LMFDB - Embedded newform 3332.1.cc.c.1495.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(135,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 32, 21]))

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

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

chi := DirichletCharacter("3332.135");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3332.cc (of order $42$, degree $12$, 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:	$1.66288462209$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{42})$
Coefficient field:	$\Q(\zeta_{84})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 $ x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Embedding invariants

Embedding label			1495.1
Root			$-0.149042 + 0.988831i$ of defining polynomial
Character	$\chi$	$=$	3332.1495
Dual form			3332.1.cc.c.2991.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.955573 + 0.294755i) q^{2} +(-0.680173 - 1.73305i) q^{3} +(0.826239 - 0.563320i) q^{4} +(1.16078 + 1.45557i) q^{6} +(-0.781831 - 0.623490i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-1.80778 + 1.67738i) q^{9} +O(q^{10})$	\(q+(-0.955573 + 0.294755i) q^{2} +(-0.680173 - 1.73305i) q^{3} +(0.826239 - 0.563320i) q^{4} +(1.16078 + 1.45557i) q^{6} +(-0.781831 - 0.623490i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-1.80778 + 1.67738i) q^{9} +(-0.432142 - 0.400969i) q^{11} +(-1.53825 - 1.04876i) q^{12} +(0.425270 - 1.86323i) q^{13} +(0.930874 + 0.365341i) q^{14} +(0.365341 - 0.930874i) q^{16} +(0.0747301 - 0.997204i) q^{17} +(1.23305 - 2.13571i) q^{18} +(-0.548760 + 1.77904i) q^{21} +(0.531130 + 0.255779i) q^{22} +(-0.145713 - 1.94440i) q^{23} +(1.77904 + 0.548760i) q^{24} +(0.955573 + 0.294755i) q^{25} +(0.142820 + 1.90580i) q^{26} +(2.45921 + 1.18429i) q^{27} +(-0.997204 - 0.0747301i) q^{28} +(0.781831 - 1.35417i) q^{31} +(-0.0747301 + 0.997204i) q^{32} +(-0.400969 + 1.02165i) q^{33} +(0.222521 + 0.974928i) q^{34} +(-0.548760 + 2.40427i) q^{36} +(-3.51833 + 0.530303i) q^{39} -1.86175i q^{42} +(-0.582926 - 0.0878620i) q^{44} +(0.712362 + 1.81507i) q^{46} -1.86175 q^{48} +(0.222521 + 0.974928i) q^{49} -1.00000 q^{50} +(-1.77904 + 0.548760i) q^{51} +(-0.698220 - 1.77904i) q^{52} +(-1.63402 + 1.11406i) q^{53} +(-2.69903 - 0.406813i) q^{54} +(0.974928 - 0.222521i) q^{56} +(-0.347948 + 1.52446i) q^{62} +(2.45921 - 0.184292i) q^{63} +(-0.222521 - 0.974928i) q^{64} +(0.0820177 - 1.09445i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-3.27064 + 1.57506i) q^{69} +(0.268565 + 0.129334i) q^{71} +(-0.184292 - 2.45921i) q^{72} +(-0.139129 - 1.85654i) q^{75} +(0.0878620 + 0.582926i) q^{77} +(3.20571 - 1.54379i) q^{78} +(0.680173 + 1.17809i) q^{79} +(0.195461 - 2.60825i) q^{81} +(0.548760 + 1.77904i) q^{84} +(0.582926 - 0.0878620i) q^{88} +(0.535628 - 0.496990i) q^{89} +(-1.49419 + 1.19158i) q^{91} +(-1.21572 - 1.52446i) q^{92} +(-2.87863 - 0.433884i) q^{93} +(1.77904 - 0.548760i) q^{96} +(-0.500000 - 0.866025i) q^{98} +1.45379 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9}+O(q^{10}) $ 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9	$ 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52} - 2 q^{53} - 4 q^{64} - 22 q^{66} - 12 q^{68} - 14 q^{69} - 4 q^{72} - 6 q^{77} + 30 q^{81} - 6 q^{84} + 2 q^{89} - 12 q^{98}+O(q^{100}) $ 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9 + 10 * q^13 + 2 * q^16 + 2 * q^17 + 10 * q^18 + 6 * q^21 + 2 * q^25 - 2 * q^26 - 2 * q^32 + 8 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^49 - 24 * q^50 + 2 * q^52 - 2 * q^53 - 4 * q^64 - 22 * q^66 - 12 * q^68 - 14 * q^69 - 4 * q^72 - 6 * q^77 + 30 * q^81 - 6 * q^84 + 2 * q^89 - 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$.

