LMFDB - Embedded newform 2200.1.cl.d.1851.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,1,Mod(251,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 0, 6]))

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

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

chi := DirichletCharacter("2200.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2200.cl (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:	$1.09794302779$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{15} - \cdots)$

Embedding invariants

Embedding label			1851.1
Root			$-0.978148 + 0.207912i$ of defining polynomial
Character	$\chi$	$=$	2200.1851
Dual form			2200.1.cl.d.851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.809017 - 0.587785i) q^{2} +(-0.564602 - 1.73767i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-1.47815 - 1.07394i) q^{6} +(-0.309017 - 0.951057i) q^{8} +(-1.89169 + 1.37440i) q^{9} +O(q^{10})$	\(q+(0.809017 - 0.587785i) q^{2} +(-0.564602 - 1.73767i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-1.47815 - 1.07394i) q^{6} +(-0.309017 - 0.951057i) q^{8} +(-1.89169 + 1.37440i) q^{9} +(0.669131 - 0.743145i) q^{11} -1.82709 q^{12} +(-0.809017 - 0.587785i) q^{16} +(-1.58268 - 1.14988i) q^{17} +(-0.722562 + 2.22382i) q^{18} +(0.413545 + 1.27276i) q^{19} +(0.104528 - 0.994522i) q^{22} +(-1.47815 + 1.07394i) q^{24} +(1.97815 + 1.43721i) q^{27} -1.00000 q^{32} +(-1.66913 - 0.743145i) q^{33} -1.95630 q^{34} +(0.722562 + 2.22382i) q^{36} +(1.08268 + 0.786610i) q^{38} +(-0.0646021 - 0.198825i) q^{41} +1.61803 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-0.564602 + 1.73767i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-1.10453 + 3.39939i) q^{51} +2.44512 q^{54} +(1.97815 - 1.43721i) q^{57} +(0.190983 - 0.587785i) q^{59} +(-0.809017 + 0.587785i) q^{64} +(-1.78716 + 0.379874i) q^{66} +1.95630 q^{67} +(-1.58268 + 1.14988i) q^{68} +(1.89169 + 1.37440i) q^{72} +(0.0646021 - 0.198825i) q^{73} +1.33826 q^{76} +(0.657960 - 2.02499i) q^{81} +(-0.169131 - 0.122881i) q^{82} +(-0.169131 - 0.122881i) q^{83} +(1.30902 - 0.951057i) q^{86} +(-0.913545 - 0.406737i) q^{88} +1.82709 q^{89} +(0.564602 + 1.73767i) q^{96} +(-1.30902 + 0.951057i) q^{97} -1.00000 q^{98} +(-0.244415 + 2.32545i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{6} + 2 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 3 * q^6 + 2 * q^8	$ 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{6} + 2 q^{8} + q^{11} - 2 q^{12} - 2 q^{16} - 2 q^{17} + 5 q^{18} - 3 q^{19} - q^{22} - 3 q^{24} + 7 q^{27} - 8 q^{32} - 9 q^{33} + 2 q^{34} - 5 q^{36} - 2 q^{38} + 2 q^{41} + 4 q^{43} - 4 q^{44} - 2 q^{48} - 2 q^{49} - 7 q^{51} - 2 q^{54} + 7 q^{57} + 6 q^{59} - 2 q^{64} - q^{66} - 2 q^{67} - 2 q^{68} - 2 q^{73} + 2 q^{76} - 3 q^{81} + 3 q^{82} + 3 q^{83} + 6 q^{86} - q^{88} + 2 q^{89} + 2 q^{96} - 6 q^{97} - 8 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 3 * q^6 + 2 * q^8 + q^11 - 2 * q^12 - 2 * q^16 - 2 * q^17 + 5 * q^18 - 3 * q^19 - q^22 - 3 * q^24 + 7 * q^27 - 8 * q^32 - 9 * q^33 + 2 * q^34 - 5 * q^36 - 2 * q^38 + 2 * q^41 + 4 * q^43 - 4 * q^44 - 2 * q^48 - 2 * q^49 - 7 * q^51 - 2 * q^54 + 7 * q^57 + 6 * q^59 - 2 * q^64 - q^66 - 2 * q^67 - 2 * q^68 - 2 * q^73 + 2 * q^76 - 3 * q^81 + 3 * q^82 + 3 * q^83 + 6 * q^86 - q^88 + 2 * q^89 + 2 * q^96 - 6 * q^97 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{4}{5}\right)$

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 $

$2$	0.809017	−	0.587785i	0.809017	−	0.587785i
$2$	0.809017	−	0.587785i	0.809017	−	0.587785i
$3$	−0.564602	−	1.73767i	−0.564602	−	1.73767i	−0.669131	−	0.743145i	$-0.733333\pi$
$3$	−0.564602	−	1.73767i	−0.564602	−	1.73767i	0.104528	−	0.994522i	$-0.466667\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$		0			0
$5$		0			0
$6$	−1.47815	−	1.07394i	−1.47815	−	1.07394i
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$9$	−1.89169	+	1.37440i	−1.89169	+	1.37440i
$10$		0			0
$11$	0.669131	−	0.743145i	0.669131	−	0.743145i
$11$	0.669131	−	0.743145i	0.669131	−	0.743145i
$12$	−1.82709			−1.82709
$13$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$13$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$14$		0			0
$15$		0			0
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$	−1.58268	−	1.14988i	−1.58268	−	1.14988i	−0.913545	−	0.406737i	$-0.866667\pi$
$17$	−1.58268	−	1.14988i	−1.58268	−	1.14988i	−0.669131	−	0.743145i	$-0.733333\pi$
$18$	−0.722562	+	2.22382i	−0.722562	+	2.22382i
$19$	0.413545	+	1.27276i	0.413545	+	1.27276i	0.913545	+	0.406737i	$0.133333\pi$
$19$	0.413545	+	1.27276i	0.413545	+	1.27276i	−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$	0.104528	−	0.994522i	0.104528	−	0.994522i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	−1.47815	+	1.07394i	−1.47815	+	1.07394i
