761$	0.734572	−	0.533698i	0.734572	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$761$	0.734572	−	0.533698i	0.734572	−	0.533698i	0.891007	+	0.453990i	$0.150000\pi$
$762$		0			0
$763$		0			0
$764$	−0.662667	−	0.481456i	−0.662667	−	0.481456i
$765$		0			0
$766$	0.172288	−	0.530249i	0.172288	−	0.530249i
$767$	0.329843	+	1.01515i	0.329843	+	1.01515i
$768$	0.128136	−	0.0930960i	0.128136	−	0.0930960i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		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
$778$	0.136729	+	0.420808i	0.136729	+	0.420808i
$779$		0			0
$780$		0			0
$781$	−1.39680	−	0.221232i	−1.39680	−	0.221232i
$782$	1.10134			1.10134
$783$	1.59811	−	1.16110i	1.59811	−	1.16110i
$784$	0.221232	+	0.680881i	0.221232	+	0.680881i
$785$		0			0
$786$		0			0
$787$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$787$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.587785	+	0.0930960i	0.587785	+	0.0930960i
$793$	−1.17557			−1.17557
$794$		0			0
$795$		0			0
$796$	−0.172288	+	0.530249i	−0.172288	+	0.530249i
$797$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$797$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$798$		0			0
$799$		0			0
$800$	0.253116	+	0.779012i	0.253116	+	0.779012i
$801$	1.44168	−	1.04744i	1.44168	−	1.04744i
$802$	0.0978870			0.0978870
$803$	−0.253116	−	1.59811i	−0.253116	−	1.59811i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.680881	−	0.494689i	−0.680881	−	0.494689i
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$811$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$	0.618034			0.618034
$814$		0			0
$815$		0			0
$816$	1.03213	−	0.749885i	1.03213	−	0.749885i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.610425	+	1.87869i	−0.610425	+	1.87869i	−0.156434	+	0.987688i	$0.550000\pi$
$821$	−0.610425	+	1.87869i	−0.610425	+	1.87869i	−0.453990	+	0.891007i	$0.650000\pi$
$822$		0			0
$823$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$823$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$824$	0.699596			0.699596
$825$	0.453990	−	0.891007i	0.453990	−	0.891007i
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$829$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$829$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−0.166977	+	0.513904i	−0.166977	+	0.513904i
$833$	0.550672	+	1.69480i	0.550672	+	1.69480i
$834$		0			0
$835$		0			0
$836$	1.52890	−	0.779012i	1.52890	−	0.779012i
$837$		0			0
$838$	0.0791922	−	0.0575365i	0.0791922	−	0.0575365i
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	−2.34786	−	1.70582i	−2.34786	−	1.70582i
$842$		0			0
$843$	0.610425	−	1.87869i	0.610425	−	1.87869i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−0.223989			−0.223989
$849$	0.500000	−	0.363271i	0.500000	−	0.363271i
$850$	0.172288	+	0.530249i	0.172288	+	0.530249i
$851$		0			0
$852$	−1.03213	−	0.749885i	−1.03213	−	0.749885i
$853$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$853$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	−0.327712	+	0.166977i	−0.327712	+	0.166977i
$859$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$859$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$863$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$864$	0.662667	+	0.481456i	0.662667	+	0.481456i
$865$		0			0
$866$		0			0
$867$	1.76007	−	1.27877i	1.76007	−	1.27877i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.113656	+	0.349798i	0.113656	+	0.349798i
$873$		0			0
$874$	0.951057	+	0.690983i	0.951057	+	0.690983i
$875$		0			0
$876$	0.451057	−	1.38821i	0.451057	−	1.38821i
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$	0.409551	−	0.297556i	0.409551	−	0.297556i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.253116	+	0.183900i	−0.253116	+	0.183900i