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$e\left(\frac{8}{21}\right)$	$-1$

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 $

$2$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$2$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$3$	−0.680173	−	1.73305i	−0.680173	−	1.73305i	−0.680173	−	0.733052i	$-0.738095\pi$
$3$	−0.680173	−	1.73305i	−0.680173	−	1.73305i		−	1.00000i	$-0.5\pi$
$4$	0.826239	−	0.563320i	0.826239	−	0.563320i
$5$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$5$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$6$	1.16078	+	1.45557i	1.16078	+	1.45557i
$7$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$7$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$8$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$9$	−1.80778	+	1.67738i	−1.80778	+	1.67738i
$10$		0			0
$11$	−0.432142	−	0.400969i	−0.432142	−	0.400969i	0.433884	−	0.900969i	$-0.357143\pi$
$11$	−0.432142	−	0.400969i	−0.432142	−	0.400969i	−0.866025	+	0.500000i	$0.833333\pi$
$12$	−1.53825	−	1.04876i	−1.53825	−	1.04876i
$13$	0.425270	−	1.86323i	0.425270	−	1.86323i	−0.0747301	−	0.997204i	$-0.523810\pi$
$13$	0.425270	−	1.86323i	0.425270	−	1.86323i	0.500000	−	0.866025i	$-0.333333\pi$
$14$	0.930874	+	0.365341i	0.930874	+	0.365341i
$15$		0			0
$16$	0.365341	−	0.930874i	0.365341	−	0.930874i
$17$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$17$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$18$	1.23305	−	2.13571i	1.23305	−	2.13571i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$	−0.548760	+	1.77904i	−0.548760	+	1.77904i
$22$	0.531130	+	0.255779i	0.531130	+	0.255779i
$23$	−0.145713	−	1.94440i	−0.145713	−	1.94440i	−0.294755	−	0.955573i	$-0.595238\pi$
$23$	−0.145713	−	1.94440i	−0.145713	−	1.94440i	0.149042	−	0.988831i	$-0.452381\pi$
$24$	1.77904	+	0.548760i	1.77904	+	0.548760i
$25$	0.955573	+	0.294755i	0.955573	+	0.294755i
$26$	0.142820	+	1.90580i	0.142820	+	1.90580i
$27$	2.45921	+	1.18429i	2.45921	+	1.18429i
$28$	−0.997204	−	0.0747301i	−0.997204	−	0.0747301i
$29$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$29$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$30$		0			0
$31$	0.781831	−	1.35417i	0.781831	−	1.35417i	−0.149042	−	0.988831i	$-0.547619\pi$
$31$	0.781831	−	1.35417i	0.781831	−	1.35417i	0.930874	−	0.365341i	$-0.119048\pi$
$32$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$33$	−0.400969	+	1.02165i	−0.400969	+	1.02165i
$34$	0.222521	+	0.974928i	0.222521	+	0.974928i
$35$		0			0
$36$	−0.548760	+	2.40427i	−0.548760	+	2.40427i
$37$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$37$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$38$		0			0
$39$	−3.51833	+	0.530303i	−3.51833	+	0.530303i
$40$		0			0
$41$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$41$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$42$		−	1.86175i		−	1.86175i
$43$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$43$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$44$	−0.582926	−	0.0878620i	−0.582926	−	0.0878620i
$45$		0			0
$46$	0.712362	+	1.81507i	0.712362	+	1.81507i
$47$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$47$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$48$	−1.86175			−1.86175
$49$	0.222521	+	0.974928i	0.222521	+	0.974928i
$50$	−1.00000			−1.00000
$51$	−1.77904	+	0.548760i	−1.77904	+	0.548760i
$52$	−0.698220	−	1.77904i	−0.698220	−	1.77904i
$53$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.733052	+	0.680173i	$0.761905\pi$
$53$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.900969	+	0.433884i	$0.857143\pi$
$54$	−2.69903	−	0.406813i	−2.69903	−	0.406813i
$55$		0			0
$56$	0.974928	−	0.222521i	0.974928	−	0.222521i
$57$		0			0
$58$		0			0
$59$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$59$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$60$		0			0
$61$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$61$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$62$	−0.347948	+	1.52446i	−0.347948	+	1.52446i
$63$	2.45921	−	0.184292i	2.45921	−	0.184292i
$64$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$65$		0			0
$66$	0.0820177	−	1.09445i	0.0820177	−	1.09445i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$69$	−3.27064	+	1.57506i	−3.27064	+	1.57506i
$70$		0			0
$71$	0.268565	+	0.129334i	0.268565	+	0.129334i	0.563320	−	0.826239i	$-0.309524\pi$
$71$	0.268565	+	0.129334i	0.268565	+	0.129334i	−0.294755	+	0.955573i	$0.595238\pi$
$72$	−0.184292	−	2.45921i	−0.184292	−	2.45921i
$73$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$73$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$74$		0			0
$75$	−0.139129	−	1.85654i	−0.139129	−	1.85654i
$76$		0			0
$77$	0.0878620	+	0.582926i	0.0878620	+	0.582926i
$78$	3.20571	−	1.54379i	3.20571	−	1.54379i
$79$	0.680173	+	1.17809i	0.680173	+	1.17809i	0.974928	+	0.222521i	$0.0714286\pi$
$79$	0.680173	+	1.17809i	0.680173	+	1.17809i	−0.294755	+	0.955573i	$0.595238\pi$
$80$		0			0
$81$	0.195461	−	2.60825i	0.195461	−	2.60825i
$82$		0			0
$83$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$83$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$84$	0.548760	+	1.77904i	0.548760	+	1.77904i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.582926	−	0.0878620i	0.582926	−	0.0878620i
$89$	0.535628	−	0.496990i	0.535628	−	0.496990i	−0.365341	−	0.930874i	$-0.619048\pi$
$89$	0.535628	−	0.496990i	0.535628	−	0.496990i	0.900969	+	0.433884i	$0.142857\pi$