$25$		0			0
$26$		0			0
$27$	1.97815	+	1.43721i	1.97815	+	1.43721i
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$	−1.00000			−1.00000
$33$	−1.66913	−	0.743145i	−1.66913	−	0.743145i
$34$	−1.95630			−1.95630
$35$		0			0
$36$	0.722562	+	2.22382i	0.722562	+	2.22382i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$	1.08268	+	0.786610i	1.08268	+	0.786610i
$39$		0			0
$40$		0			0
$41$	−0.0646021	−	0.198825i	−0.0646021	−	0.198825i	0.913545	−	0.406737i	$-0.133333\pi$
$41$	−0.0646021	−	0.198825i	−0.0646021	−	0.198825i	−0.978148	+	0.207912i	$0.933333\pi$
$42$		0			0
$43$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$43$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$44$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$45$		0			0
$46$		0			0
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	−0.564602	+	1.73767i	−0.564602	+	1.73767i
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$		0			0
$51$	−1.10453	+	3.39939i	−1.10453	+	3.39939i
$52$		0			0
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$	2.44512			2.44512
$55$		0			0
$56$		0			0
$57$	1.97815	−	1.43721i	1.97815	−	1.43721i
$58$		0			0
$59$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$59$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$60$		0			0
$61$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$61$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$		0			0
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$	−1.78716	+	0.379874i	−1.78716	+	0.379874i
$67$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$67$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$68$	−1.58268	+	1.14988i	−1.58268	+	1.14988i
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$	1.89169	+	1.37440i	1.89169	+	1.37440i
$73$	0.0646021	−	0.198825i	0.0646021	−	0.198825i	−0.913545	−	0.406737i	$-0.866667\pi$
$73$	0.0646021	−	0.198825i	0.0646021	−	0.198825i	0.978148	+	0.207912i	$0.0666667\pi$
$74$		0			0
$75$		0			0
$76$	1.33826			1.33826
$77$		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			0
$81$	0.657960	−	2.02499i	0.657960	−	2.02499i
$82$	−0.169131	−	0.122881i	−0.169131	−	0.122881i
$83$	−0.169131	−	0.122881i	−0.169131	−	0.122881i	0.500000	−	0.866025i	$-0.333333\pi$
$83$	−0.169131	−	0.122881i	−0.169131	−	0.122881i	−0.669131	+	0.743145i	$0.733333\pi$
$84$		0			0
$85$		0			0
$86$	1.30902	−	0.951057i	1.30902	−	0.951057i
$87$		0			0
$88$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$89$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$89$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.564602	+	1.73767i	0.564602	+	1.73767i
$97$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$97$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$98$	−1.00000			−1.00000
$99$	−0.244415	+	2.32545i	−0.244415	+	2.32545i
$100$		0			0
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$	1.10453	+	3.39939i	1.10453	+	3.39939i
$103$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$103$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.564602	−	1.73767i	−0.564602	−	1.73767i	−0.669131	−	0.743145i	$-0.733333\pi$
$107$	−0.564602	−	1.73767i	−0.564602	−	1.73767i	0.104528	−	0.994522i	$-0.466667\pi$
$108$	1.97815	−	1.43721i	1.97815	−	1.43721i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.604528	+	1.86055i	0.604528	+	1.86055i	0.500000	+	0.866025i	$0.333333\pi$
$113$	0.604528	+	1.86055i	0.604528	+	1.86055i	0.104528	+	0.994522i	$0.466667\pi$
$114$	0.755585	−	2.32545i	0.755585	−	2.32545i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$119$		0			0
$120$		0			0
$121$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$122$		0			0
$123$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$129$	−0.913545	−	2.81160i	−0.913545	−	2.81160i
$130$		0			0
$131$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$131$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$132$	−1.22256	+	1.35779i	−1.22256	+	1.35779i
$133$		0			0
$134$	1.58268	−	1.14988i	1.58268	−	1.14988i
$135$		0			0
$136$	−0.604528	+	1.86055i	−0.604528	+	1.86055i
$137$	−0.169131	−	0.122881i	−0.169131	−	0.122881i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	−0.169131	−	0.122881i	−0.169131	−	0.122881i	−0.669131	+	0.743145i	$0.733333\pi$
$138$		0			0
$139$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$139$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	2.33826			2.33826
$145$		0			0
$146$	−0.0646021	−	0.198825i	−0.0646021	−	0.198825i
$147$	−0.564602	+	1.73767i	−0.564602	+	1.73767i
$148$		0			0
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$		0			0
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$	1.08268	−	0.786610i	1.08268	−	0.786610i
$153$	4.57433			4.57433