$883$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$883$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$884$	−0.583987	+	1.79733i	−0.583987	+	1.79733i
$885$		0			0
$886$		0			0
$887$	−0.280582	+	0.863541i	−0.280582	+	0.863541i	0.707107	+	0.707107i	$0.250000\pi$
$887$	−0.280582	+	0.863541i	−0.280582	+	0.863541i	−0.987688	+	0.156434i	$0.950000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.87869	+	1.36495i	1.87869	+	1.36495i
$898$		0			0
$899$		0			0
$900$	0.729825	−	0.530249i	0.729825	−	0.530249i
$901$	−0.557537			−0.557537
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$907$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$908$	−1.03213	−	0.749885i	−1.03213	−	0.749885i
$909$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$910$		0			0
$911$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$911$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$912$	1.36176			1.36176
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.327712	+	1.00859i	0.327712	+	1.00859i
$917$		0			0
$918$	0.451057	+	0.327712i	0.451057	+	0.327712i
$919$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$919$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.66251			1.66251
$924$		0			0
$925$		0			0
$926$	−0.481456	+	0.349798i	−0.481456	+	0.349798i
$927$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$928$	0.500000	−	1.53884i	0.500000	−	1.53884i
$929$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$932$	−0.253116	−	0.779012i	−0.253116	−	0.779012i
$933$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$934$	−0.284079			−0.284079
$935$		0			0
$936$	−0.699596			−0.699596
$937$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$937$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.253116	+	0.183900i	0.253116	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$941$	0.253116	+	0.183900i	0.253116	+	0.183900i	−0.453990	+	0.891007i	$0.650000\pi$
$942$		0			0
$943$		0			0
$944$	−0.525896	+	0.382085i	−0.525896	+	0.382085i
$945$		0			0
$946$		0			0
$947$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$947$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$948$		0			0
$949$	0.587785	+	1.80902i	0.587785	+	1.80902i
$950$	−0.183900	+	0.565985i	−0.183900	+	0.565985i
$951$		0			0
$952$		0			0
$953$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	−0.987688	−	0.156434i	$-0.950000\pi$
$953$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	0.891007	+	0.453990i	$0.150000\pi$
$954$	−0.0302487	−	0.0930960i	−0.0302487	−	0.0930960i
$955$		0			0
$956$		0			0
$957$	−1.76007	+	0.896802i	−1.76007	+	0.896802i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.309017	−	0.951057i	0.309017	−	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$967$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$968$	−0.565985	−	0.183900i	−0.565985	−	0.183900i
$969$	3.38959			3.38959
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$	0.278768	−	0.857960i	0.278768	−	0.857960i
$973$		0			0
$974$	−0.409551	−	0.297556i	−0.409551	−	0.297556i
$975$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$976$	−0.221232	−	0.680881i	−0.221232	−	0.680881i
$977$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$977$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$978$	−0.367799			−0.367799
$979$	−1.58779	+	0.809017i	−1.58779	+	0.809017i
$980$		0			0
$981$	0.500000	−	0.363271i	0.500000	−	0.363271i
$982$		0			0
$983$	0.550672	−	1.69480i	0.550672	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$983$	0.550672	−	1.69480i	0.550672	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$984$		0			0