$90$		0			0
$91$	−1.49419	+	1.19158i	−1.49419	+	1.19158i
$92$	−1.21572	−	1.52446i	−1.21572	−	1.52446i
$93$	−2.87863	−	0.433884i	−2.87863	−	0.433884i
$94$		0			0
$95$		0			0
$96$	1.77904	−	0.548760i	1.77904	−	0.548760i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$99$	1.45379			1.45379
$100$	0.955573	−	0.294755i	0.955573	−	0.294755i
$101$	0.162592	+	0.414278i	0.162592	+	0.414278i	0.988831	−	0.149042i	$-0.0476190\pi$
$101$	0.162592	+	0.414278i	0.162592	+	0.414278i	−0.826239	+	0.563320i	$0.809524\pi$
$102$	1.53825	−	1.04876i	1.53825	−	1.04876i
$103$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$103$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$104$	1.19158	+	1.49419i	1.19158	+	1.49419i
$105$		0			0
$106$	1.23305	−	1.54620i	1.23305	−	1.54620i
$107$	−0.825886	+	0.766310i	−0.825886	+	0.766310i	−0.974928	−	0.222521i	$-0.928571\pi$
$107$	−0.825886	+	0.766310i	−0.825886	+	0.766310i	0.149042	+	0.988831i	$0.452381\pi$
$108$	2.69903	−	0.406813i	2.69903	−	0.406813i
$109$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$109$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$110$		0			0
$111$		0			0
$112$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$113$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$113$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.35654	+	4.08165i	2.35654	+	4.08165i
$118$		0			0
$119$	−0.680173	+	0.733052i	−0.680173	+	0.733052i
$120$		0			0
$121$	−0.0487597	−	0.650653i	−0.0487597	−	0.650653i
$122$		0			0
$123$		0			0
$124$	−0.116853	−	1.55929i	−0.116853	−	1.55929i
$125$		0			0
$126$	−2.29563	+	0.900969i	−2.29563	+	0.900969i
$127$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$127$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$	−0.712362	+	1.81507i	−0.712362	+	1.81507i	−0.149042	+	0.988831i	$0.547619\pi$
$131$	−0.712362	+	1.81507i	−0.712362	+	1.81507i	−0.563320	+	0.826239i	$0.690476\pi$
$132$	0.244221	+	1.07000i	0.244221	+	1.07000i
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.733052	+	0.680173i	0.733052	+	0.680173i
$137$	0.722521	−	0.108903i	0.722521	−	0.108903i	0.222521	−	0.974928i	$-0.428571\pi$
$137$	0.722521	−	0.108903i	0.722521	−	0.108903i	0.500000	+	0.866025i	$0.333333\pi$
$138$	2.66108	−	2.46912i	2.66108	−	2.46912i
$139$	−0.848162	+	1.06356i	−0.848162	+	1.06356i	0.149042	+	0.988831i	$0.452381\pi$
$139$	−0.848162	+	1.06356i	−0.848162	+	1.06356i	−0.997204	+	0.0747301i	$0.976190\pi$
$140$		0			0
$141$		0			0
$142$	−0.294755	−	0.0444272i	−0.294755	−	0.0444272i
$143$	−0.930874	+	0.634659i	−0.930874	+	0.634659i
$144$	0.900969	+	2.29563i	0.900969	+	2.29563i
$145$		0			0
$146$		0			0
$147$	1.53825	−	1.04876i	1.53825	−	1.04876i
$148$		0			0
$149$	0.698220	−	0.215372i	0.698220	−	0.215372i	0.0747301	−	0.997204i	$-0.476190\pi$
$149$	0.698220	−	0.215372i	0.698220	−	0.215372i	0.623490	+	0.781831i	$0.285714\pi$
$150$	0.680173	+	1.73305i	0.680173	+	1.73305i
$151$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$151$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$152$		0			0
$153$	1.53759	+	1.92808i	1.53759	+	1.92808i
$154$	−0.255779	−	0.531130i	−0.255779	−	0.531130i
$155$		0			0
$156$	−2.60825	+	2.42010i	−2.60825	+	2.42010i
$157$	1.44973	−	0.218511i	1.44973	−	0.218511i	0.623490	−	0.781831i	$-0.285714\pi$
$157$	1.44973	−	0.218511i	1.44973	−	0.218511i	0.826239	+	0.563320i	$0.190476\pi$
$158$	−0.997204	−	0.925270i	−0.997204	−	0.925270i
$159$	3.04213	+	2.07409i	3.04213	+	2.07409i
$160$		0			0
$161$	−1.09839	+	1.61105i	−1.09839	+	1.61105i
$162$	0.582018	+	2.54999i	0.582018	+	2.54999i
$163$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$163$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.79690	−	0.865341i	1.79690	−	0.865341i	0.866025	−	0.500000i	$-0.166667\pi$
$167$	1.79690	−	0.865341i	1.79690	−	0.865341i	0.930874	−	0.365341i	$-0.119048\pi$
$168$	−1.04876	−	1.53825i	−1.04876	−	1.53825i
$169$	−2.38980	−	1.15087i	−2.38980	−	1.15087i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$173$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$174$		0			0
$175$	−0.563320	−	0.826239i	−0.563320	−	0.826239i
$176$	−0.531130	+	0.255779i	−0.531130	+	0.255779i
$177$		0			0
$178$	−0.365341	+	0.632789i	−0.365341	+	0.632789i
$179$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$179$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$180$		0			0
$181$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$181$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$182$	1.07659	−	1.57906i	1.07659	−	1.57906i
$183$		0			0
$184$	1.61105	+	1.09839i	1.61105	+	1.09839i
$185$		0			0
$186$	2.87863	−	0.433884i	2.87863	−	0.433884i
$187$	−0.432142	+	0.400969i	−0.432142	+	0.400969i
$188$		0			0
$189$	−1.18429	−	2.45921i	−1.18429	−	2.45921i
$190$		0			0
$191$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$191$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$192$	−1.53825	+	1.04876i	−1.53825	+	1.04876i
$193$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$193$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$194$		0			0
$195$		0			0
$196$	0.733052	+	0.680173i	0.733052	+	0.680173i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−1.38921	+	0.428513i	−1.38921	+	0.428513i
$199$	0.108903	+	0.277479i	0.108903	+	0.277479i	0.974928	−	0.222521i	$-0.0714286\pi$
$199$	0.108903	+	0.277479i	0.108903	+	0.277479i	−0.866025	+	0.500000i	$0.833333\pi$