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.657960	−	2.02499i	−0.657960	−	2.02499i
$163$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.104528	−	0.994522i	$-0.466667\pi$
$163$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.978148	+	0.207912i	$0.0666667\pi$
$164$	−0.209057			−0.209057
$165$		0			0
$166$	−0.209057			−0.209057
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	0.309017	−	0.951057i	0.309017	−	0.951057i
$170$		0			0
$171$	−2.53158	−	1.83930i	−2.53158	−	1.83930i
$172$	0.500000	−	1.53884i	0.500000	−	1.53884i
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$177$	−1.12920			−1.12920
$178$	1.47815	−	1.07394i	1.47815	−	1.07394i
$179$	−0.604528	−	1.86055i	−0.604528	−	1.86055i	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−0.604528	−	1.86055i	−0.604528	−	1.86055i	−0.104528	−	0.994522i	$-0.533333\pi$
$180$		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			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.91355	+	0.406737i	−1.91355	+	0.406737i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$191$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$192$	1.47815	+	1.07394i	1.47815	+	1.07394i
$193$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$193$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$194$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$195$		0			0
$196$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	1.16913	+	2.02499i	1.16913	+	2.02499i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−1.10453	−	3.39939i	−1.10453	−	3.39939i
$202$		0			0
$203$		0			0
$204$	2.89169	+	2.10094i	2.89169	+	2.10094i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.22256	+	0.544320i	1.22256	+	0.544320i
$210$		0			0
$211$	0.169131	−	0.122881i	0.169131	−	0.122881i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	0.169131	−	0.122881i	0.169131	−	0.122881i	0.669131	+	0.743145i	$0.266667\pi$
$212$		0			0
$213$		0			0
$214$	−1.47815	−	1.07394i	−1.47815	−	1.07394i
$215$		0			0
$216$	0.755585	−	2.32545i	0.755585	−	2.32545i
$217$		0			0
$218$		0			0
$219$	−0.381966			−0.381966
$220$		0			0
$221$		0			0
$222$		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			0
$226$	1.58268	+	1.14988i	1.58268	+	1.14988i
$227$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$227$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$228$	−0.755585	−	2.32545i	−0.755585	−	2.32545i
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$233$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$237$		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
$241$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$241$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$242$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$243$	−1.44512			−1.44512
$244$		0			0
$245$		0			0
$246$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$247$		0			0
$248$		0			0
$249$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$250$		0			0
$251$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$251$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$257$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$258$	−2.39169	−	1.73767i	−2.39169	−	1.73767i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.500000	−	0.363271i	0.500000	−	0.363271i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−0.190983	+	1.81708i	−0.190983	+	1.81708i
$265$		0			0
$266$		0			0
$267$	−1.03158	−	3.17488i	−1.03158	−	3.17488i
$268$	0.604528	−	1.86055i	0.604528	−	1.86055i
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$	0.604528	+	1.86055i	0.604528	+	1.86055i
$273$		0			0
$274$	−0.209057			−0.209057
$275$		0			0
$276$		0			0
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$	0.309017	+	0.951057i	0.309017	+	0.951057i
$279$		0			0
$280$		0			0
$281$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$281$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$283$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.89169	−	1.37440i	1.89169	−	1.37440i
$289$	0.873619	+	2.68872i	0.873619	+	2.68872i
$290$		0			0
$291$	2.39169	+	1.73767i	2.39169	+	1.73767i
$292$	−0.169131	−	0.122881i	−0.169131	−	0.122881i
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$	0.564602	+	1.73767i	0.564602	+	1.73767i
$295$		0			0
$296$		0			0
$297$	2.39169	−	0.508370i	2.39169	−	0.508370i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.413545	−	1.27276i	0.413545	−	1.27276i
$305$		0			0
$306$	3.70071	−	2.68872i	3.70071	−	2.68872i
$307$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$307$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$313$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$314$		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			0
$319$		0			0
$320$		0			0