$985$		0			0
$986$	0.340334	−	1.04744i	0.340334	−	1.04744i
$987$		0			0
$988$	−1.63194	+	1.18567i	−1.63194	+	1.18567i
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$997$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.1.bm.d.1829.2	yes	16
3.2	odd	2	inner	2013.1.bm.d.1829.3	yes	16
11.4	even	5	inner	2013.1.bm.d.1280.2	✓	16
33.26	odd	10	inner	2013.1.bm.d.1280.3	yes	16
61.60	even	2	inner	2013.1.bm.d.1829.3	yes	16
183.182	odd	2	CM	2013.1.bm.d.1829.2	yes	16
671.609	even	10	inner	2013.1.bm.d.1280.3	yes	16
2013.1280	odd	10	inner	2013.1.bm.d.1280.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.1.bm.d.1280.2	✓	16	11.4	even	5	inner
2013.1.bm.d.1280.2	✓	16	2013.1280	odd	10	inner
2013.1.bm.d.1280.3	yes	16	33.26	odd	10	inner
2013.1.bm.d.1280.3	yes	16	671.609	even	10	inner
2013.1.bm.d.1829.2	yes	16	1.1	even	1	trivial
2013.1.bm.d.1829.2	yes	16	183.182	odd	2	CM
2013.1.bm.d.1829.3	yes	16	3.2	odd	2	inner
2013.1.bm.d.1829.3	yes	16	61.60	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2013.bm (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.253116 + 0.183900i) q^{2} +(-0.309017 - 0.951057i) q^{3} +(-0.278768 + 0.857960i) q^{4} +(0.253116 + 0.183900i) q^{6} +(-0.183900 - 0.565985i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\)	\(q+(-0.253116 + 0.183900i) q^{2} +(-0.309017 - 0.951057i) q^{3} +(-0.278768 + 0.857960i) q^{4} +(0.253116 + 0.183900i) q^{6} +(-0.183900 - 0.565985i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.891007 - 0.453990i) q^{11} +0.902113 q^{12} +(-0.951057 + 0.690983i) q^{13} +(-0.579192 - 0.420808i) q^{16} +(-1.44168 - 1.04744i) q^{17} +(0.0966818 - 0.297556i) q^{18} +(-0.587785 - 1.80902i) q^{19} +(-0.142040 + 0.278768i) q^{22} +1.97538 q^{23} +(-0.481456 + 0.349798i) q^{24} +(0.309017 + 0.951057i) q^{25} +(0.113656 - 0.349798i) q^{26} +(0.809017 + 0.587785i) q^{27} +(0.610425 - 1.87869i) q^{29} +0.819101 q^{32} +(-0.707107 - 0.707107i) q^{33} +0.557537 q^{34} +(-0.278768 - 0.857960i) q^{36} +(0.481456 + 0.349798i) q^{38} +(0.951057 + 0.690983i) q^{39} +(0.141122 + 0.891007i) q^{44} +(-0.500000 + 0.363271i) q^{46} +(-0.221232 + 0.680881i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-0.253116 - 0.183900i) q^{50} +(-0.550672 + 1.69480i) q^{51} +(-0.327712 - 1.00859i) q^{52} +(0.253116 - 0.183900i) q^{53} -0.312869 q^{54} +(-1.53884 + 1.11803i) q^{57} +(0.190983 + 0.587785i) q^{58} +(0.280582 - 0.863541i) q^{59} +(0.809017 + 0.587785i) q^{61} +(0.371864 - 0.270175i) q^{64} +(0.309017 + 0.0489435i) q^{66} +(1.30056 - 0.944910i) q^{68} +(-0.610425 - 1.87869i) q^{69} +(-1.14412 - 0.831254i) q^{71} +(0.481456 + 0.349798i) q^{72} +(0.500000 - 1.53884i) q^{73} +(0.809017 - 0.587785i) q^{75} +1.71592 q^{76} -0.367799 q^{78} +(0.309017 - 0.951057i) q^{81} -1.97538 q^{87} +(-0.420808 - 0.420808i) q^{88} -1.78201 q^{89} +(-0.550672 + 1.69480i) q^{92} +(-0.253116 - 0.779012i) q^{96} +0.312869 q^{98} +(-0.453990 + 0.891007i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9	\( 16 q + 4 q^{3} - 4 q^{4} - 4 q^{9} - 16 q^{12} - 16 q^{16} + 4 q^{22} - 4 q^{25} + 4 q^{27} + 8 q^{34} - 4 q^{36} - 8 q^{46} - 4 q^{48} - 4 q^{49} - 20 q^{52} + 12 q^{58} + 4 q^{61} - 16 q^{64} - 4 q^{66} + 8 q^{73} + 4 q^{75} + 40 q^{76} - 4 q^{81}+O(q^{100}) \) 16 * q + 4 * q^3 - 4 * q^4 - 4 * q^9 - 16 * q^12 - 16 * q^16 + 4 * q^22 - 4 * q^25 + 4 * q^27 + 8 * q^34 - 4 * q^36 - 8 * q^46 - 4 * q^48 - 4 * q^49 - 20 * q^52 + 12 * q^58 + 4 * q^61 - 16 * q^64 - 4 * q^66 + 8 * q^73 + 4 * q^75 + 40 * q^76 - 4 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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