$200$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$201$		0			0
$202$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$203$		0			0
$204$	−1.16078	+	1.45557i	−1.16078	+	1.45557i
$205$		0			0
$206$		0			0
$207$	3.52491	+	3.27064i	3.52491	+	3.27064i
$208$	−1.57906	−	1.07659i	−1.57906	−	1.07659i
$209$		0			0
$210$		0			0
$211$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	−0.974928	−	0.222521i	$-0.928571\pi$
$211$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	0.781831	−	0.623490i	$-0.214286\pi$
$212$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$213$	0.0414721	−	0.553406i	0.0414721	−	0.553406i
$214$	0.563320	−	0.975699i	0.563320	−	0.975699i
$215$		0			0
$216$	−2.45921	+	1.18429i	−2.45921	+	1.18429i
$217$	−1.45557	+	0.571270i	−1.45557	+	0.571270i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.82624	−	0.563320i	−1.82624	−	0.563320i
$222$		0			0
$223$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$223$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$224$	0.680173	−	0.733052i	0.680173	−	0.733052i
$225$	−2.22188	+	1.07000i	−2.22188	+	1.07000i
$226$		0			0
$227$	−0.149042	+	0.258149i	−0.149042	+	0.258149i	−0.930874	−	0.365341i	$-0.880952\pi$
$227$	−0.149042	+	0.258149i	−0.149042	+	0.258149i	0.781831	+	0.623490i	$0.214286\pi$
$228$		0			0
$229$	0.658322	−	1.67738i	0.658322	−	1.67738i	−0.0747301	−	0.997204i	$-0.523810\pi$
$229$	0.658322	−	1.67738i	0.658322	−	1.67738i	0.733052	−	0.680173i	$-0.238095\pi$
$230$		0			0
$231$	0.950480	−	0.548760i	0.950480	−	0.548760i
$232$		0			0
$233$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$233$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$234$	−3.45493	−	3.20571i	−3.45493	−	3.20571i
$235$		0			0
$236$		0			0
$237$	1.57906	−	1.98008i	1.57906	−	1.98008i
$238$	0.433884	−	0.900969i	0.433884	−	0.900969i
$239$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$239$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$240$		0			0
$241$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$241$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$242$	0.238377	+	0.607374i	0.238377	+	0.607374i
$243$	−2.04493	+	0.630777i	−2.04493	+	0.630777i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.571270	+	1.45557i	0.571270	+	1.45557i
$249$		0			0
$250$		0			0
$251$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$251$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$252$	1.92808	−	1.53759i	1.92808	−	1.53759i
$253$	−0.716677	+	0.898684i	−0.716677	+	0.898684i
$254$		0			0
$255$		0			0
$256$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$257$	0.826239	+	0.563320i	0.826239	+	0.563320i	0.900969	−	0.433884i	$-0.142857\pi$
$257$	0.826239	+	0.563320i	0.826239	+	0.563320i	−0.0747301	+	0.997204i	$0.523810\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.145713	−	1.94440i	0.145713	−	1.94440i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	−0.548760	−	0.950480i	−0.548760	−	0.950480i
$265$		0			0
$266$		0			0
$267$	−1.22563	−	0.590232i	−1.22563	−	0.590232i
$268$		0			0
$269$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$269$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$270$		0			0
$271$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$271$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$272$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$273$	3.08138	+	1.77904i	3.08138	+	1.77904i
$274$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$275$	−0.294755	−	0.510531i	−0.294755	−	0.510531i
$276$	−1.81507	+	3.14379i	−1.81507	+	3.14379i
$277$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$277$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$278$	0.496990	−	1.26631i	0.496990	−	1.26631i
$279$	0.858075	+	3.75947i	0.858075	+	3.75947i
$280$		0			0
$281$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$281$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$282$		0			0
$283$	0.432142	+	0.400969i	0.432142	+	0.400969i	0.866025	−	0.500000i	$-0.166667\pi$
$283$	0.432142	+	0.400969i	0.432142	+	0.400969i	−0.433884	+	0.900969i	$0.642857\pi$
$284$	0.294755	−	0.0444272i	0.294755	−	0.0444272i
$285$		0			0
$286$	0.702449	−	0.880843i	0.702449	−	0.880843i
$287$		0			0
$288$	−1.53759	−	1.92808i	−1.53759	−	1.92808i
$289$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$293$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$294$	−1.16078	+	1.45557i	−1.16078	+	1.45557i
$295$		0			0
$296$		0			0
$297$	−0.587862	−	1.49785i	−0.587862	−	1.49785i
$298$	−0.603718	+	0.411608i	−0.603718	+	0.411608i
$299$	−3.68484	−	0.555400i	−3.68484	−	0.555400i
$300$	−1.16078	−	1.45557i	−1.16078	−	1.45557i
$301$		0			0
$302$		0			0
$303$	0.607374	−	0.563561i	0.607374	−	0.563561i
$304$		0			0
$305$		0			0
$306$	−2.03759	−	1.38921i	−2.03759	−	1.38921i
$307$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$307$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$308$	0.400969	+	0.432142i	0.400969	+	0.432142i
$309$		0			0
$310$		0			0
$311$	−0.129436	+	1.72721i	−0.129436	+	1.72721i	0.433884	+	0.900969i	$0.357143\pi$
$311$	−0.129436	+	1.72721i	−0.129436	+	1.72721i	−0.563320	+	0.826239i	$0.690476\pi$
$312$	1.77904	−	3.08138i	1.77904	−	3.08138i
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	−1.32091	+	0.636119i	−1.32091	+	0.636119i
$315$		0			0
$316$	1.22563	+	0.590232i	1.22563	+	0.590232i
$317$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$317$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$318$	−3.51833	−	1.08526i	−3.51833	−	1.08526i