$321$	−2.70071	+	1.96218i	−2.70071	+	1.96218i
$322$		0			0
$323$	0.809017	−	2.48990i	0.809017	−	2.48990i
$324$	−1.72256	−	1.25151i	−1.72256	−	1.25151i
$325$		0			0
$326$	0.413545	−	1.27276i	0.413545	−	1.27276i
$327$		0			0
$328$	−0.169131	+	0.122881i	−0.169131	+	0.122881i
$329$		0			0
$330$		0			0
$331$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$331$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$332$	−0.169131	+	0.122881i	−0.169131	+	0.122881i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.413545	+	1.27276i	−0.413545	+	1.27276i	0.500000	+	0.866025i	$0.333333\pi$
$337$	−0.413545	+	1.27276i	−0.413545	+	1.27276i	−0.913545	+	0.406737i	$0.866667\pi$
$338$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$339$	2.89169	−	2.10094i	2.89169	−	2.10094i
$340$		0			0
$341$		0			0
$342$	−3.12920			−3.12920
$343$		0			0
$344$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$345$		0			0
$346$		0			0
$347$	1.08268	+	0.786610i	1.08268	+	0.786610i	0.978148	−	0.207912i	$-0.0666667\pi$
$347$	1.08268	+	0.786610i	1.08268	+	0.786610i	0.104528	+	0.994522i	$0.466667\pi$
$348$		0			0
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$		0			0
$352$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$353$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$353$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$354$	−0.913545	+	0.663730i	−0.913545	+	0.663730i
$355$		0			0
$356$	0.564602	−	1.73767i	0.564602	−	1.73767i
$357$		0			0
$358$	−1.58268	−	1.14988i	−1.58268	−	1.14988i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−0.639886	+	0.464905i	−0.639886	+	0.464905i
$362$		0			0
$363$	−1.66913	+	0.743145i	−1.66913	+	0.743145i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$367$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$368$		0			0
$369$	0.395472	+	0.287327i	0.395472	+	0.287327i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	−1.30902	+	1.45381i	−1.30902	+	1.45381i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.169131	+	0.122881i	0.169131	+	0.122881i	0.669131	−	0.743145i	$-0.266667\pi$
$379$	0.169131	+	0.122881i	0.169131	+	0.122881i	−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$383$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$384$	1.82709			1.82709
$385$		0			0
$386$	−1.00000			−1.00000
$387$	−3.06082	+	2.22382i	−3.06082	+	2.22382i
$388$	0.500000	+	1.53884i	0.500000	+	1.53884i
$389$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$389$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$390$		0			0
$391$		0			0
$392$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$393$	−0.348943	−	1.07394i	−0.348943	−	1.07394i
$394$		0			0
$395$		0			0
$396$	2.13611	+	0.951057i	2.13611	+	0.951057i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.169131	+	0.122881i	0.169131	+	0.122881i	0.669131	−	0.743145i	$-0.266667\pi$
$401$	0.169131	+	0.122881i	0.169131	+	0.122881i	−0.500000	+	0.866025i	$0.666667\pi$
$402$	−2.89169	−	2.10094i	−2.89169	−	2.10094i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	3.57433			3.57433
$409$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$409$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$410$		0			0
$411$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.82709			1.82709
$418$	1.30902	−	0.278240i	1.30902	−	0.278240i
$419$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$419$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$	0.0646021	−	0.198825i	0.0646021	−	0.198825i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.82709			−1.82709
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	−0.755585	−	2.32545i	−0.755585	−	2.32545i
$433$	−0.413545	+	1.27276i	−0.413545	+	1.27276i	0.500000	+	0.866025i	$0.333333\pi$
$433$	−0.413545	+	1.27276i	−0.413545	+	1.27276i	−0.913545	+	0.406737i	$0.866667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	2.33826			2.33826
$442$		0			0
$443$	0.0646021	+	0.198825i	0.0646021	+	0.198825i	0.978148	−	0.207912i	$-0.0666667\pi$
$443$	0.0646021	+	0.198825i	0.0646021	+	0.198825i	−0.913545	+	0.406737i	$0.866667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.58268	−	1.14988i	1.58268	−	1.14988i	0.669131	−	0.743145i	$-0.266667\pi$
$449$	1.58268	−	1.14988i	1.58268	−	1.14988i	0.913545	−	0.406737i	$-0.133333\pi$
$450$		0			0
$451$	−0.190983	−	0.0850311i	−0.190983	−	0.0850311i
$452$	1.95630			1.95630
$453$		0			0
$454$	0.190983	+	0.587785i	0.190983	+	0.587785i
$455$		0			0
$456$	−1.97815	−	1.43721i	−1.97815	−	1.43721i
$457$	1.47815	+	1.07394i	1.47815	+	1.07394i	0.978148	+	0.207912i	$0.0666667\pi$
$457$	1.47815	+	1.07394i	1.47815	+	1.07394i	0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$	−1.47815	−	4.54927i	−1.47815	−	4.54927i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$467$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$467$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.618034			−0.618034