$319$		0			0
$320$		0			0
$321$	1.88980	+	0.910080i	1.88980	+	0.910080i
$322$	0.574730	−	1.86323i	0.574730	−	1.86323i
$323$		0			0
$324$	−1.30778	−	2.26514i	−1.30778	−	2.26514i
$325$	0.955573	−	1.65510i	0.955573	−	1.65510i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$331$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$332$		0			0
$333$		0			0
$334$	−1.46200	+	1.35654i	−1.46200	+	1.35654i
$335$		0			0
$336$	1.45557	+	1.16078i	1.45557	+	1.16078i
$337$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$337$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$338$	2.62285	+	0.395331i	2.62285	+	0.395331i
$339$		0			0
$340$		0			0
$341$	−0.880843	+	0.271704i	−0.880843	+	0.271704i
$342$		0			0
$343$	0.433884	−	0.900969i	0.433884	−	0.900969i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.716983	−	0.488831i	0.716983	−	0.488831i	−0.149042	−	0.988831i	$-0.547619\pi$
$347$	0.716983	−	0.488831i	0.716983	−	0.488831i	0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$349$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$350$	0.781831	+	0.623490i	0.781831	+	0.623490i
$351$	3.25243	−	4.07842i	3.25243	−	4.07842i
$352$	0.432142	−	0.400969i	0.432142	−	0.400969i
$353$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.365341	−	0.930874i	$-0.380952\pi$
$353$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.623490	+	0.781831i	$0.285714\pi$
$354$		0			0
$355$		0			0
$356$	0.162592	−	0.712362i	0.162592	−	0.712362i
$357$	1.73305	+	0.680173i	1.73305	+	0.680173i
$358$		0			0
$359$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$359$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$	−1.09445	+	0.527060i	−1.09445	+	0.527060i
$364$	−0.563320	+	1.82624i	−0.563320	+	1.82624i
$365$		0			0
$366$		0			0
$367$	−1.07659	−	0.332083i	−1.07659	−	0.332083i	−0.294755	−	0.955573i	$-0.595238\pi$
$367$	−1.07659	−	0.332083i	−1.07659	−	0.332083i	−0.781831	+	0.623490i	$0.785714\pi$
$368$	−1.86323	−	0.574730i	−1.86323	−	0.574730i
$369$		0			0
$370$		0			0
$371$	1.97213	+	0.147791i	1.97213	+	0.147791i
$372$	−2.62285	+	1.26310i	−2.62285	+	1.26310i
$373$	0.955573	+	1.65510i	0.955573	+	1.65510i	0.733052	+	0.680173i	$0.238095\pi$
$373$	0.955573	+	1.65510i	0.955573	+	1.65510i	0.222521	+	0.974928i	$0.428571\pi$
$374$	0.294755	−	0.510531i	0.294755	−	0.510531i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	1.85654	+	2.00088i	1.85654	+	2.00088i
$379$	0.385418	−	1.68862i	0.385418	−	1.68862i	−0.294755	−	0.955573i	$-0.595238\pi$
$379$	0.385418	−	1.68862i	0.385418	−	1.68862i	0.680173	−	0.733052i	$-0.261905\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$383$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$384$	1.16078	−	1.45557i	1.16078	−	1.45557i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.0546039	−	0.139129i	−0.0546039	−	0.139129i	0.900969	−	0.433884i	$-0.142857\pi$
$389$	−0.0546039	−	0.139129i	−0.0546039	−	0.139129i	−0.955573	+	0.294755i	$0.904762\pi$
$390$		0			0
$391$	−1.94986			−1.94986
$392$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$393$	3.63014			3.63014
$394$		0			0
$395$		0			0
$396$	1.20118	−	0.818951i	1.20118	−	0.818951i
$397$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$397$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$398$	−0.185853	−	0.233052i	−0.185853	−	0.233052i
$399$		0			0
$400$	0.623490	−	0.781831i	0.623490	−	0.781831i
$401$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$401$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$402$		0			0
$403$	−2.19064	−	2.03262i	−2.19064	−	2.03262i
$404$	0.367711	+	0.250701i	0.367711	+	0.250701i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.680173	−	1.73305i	0.680173	−	1.73305i
$409$	−0.147791	+	1.97213i	−0.147791	+	1.97213i	0.0747301	+	0.997204i	$0.476190\pi$
$409$	−0.147791	+	1.97213i	−0.147791	+	1.97213i	−0.222521	+	0.974928i	$0.571429\pi$
$410$		0			0
$411$	−0.680173	−	1.17809i	−0.680173	−	1.17809i
$412$		0			0
$413$		0			0
$414$	−4.33235	−	2.08635i	−4.33235	−	2.08635i
$415$		0			0
$416$	1.82624	+	0.563320i	1.82624	+	0.563320i
$417$	2.42010	+	0.746503i	2.42010	+	0.746503i
$418$		0			0
$419$	1.22563	+	0.590232i	1.22563	+	0.590232i	0.930874	−	0.365341i	$-0.119048\pi$
$419$	1.22563	+	0.590232i	1.22563	+	0.590232i	0.294755	+	0.955573i	$0.404762\pi$
$420$		0			0
$421$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.222521	−	0.974928i	$-0.428571\pi$
$421$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.900969	+	0.433884i	$0.142857\pi$
$422$	0.433884	+	0.751509i	0.433884	+	0.751509i
$423$		0			0
$424$	0.147791	−	1.97213i	0.147791	−	1.97213i
$425$	0.365341	−	0.930874i	0.365341	−	0.930874i
$426$	0.123490	+	0.541044i	0.123490	+	0.541044i
$427$		0			0
$428$	−0.250701	+	1.09839i	−0.250701	+	1.09839i
$429$	1.73305	+	1.18157i	1.73305	+	1.18157i
$430$		0			0
$431$	1.71271	−	0.258149i	1.71271	−	0.258149i	0.781831	−	0.623490i	$-0.214286\pi$
$431$	1.71271	−	0.258149i	1.71271	−	0.258149i	0.930874	+	0.365341i	$0.119048\pi$
$432$	2.00088	−	1.85654i	2.00088	−	1.85654i
$433$	0.277479	−	0.347948i	0.277479	−	0.347948i	−0.623490	−	0.781831i	$-0.714286\pi$
$433$	0.277479	−	0.347948i	0.277479	−	0.347948i	0.900969	+	0.433884i	$0.142857\pi$
$434$	1.22252	−	0.974928i	1.22252	−	0.974928i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.29991	−	0.400969i	1.29991	−	0.400969i	0.433884	−	0.900969i	$-0.357143\pi$