$473$	1.08268	−	1.20243i	1.08268	−	1.20243i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		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
$482$	−1.58268	+	1.14988i	−1.58268	+	1.14988i
$483$		0			0
$484$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$485$		0			0
$486$	−1.16913	+	0.849423i	−1.16913	+	0.849423i
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$		0			0
$489$	−1.97815	−	1.43721i	−1.97815	−	1.43721i
$490$		0			0
$491$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$491$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$492$	0.118034	+	0.363271i	0.118034	+	0.363271i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.118034	+	0.363271i	0.118034	+	0.363271i
$499$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$499$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$500$		0			0
$501$		0			0
$502$	0.309017	−	0.951057i	0.309017	−	0.951057i
$503$		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			0
$507$	−1.82709			−1.82709
$508$		0			0
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$	0.809017	+	0.587785i	0.809017	+	0.587785i
$513$	−1.01117	+	3.11206i	−1.01117	+	3.11206i
$514$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$515$		0			0
$516$	−2.95630			−2.95630
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.604528	+	1.86055i	−0.604528	+	1.86055i	−0.104528	+	0.994522i	$0.533333\pi$
$521$	−0.604528	+	1.86055i	−0.604528	+	1.86055i	−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	1.08268	+	0.786610i	1.08268	+	0.786610i	0.978148	−	0.207912i	$-0.0666667\pi$
$523$	1.08268	+	0.786610i	1.08268	+	0.786610i	0.104528	+	0.994522i	$0.466667\pi$
$524$	0.190983	−	0.587785i	0.190983	−	0.587785i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.913545	+	1.58231i	0.913545	+	1.58231i
$529$	1.00000			1.00000
$530$		0			0
$531$	0.446568	+	1.37440i	0.446568	+	1.37440i
$532$		0			0
$533$		0			0
$534$	−2.70071	−	1.96218i	−2.70071	−	1.96218i
$535$		0			0
$536$	−0.604528	−	1.86055i	−0.604528	−	1.86055i
$537$	−2.89169	+	2.10094i	−2.89169	+	2.10094i
$538$		0			0
$539$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$	1.58268	+	1.14988i	1.58268	+	1.14988i
$545$		0			0
$546$		0			0
$547$	0.604528	+	1.86055i	0.604528	+	1.86055i	0.500000	+	0.866025i	$0.333333\pi$
$547$	0.604528	+	1.86055i	0.604528	+	1.86055i	0.104528	+	0.994522i	$0.466667\pi$
$548$	−0.169131	+	0.122881i	−0.169131	+	0.122881i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.809017	+	0.587785i	0.809017	+	0.587785i
$557$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$557$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.78716	+	3.09546i	1.78716	+	3.09546i
$562$	1.61803			1.61803
$563$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$563$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$	0.809017	+	0.587785i	0.809017	+	0.587785i
$567$		0			0
$568$		0			0
$569$	0.413545	+	1.27276i	0.413545	+	1.27276i	0.913545	+	0.406737i	$0.133333\pi$
$569$	0.413545	+	1.27276i	0.413545	+	1.27276i	−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	2.00000			2.00000			1.00000			$0$
$571$	2.00000			2.00000			1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.722562	−	2.22382i	0.722562	−	2.22382i
$577$	1.08268	+	0.786610i	1.08268	+	0.786610i	0.978148	−	0.207912i	$-0.0666667\pi$
$577$	1.08268	+	0.786610i	1.08268	+	0.786610i	0.104528	+	0.994522i	$0.466667\pi$
$578$	2.28716	+	1.66172i	2.28716	+	1.66172i
$579$	−0.564602	+	1.73767i	−0.564602	+	1.73767i
$580$		0			0
$581$		0			0
$582$	2.95630			2.95630
$583$		0			0
$584$	−0.209057			−0.209057
$585$		0			0
$586$		0			0
$587$	0.604528	−	1.86055i	0.604528	−	1.86055i	0.104528	−	0.994522i	$-0.466667\pi$
$587$	0.604528	−	1.86055i	0.604528	−	1.86055i	0.500000	−	0.866025i	$-0.333333\pi$
$588$	1.47815	+	1.07394i	1.47815	+	1.07394i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$593$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$594$	1.63611	−	1.81708i	1.63611	−	1.81708i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$599$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$600$		0			0
$601$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i	−0.978148	−	0.207912i	$-0.933333\pi$
$601$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i	0.913545	+	0.406737i	$0.133333\pi$
$602$		0			0
$603$	−3.70071	+	2.68872i	−3.70071	+	2.68872i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$	−0.413545	−	1.27276i	−0.413545	−	1.27276i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.41355	−	4.35045i	1.41355	−	4.35045i
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$	−1.47815	+	1.07394i	−1.47815	+	1.07394i
$615$		0			0
$616$		0			0
$617$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$617$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$618$		0			0
$619$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$619$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	0.618034			0.618034
$627$	0.255585	−	2.43173i	0.255585	−	2.43173i
$628$		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
$633$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$641$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$642$	−1.03158	+	3.17488i	−1.03158	+	3.17488i
$643$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$643$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$	−2.12920			−2.12920
$649$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$650$		0			0
$651$		0			0
$652$	−0.413545	−	1.27276i	−0.413545	−	1.27276i
$653$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$653$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i
$657$	0.151057	+	0.464905i	0.151057	+	0.464905i
$658$		0			0
$659$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$659$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	1.47815	−	1.07394i	1.47815	−	1.07394i
$663$		0			0
$664$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$673$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$674$	0.413545	+	1.27276i	0.413545	+	1.27276i
$675$		0			0
$676$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$	1.10453	−	3.39939i	1.10453	−	3.39939i
$679$		0			0
$680$		0			0
$681$	1.12920			1.12920
$682$		0			0
$683$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$684$	−2.53158	+	1.83930i	−2.53158	+	1.83930i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$689$		0			0
$690$		0			0
$691$	−1.08268	+	0.786610i	−1.08268	+	0.786610i	−0.978148	−	0.207912i	$-0.933333\pi$
$691$	−1.08268	+	0.786610i	−1.08268	+	0.786610i	−0.104528	+	0.994522i	$0.533333\pi$
$692$		0			0
$693$		0			0
$694$	1.33826			1.33826
$695$		0			0
$696$		0			0
$697$	−0.126381	+	0.388960i	−0.126381	+	0.388960i
$698$		0			0
$699$	2.39169	+	1.73767i	2.39169	+	1.73767i
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$705$		0			0
$706$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$707$		0			0
$708$	−0.348943	+	1.07394i	−0.348943	+	1.07394i
$709$		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
$712$	−0.564602	−	1.73767i	−0.564602	−	1.73767i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.95630			−1.95630
$717$		0			0
$718$		0			0
$719$		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
$722$	−0.244415	+	0.752232i	−0.244415	+	0.752232i
$723$	1.10453	+	3.39939i	1.10453	+	3.39939i
$724$		0			0
$725$		0			0
$726$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.157960	+	0.486152i	0.157960	+	0.486152i
$730$		0			0
$731$	−2.56082	−	1.86055i	−2.56082	−	1.86055i
$732$		0			0
$733$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$733$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.30902	−	1.45381i	1.30902	−	1.45381i
$738$	0.488830			0.488830
$739$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$739$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.488830			0.488830
$748$	−0.204489	+	1.94558i	−0.204489	+	1.94558i
$749$		0			0
$750$		0			0
$751$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$751$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$752$		0			0
$753$	−1.47815	−	1.07394i	−1.47815	−	1.07394i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$757$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$758$	0.209057			0.209057
$759$		0			0
$760$		0			0
$761$	1.58268	−	1.14988i	1.58268	−	1.14988i	0.669131	−	0.743145i	$-0.266667\pi$
$761$	1.58268	−	1.14988i	1.58268	−	1.14988i	0.913545	−	0.406737i	$-0.133333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.47815	−	1.07394i	1.47815	−	1.07394i
$769$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−2.95630			−2.95630
$772$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$	−1.16913	+	3.59821i	−1.16913	+	3.59821i
$775$		0			0
$776$	1.30902	+	0.951057i	1.30902	+	0.951057i
$777$		0			0
$778$		0			0
$779$	0.226341	−	0.164446i	0.226341	−	0.164446i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.309017	+	0.951057i	0.309017	+	0.951057i
$785$		0			0
$786$	−0.913545	−	0.663730i	−0.913545	−	0.663730i
$787$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$787$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	2.28716	−	0.486152i	2.28716	−	0.486152i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		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			0
$801$	−3.45630	+	2.51115i	−3.45630	+	2.51115i
$802$	0.209057			0.209057
$803$	−0.104528	−	0.181049i	−0.104528	−	0.181049i
$804$	−3.57433			−3.57433