$439$	1.29991	−	0.400969i	1.29991	−	0.400969i	0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	−2.03759	−	1.38921i	−2.03759	−	1.38921i
$442$	1.91115			1.91115
$443$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$443$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−0.848162	−	1.06356i	−0.848162	−	1.06356i
$448$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$449$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$449$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$450$	1.80778	−	1.67738i	1.80778	−	1.67738i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	0.0663300	−	0.290611i	0.0663300	−	0.290611i
$455$		0			0
$456$		0			0
$457$	0.162592	−	0.414278i	0.162592	−	0.414278i	−0.826239	−	0.563320i	$-0.809524\pi$
$457$	0.162592	−	0.414278i	0.162592	−	0.414278i	0.988831	+	0.149042i	$0.0476190\pi$
$458$	−0.134659	+	1.79690i	−0.134659	+	1.79690i
$459$	1.36476	−	2.36383i	1.36476	−	2.36383i
$460$		0			0
$461$	1.78181	−	0.858075i	1.78181	−	0.858075i	0.826239	−	0.563320i	$-0.190476\pi$
$461$	1.78181	−	0.858075i	1.78181	−	0.858075i	0.955573	−	0.294755i	$-0.0952381\pi$
$462$	−0.746503	+	0.804539i	−0.746503	+	0.804539i
$463$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$463$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$467$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$468$	4.24634	+	2.04493i	4.24634	+	2.04493i
$469$		0			0
$470$		0			0
$471$	−1.36476	−	2.36383i	−1.36476	−	2.36383i
$472$		0			0
$473$		0			0
$474$	−0.925270	+	2.35755i	−0.925270	+	2.35755i
$475$		0			0
$476$	−0.149042	+	0.988831i	−0.149042	+	0.988831i
$477$	1.08526	−	4.75484i	1.08526	−	4.75484i
$478$		0			0
$479$	−1.42935	−	1.32624i	−1.42935	−	1.32624i	−0.866025	−	0.500000i	$-0.833333\pi$
$479$	−1.42935	−	1.32624i	−1.42935	−	1.32624i	−0.563320	−	0.826239i	$-0.690476\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	3.53912	+	0.807782i	3.53912	+	0.807782i
$484$	−0.406813	−	0.510127i	−0.406813	−	0.510127i
$485$		0			0
$486$	1.76815	−	1.20551i	1.76815	−	1.20551i
$487$	0.632789	+	1.61232i	0.632789	+	1.61232i	0.781831	+	0.623490i	$0.214286\pi$
$487$	0.632789	+	1.61232i	0.632789	+	1.61232i	−0.149042	+	0.988831i	$0.547619\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.974928	−	1.22252i	−0.974928	−	1.22252i
$497$	−0.129334	−	0.268565i	−0.129334	−	0.268565i
$498$		0			0
$499$	−0.997204	+	0.925270i	−0.997204	+	0.925270i	−0.997204	−	0.0747301i	$-0.976190\pi$
$499$	−0.997204	+	0.925270i	−0.997204	+	0.925270i			1.00000i	$0.5\pi$
$500$		0			0
$501$	−2.72188	−	2.52554i	−2.72188	−	2.52554i
$502$		0			0
$503$	−0.250701	+	1.09839i	−0.250701	+	1.09839i	0.680173	+	0.733052i	$0.261905\pi$
$503$	−0.250701	+	1.09839i	−0.250701	+	1.09839i	−0.930874	+	0.365341i	$0.880952\pi$
$504$	−1.38921	+	2.03759i	−1.38921	+	2.03759i
$505$		0			0
$506$	0.419945	−	1.07000i	0.419945	−	1.07000i
$507$	−0.369035	+	4.92444i	−0.369035	+	4.92444i
$508$		0			0
$509$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$509$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$510$		0			0
$511$		0			0
$512$	0.900969	+	0.433884i	0.900969	+	0.433884i
$513$		0			0
$514$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$523$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$524$	0.433884	+	1.90097i	0.433884	+	1.90097i
$525$	−1.04876	+	1.53825i	−1.04876	+	1.53825i
$526$		0			0
$527$	−1.29196	−	0.880843i	−1.29196	−	0.880843i
$528$	0.804539	+	0.746503i	0.804539	+	0.746503i
$529$	−2.77064	+	0.417607i	−2.77064	+	0.417607i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	1.34515	+	0.202749i	1.34515	+	0.202749i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.294755	−	0.510531i	0.294755	−	0.510531i
$540$		0			0
$541$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$541$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$542$		0			0
$543$		0			0
$544$	0.988831	+	0.149042i	0.988831	+	0.149042i
$545$		0			0
$546$	−3.46886	−	0.791745i	−3.46886	−	0.791745i
$547$	−1.24349	+	1.55929i	−1.24349	+	1.55929i	−0.563320	+	0.826239i	$0.690476\pi$
$547$	−1.24349	+	1.55929i	−1.24349	+	1.55929i	−0.680173	+	0.733052i	$0.738095\pi$
$548$	0.535628	−	0.496990i	0.535628	−	0.496990i
$549$		0			0
$550$	0.432142	+	0.400969i	0.432142	+	0.400969i
$551$		0			0
$552$	0.807782	−	3.53912i	0.807782	−	3.53912i
$553$	0.202749	−	1.34515i	0.202749	−	1.34515i
$554$		0			0
$555$		0			0
$556$	−0.101659	+	1.35654i	−0.101659	+	1.35654i
$557$	−0.988831	+	1.71271i	−0.988831	+	1.71271i	−0.365341	+	0.930874i	$0.619048\pi$
$557$	−0.988831	+	1.71271i	−0.988831	+	1.71271i	−0.623490	+	0.781831i	$0.714286\pi$
$558$	−1.92808	−	3.33953i	−1.92808	−	3.33953i
$559$		0			0
$560$		0			0
$561$	0.988831	+	0.476196i	0.988831	+	0.476196i
$562$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$563$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$563$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$564$		0			0
$565$		0			0
$566$	−0.531130	−	0.255779i	−0.531130	−	0.255779i
$567$	−1.77904	+	1.91734i	−1.77904	+	1.91734i
$568$	−0.268565	+	0.129334i	−0.268565	+	0.129334i
$569$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$569$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$570$		0			0
$571$	−0.116853	+	1.55929i	−0.116853	+	1.55929i	0.563320	+	0.826239i	$0.309524\pi$
$571$	−0.116853	+	1.55929i	−0.116853	+	1.55929i	−0.680173	+	0.733052i	$0.738095\pi$
$572$	−0.411608	+	1.04876i	−0.411608	+	1.04876i
$573$		0			0
$574$		0			0