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$809$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$810$		0			0
$811$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	2.89169	−	2.10094i	2.89169	−	2.10094i
$817$	0.669131	+	2.05937i	0.669131	+	2.05937i
$818$	0.309017	−	0.951057i	0.309017	−	0.951057i
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$	0.118034	+	0.363271i	0.118034	+	0.363271i
$823$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$823$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.104528	−	0.994522i	$-0.466667\pi$
$827$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.978148	+	0.207912i	$0.0666667\pi$
$828$		0			0
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.604528	+	1.86055i	0.604528	+	1.86055i
$834$	1.47815	−	1.07394i	1.47815	−	1.07394i
$835$		0			0
$836$	0.895472	−	0.994522i	0.895472	−	0.994522i
$837$		0			0
$838$	−0.169131	+	0.122881i	−0.169131	+	0.122881i
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$	0.913545	−	2.81160i	0.913545	−	2.81160i
$844$	−0.0646021	−	0.198825i	−0.0646021	−	0.198825i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.47815	−	1.07394i	1.47815	−	1.07394i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$	−1.47815	+	1.07394i	−1.47815	+	1.07394i
$857$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$857$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$858$		0			0
$859$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$859$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$860$		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	$0.200000\pi$
$864$	−1.97815	−	1.43721i	−1.97815	−	1.43721i
$865$		0			0
$866$	0.413545	+	1.27276i	0.413545	+	1.27276i
$867$	4.17886	−	3.03612i	4.17886	−	3.03612i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.16913	−	3.59821i	1.16913	−	3.59821i
$874$		0			0
$875$		0			0
$876$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$881$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$882$	1.89169	−	1.37440i	1.89169	−	1.37440i
$883$	0.0646021	+	0.198825i	0.0646021	+	0.198825i	0.978148	−	0.207912i	$-0.0666667\pi$
$883$	0.0646021	+	0.198825i	0.0646021	+	0.198825i	−0.913545	+	0.406737i	$0.866667\pi$
$884$		0			0
$885$		0			0
$886$	0.169131	+	0.122881i	0.169131	+	0.122881i
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.06460	−	1.84395i	−1.06460	−	1.84395i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.604528	−	1.86055i	0.604528	−	1.86055i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.204489	+	0.0434654i	−0.204489	+	0.0434654i
$903$		0			0
$904$	1.58268	−	1.14988i	1.58268	−	1.14988i
$905$		0			0
$906$		0			0
$907$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$907$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$908$	0.500000	+	0.363271i	0.500000	+	0.363271i
$909$		0			0
$910$		0			0
$911$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$911$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$912$	−2.44512			−2.44512
$913$	−0.204489	+	0.0434654i	−0.204489	+	0.0434654i
$914$	1.82709			1.82709
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−3.86984	−	2.81160i	−3.86984	−	2.81160i
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$	1.03158	+	3.17488i	1.03158	+	3.17488i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$929$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	0.413545	−	1.27276i	0.413545	−	1.27276i
$932$	0.500000	+	1.53884i	0.500000	+	1.53884i
$933$		0			0
$934$	2.00000			2.00000
$935$		0			0
$936$		0			0
$937$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.104528	−	0.994522i	$-0.466667\pi$
$937$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.978148	+	0.207912i	$0.0666667\pi$
$938$		0			0
$939$	0.348943	−	1.07394i	0.348943	−	1.07394i
$940$		0			0
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$945$		0			0
$946$	0.169131	−	1.60917i	0.169131	−	1.60917i
$947$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$947$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.0646021	−	0.198825i	0.0646021	−	0.198825i	−0.913545	−	0.406737i	$-0.866667\pi$
$953$	0.0646021	−	0.198825i	0.0646021	−	0.198825i	0.978148	+	0.207912i	$0.0666667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.309017	−	0.951057i	0.309017	−	0.951057i
$962$		0			0
$963$	3.45630	+	2.51115i	3.45630	+	2.51115i
$964$	−0.604528	+	1.86055i	−0.604528	+	1.86055i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$969$	−4.78339			−4.78339
$970$		0			0
$971$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$971$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$972$	−0.446568	+	1.37440i	−0.446568	+	1.37440i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$977$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$978$	−2.44512			−2.44512