$575$	0.433884	−	1.90097i	0.433884	−	1.90097i
$576$	2.03759	+	1.38921i	2.03759	+	1.38921i
$577$	1.21135	+	1.12397i	1.21135	+	1.12397i	0.988831	+	0.149042i	$0.0476190\pi$
$577$	1.21135	+	1.12397i	1.21135	+	1.12397i	0.222521	+	0.974928i	$0.428571\pi$
$578$	0.988831	−	0.149042i	0.988831	−	0.149042i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.15283	+	0.173761i	1.15283	+	0.173761i
$584$		0			0
$585$		0			0
$586$	1.40097	−	0.432142i	1.40097	−	0.432142i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.680173	−	1.73305i	0.680173	−	1.73305i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.63402	−	0.246289i	−1.63402	−	0.246289i	−0.733052	−	0.680173i	$-0.761905\pi$
$593$	−1.63402	−	0.246289i	−1.63402	−	0.246289i	−0.900969	+	0.433884i	$0.857143\pi$
$594$	1.00324	+	1.25803i	1.00324	+	1.25803i
$595$		0			0
$596$	0.455573	−	0.571270i	0.455573	−	0.571270i
$597$	0.406813	−	0.377467i	0.406813	−	0.377467i
$598$	3.68484	−	0.555400i	3.68484	−	0.555400i
$599$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$599$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$600$	1.53825	+	1.04876i	1.53825	+	1.04876i
$601$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$601$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−0.414278	+	0.717550i	−0.414278	+	0.717550i
$607$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$607$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	2.35654	+	0.726897i	2.35654	+	0.726897i
$613$	0.142820	+	0.0440542i	0.142820	+	0.0440542i	0.365341	−	0.930874i	$-0.380952\pi$
$613$	0.142820	+	0.0440542i	0.142820	+	0.0440542i	−0.222521	+	0.974928i	$0.571429\pi$
$614$		0			0
$615$		0			0
$616$	−0.510531	−	0.294755i	−0.510531	−	0.294755i
$617$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$617$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$618$		0			0
$619$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$619$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$620$		0			0
$621$	1.94440	−	4.95426i	1.94440	−	4.95426i
$622$	−0.385418	−	1.68862i	−0.385418	−	1.68862i
$623$	−0.728639	+	0.0546039i	−0.728639	+	0.0546039i
$624$	−0.791745	+	3.46886i	−0.791745	+	3.46886i
$625$	0.826239	+	0.563320i	0.826239	+	0.563320i
$626$		0			0
$627$		0			0
$628$	1.07473	−	0.997204i	1.07473	−	0.997204i
$629$		0			0
$630$		0			0
$631$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$631$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$632$	−1.34515	−	0.202749i	−1.34515	−	0.202749i
$633$	−1.33484	+	0.910080i	−1.33484	+	0.910080i
$634$		0			0
$635$		0			0
$636$	3.68191			3.68191
$637$	1.91115			1.91115
$638$		0			0
$639$	−0.702449	+	0.216677i	−0.702449	+	0.216677i
$640$		0			0
$641$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$641$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$642$	−2.07409	−	0.312619i	−2.07409	−	0.312619i
$643$	−1.24349	−	1.55929i	−1.24349	−	1.55929i	−0.680173	−	0.733052i	$-0.738095\pi$
$643$	−1.24349	−	1.55929i	−1.24349	−	1.55929i	−0.563320	−	0.826239i	$-0.690476\pi$
$644$			1.94986i			1.94986i
$645$		0			0
$646$		0			0
$647$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$647$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$648$	1.91734	+	1.77904i	1.91734	+	1.77904i
$649$		0			0
$650$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$651$	1.98008	+	2.13402i	1.98008	+	2.13402i
$652$		0			0
$653$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$653$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$659$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$660$		0			0
$661$	1.19158	+	0.367554i	1.19158	+	0.367554i	0.826239	−	0.563320i	$-0.190476\pi$
$661$	1.19158	+	0.367554i	1.19158	+	0.367554i	0.365341	+	0.930874i	$0.380952\pi$
$662$		0			0
$663$	0.265895	+	3.54812i	0.265895	+	3.54812i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.997204	−	1.72721i	0.997204	−	1.72721i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−1.73305	−	0.680173i	−1.73305	−	0.680173i
$673$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$673$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$674$		0			0
$675$	2.00088	+	1.85654i	2.00088	+	1.85654i
$676$	−2.62285	+	0.395331i	−2.62285	+	0.395331i
$677$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$677$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.548760	+	0.0827122i	0.548760	+	0.0827122i
$682$	0.761623	−	0.519266i	0.761623	−	0.519266i
$683$	0.108903	+	0.277479i	0.108903	+	0.277479i	0.974928	−	0.222521i	$-0.0714286\pi$
$683$	0.108903	+	0.277479i	0.108903	+	0.277479i	−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$	−0.149042	+	0.988831i	−0.149042	+	0.988831i
$687$	−3.35475			−3.35475
$688$		0			0
$689$	1.38084	+	3.51833i	1.38084	+	3.51833i
$690$		0			0
$691$	−1.54620	−	0.233052i	−1.54620	−	0.233052i	−0.680173	−	0.733052i	$-0.738095\pi$
$691$	−1.54620	−	0.233052i	−1.54620	−	0.233052i	−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$	−1.13662	−	0.906426i	−1.13662	−	0.906426i
$694$	−0.541044	+	0.678448i	−0.541044	+	0.678448i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	1.48883	+	1.01507i	1.48883	+	1.01507i
$699$		0			0
$700$	−0.930874	−	0.365341i	−0.930874	−	0.365341i
$701$	−0.440071	−	1.92808i	−0.440071	−	1.92808i	−0.365341	−	0.930874i	$-0.619048\pi$
$701$	−0.440071	−	1.92808i	−0.440071	−	1.92808i	−0.0747301	−	0.997204i	$-0.523810\pi$
$702$	−1.90580	+	4.85590i	−1.90580	+	4.85590i
$703$		0			0
$704$	−0.294755	+	0.510531i	−0.294755	+	0.510531i