$979$	1.22256	−	1.35779i	1.22256	−	1.35779i
$980$		0			0
$981$		0			0
$982$	0.500000	+	1.53884i	0.500000	+	1.53884i
$983$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$983$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$984$	0.309017	+	0.224514i	0.309017	+	0.224514i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−1.03158	−	3.17488i	−1.03158	−	3.17488i
$994$		0			0
$995$		0			0
$996$	0.309017	+	0.224514i	0.309017	+	0.224514i
$997$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$997$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$998$	0.500000	+	1.53884i	0.500000	+	1.53884i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.1.cl.d.1851.1	yes	8
5.2	odd	4		2200.1.dd.b.1499.3		16
5.3	odd	4		2200.1.dd.b.1499.2		16
5.4	even	2		2200.1.cl.b.1851.2	yes	8
8.3	odd	2	CM	2200.1.cl.d.1851.1	yes	8
11.4	even	5	inner	2200.1.cl.d.851.1	yes	8
40.3	even	4		2200.1.dd.b.1499.2		16
40.19	odd	2		2200.1.cl.b.1851.2	yes	8
40.27	even	4		2200.1.dd.b.1499.3		16
55.4	even	10		2200.1.cl.b.851.2	✓	8
55.37	odd	20		2200.1.dd.b.499.2		16
55.48	odd	20		2200.1.dd.b.499.3		16
88.59	odd	10	inner	2200.1.cl.d.851.1	yes	8
440.59	odd	10		2200.1.cl.b.851.2	✓	8
440.147	even	20		2200.1.dd.b.499.2		16
440.323	even	20		2200.1.dd.b.499.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.1.cl.b.851.2	✓	8	55.4	even	10
2200.1.cl.b.851.2	✓	8	440.59	odd	10
2200.1.cl.b.1851.2	yes	8	5.4	even	2
2200.1.cl.b.1851.2	yes	8	40.19	odd	2
2200.1.cl.d.851.1	yes	8	11.4	even	5	inner
2200.1.cl.d.851.1	yes	8	88.59	odd	10	inner
2200.1.cl.d.1851.1	yes	8	1.1	even	1	trivial
2200.1.cl.d.1851.1	yes	8	8.3	odd	2	CM
2200.1.dd.b.499.2		16	55.37	odd	20
2200.1.dd.b.499.2		16	440.147	even	20
2200.1.dd.b.499.3		16	55.48	odd	20
2200.1.dd.b.499.3		16	440.323	even	20
2200.1.dd.b.1499.2		16	5.3	odd	4
2200.1.dd.b.1499.2		16	40.3	even	4
2200.1.dd.b.1499.3		16	5.2	odd	4
2200.1.dd.b.1499.3		16	40.27	even	4

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:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2200.cl (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 - 0.587785i) q^{2} +(-0.564602 - 1.73767i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-1.47815 - 1.07394i) q^{6} +(-0.309017 - 0.951057i) q^{8} +(-1.89169 + 1.37440i) q^{9} +O(q^{10})\)	\(q+(0.809017 - 0.587785i) q^{2} +(-0.564602 - 1.73767i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-1.47815 - 1.07394i) q^{6} +(-0.309017 - 0.951057i) q^{8} +(-1.89169 + 1.37440i) q^{9} +(0.669131 - 0.743145i) q^{11} -1.82709 q^{12} +(-0.809017 - 0.587785i) q^{16} +(-1.58268 - 1.14988i) q^{17} +(-0.722562 + 2.22382i) q^{18} +(0.413545 + 1.27276i) q^{19} +(0.104528 - 0.994522i) q^{22} +(-1.47815 + 1.07394i) q^{24} +(1.97815 + 1.43721i) q^{27} -1.00000 q^{32} +(-1.66913 - 0.743145i) q^{33} -1.95630 q^{34} +(0.722562 + 2.22382i) q^{36} +(1.08268 + 0.786610i) q^{38} +(-0.0646021 - 0.198825i) q^{41} +1.61803 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-0.564602 + 1.73767i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-1.10453 + 3.39939i) q^{51} +2.44512 q^{54} +(1.97815 - 1.43721i) q^{57} +(0.190983 - 0.587785i) q^{59} +(-0.809017 + 0.587785i) q^{64} +(-1.78716 + 0.379874i) q^{66} +1.95630 q^{67} +(-1.58268 + 1.14988i) q^{68} +(1.89169 + 1.37440i) q^{72} +(0.0646021 - 0.198825i) q^{73} +1.33826 q^{76} +(0.657960 - 2.02499i) q^{81} +(-0.169131 - 0.122881i) q^{82} +(-0.169131 - 0.122881i) q^{83} +(1.30902 - 0.951057i) q^{86} +(-0.913545 - 0.406737i) q^{88} +1.82709 q^{89} +(0.564602 + 1.73767i) q^{96} +(-1.30902 + 0.951057i) q^{97} -1.00000 q^{98} +(-0.244415 + 2.32545i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{6} + 2 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 3 * q^6 + 2 * q^8	\( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{6} + 2 q^{8} + q^{11} - 2 q^{12} - 2 q^{16} - 2 q^{17} + 5 q^{18} - 3 q^{19} - q^{22} - 3 q^{24} + 7 q^{27} - 8 q^{32} - 9 q^{33} + 2 q^{34} - 5 q^{36} - 2 q^{38} + 2 q^{41} + 4 q^{43} - 4 q^{44} - 2 q^{48} - 2 q^{49} - 7 q^{51} - 2 q^{54} + 7 q^{57} + 6 q^{59} - 2 q^{64} - q^{66} - 2 q^{67} - 2 q^{68} - 2 q^{73} + 2 q^{76} - 3 q^{81} + 3 q^{82} + 3 q^{83} + 6 q^{86} - q^{88} + 2 q^{89} + 2 q^{96} - 6 q^{97} - 8 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 3 * q^6 + 2 * q^8 + q^11 - 2 * q^12 - 2 * q^16 - 2 * q^17 + 5 * q^18 - 3 * q^19 - q^22 - 3 * q^24 + 7 * q^27 - 8 * q^32 - 9 * q^33 + 2 * q^34 - 5 * q^36 - 2 * q^38 + 2 * q^41 + 4 * q^43 - 4 * q^44 - 2 * q^48 - 2 * q^49 - 7 * q^51 - 2 * q^54 + 7 * q^57 + 6 * q^59 - 2 * q^64 - q^66 - 2 * q^67 - 2 * q^68 - 2 * q^73 + 2 * q^76 - 3 * q^81 + 3 * q^82 + 3 * q^83 + 6 * q^86 - q^88 + 2 * q^89 + 2 * q^96 - 6 * q^97 - 8 * q^98

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