$705$		0			0
$706$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$707$	0.131178	−	0.425270i	0.131178	−	0.425270i
$708$		0			0
$709$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$709$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$710$		0			0
$711$	−3.20571	−	0.988831i	−3.20571	−	0.988831i
$712$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$713$	−2.74698	−	1.32288i	−2.74698	−	1.32288i
$714$	−1.85654	−	0.139129i	−1.85654	−	0.139129i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.215372	−	0.548760i	0.215372	−	0.548760i	−0.781831	−	0.623490i	$-0.785714\pi$
$719$	0.215372	−	0.548760i	0.215372	−	0.548760i	0.997204	+	0.0747301i	$0.0238095\pi$
$720$		0			0
$721$		0			0
$722$	0.222521	−	0.974928i	0.222521	−	0.974928i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	0.890474	−	0.826239i	0.890474	−	0.826239i
$727$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$727$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$728$		−	1.91115i		−	1.91115i
$729$	0.853298	+	1.07000i	0.853298	+	1.07000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.733052	−	0.680173i	$-0.761905\pi$
$733$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.222521	+	0.974928i	$0.571429\pi$
$734$	1.12664			1.12664
$735$		0			0
$736$	1.94986			1.94986
$737$		0			0
$738$		0			0
$739$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$739$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$740$		0			0
$741$		0			0
$742$	−1.92808	+	0.440071i	−1.92808	+	0.440071i
$743$	1.16078	−	1.45557i	1.16078	−	1.45557i	0.294755	−	0.955573i	$-0.404762\pi$
$743$	1.16078	−	1.45557i	1.16078	−	1.45557i	0.866025	−	0.500000i	$-0.166667\pi$
$744$	2.13402	−	1.98008i	2.13402	−	1.98008i
$745$		0			0
$746$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$747$		0			0
$748$	−0.131178	+	0.574730i	−0.131178	+	0.574730i
$749$	1.12349	−	0.0841939i	1.12349	−	0.0841939i
$750$		0			0
$751$	0.496990	−	1.26631i	0.496990	−	1.26631i	−0.433884	−	0.900969i	$-0.642857\pi$
$751$	0.496990	−	1.26631i	0.496990	−	1.26631i	0.930874	−	0.365341i	$-0.119048\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−2.36383	−	1.36476i	−2.36383	−	1.36476i
$757$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.0747301	−	0.997204i	$-0.523810\pi$
$757$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.826239	+	0.563320i	$0.809524\pi$
$758$	0.129436	+	1.72721i	0.129436	+	1.72721i
$759$	2.04493	+	0.630777i	2.04493	+	0.630777i
$760$		0			0
$761$	−0.0111692	−	0.149042i	−0.0111692	−	0.149042i	0.988831	−	0.149042i	$-0.0476190\pi$
$761$	−0.0111692	−	0.149042i	−0.0111692	−	0.149042i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.680173	+	1.73305i	−0.680173	+	1.73305i
$769$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	−0.955573	−	0.294755i	$-0.904762\pi$
$769$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	0.733052	−	0.680173i	$-0.238095\pi$
$770$		0			0
$771$	0.414278	−	1.81507i	0.414278	−	1.81507i
$772$		0			0
$773$	0.109562	+	0.101659i	0.109562	+	0.101659i	0.733052	−	0.680173i	$-0.238095\pi$
$773$	0.109562	+	0.101659i	0.109562	+	0.101659i	−0.623490	+	0.781831i	$0.714286\pi$
$774$		0			0
$775$	1.14625	−	1.06356i	1.14625	−	1.06356i
$776$		0			0
$777$		0			0
$778$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$779$		0			0
$780$		0			0
$781$	−0.0641992	−	0.163577i	−0.0641992	−	0.163577i
$782$	1.86323	−	0.574730i	1.86323	−	0.574730i
$783$		0			0
$784$	0.988831	+	0.149042i	0.988831	+	0.149042i
$785$		0			0
$786$	−3.46886	+	1.07000i	−3.46886	+	1.07000i
$787$	0.317031	+	0.807782i	0.317031	+

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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3332.cc (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.955573 + 0.294755i) q^{2} +(-0.680173 - 1.73305i) q^{3} +(0.826239 - 0.563320i) q^{4} +(1.16078 + 1.45557i) q^{6} +(-0.781831 - 0.623490i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-1.80778 + 1.67738i) q^{9} +O(q^{10})\)	\(q+(-0.955573 + 0.294755i) q^{2} +(-0.680173 - 1.73305i) q^{3} +(0.826239 - 0.563320i) q^{4} +(1.16078 + 1.45557i) q^{6} +(-0.781831 - 0.623490i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-1.80778 + 1.67738i) q^{9} +(-0.432142 - 0.400969i) q^{11} +(-1.53825 - 1.04876i) q^{12} +(0.425270 - 1.86323i) q^{13} +(0.930874 + 0.365341i) q^{14} +(0.365341 - 0.930874i) q^{16} +(0.0747301 - 0.997204i) q^{17} +(1.23305 - 2.13571i) q^{18} +(-0.548760 + 1.77904i) q^{21} +(0.531130 + 0.255779i) q^{22} +(-0.145713 - 1.94440i) q^{23} +(1.77904 + 0.548760i) q^{24} +(0.955573 + 0.294755i) q^{25} +(0.142820 + 1.90580i) q^{26} +(2.45921 + 1.18429i) q^{27} +(-0.997204 - 0.0747301i) q^{28} +(0.781831 - 1.35417i) q^{31} +(-0.0747301 + 0.997204i) q^{32} +(-0.400969 + 1.02165i) q^{33} +(0.222521 + 0.974928i) q^{34} +(-0.548760 + 2.40427i) q^{36} +(-3.51833 + 0.530303i) q^{39} -1.86175i q^{42} +(-0.582926 - 0.0878620i) q^{44} +(0.712362 + 1.81507i) q^{46} -1.86175 q^{48} +(0.222521 + 0.974928i) q^{49} -1.00000 q^{50} +(-1.77904 + 0.548760i) q^{51} +(-0.698220 - 1.77904i) q^{52} +(-1.63402 + 1.11406i) q^{53} +(-2.69903 - 0.406813i) q^{54} +(0.974928 - 0.222521i) q^{56} +(-0.347948 + 1.52446i) q^{62} +(2.45921 - 0.184292i) q^{63} +(-0.222521 - 0.974928i) q^{64} +(0.0820177 - 1.09445i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-3.27064 + 1.57506i) q^{69} +(0.268565 + 0.129334i) q^{71} +(-0.184292 - 2.45921i) q^{72} +(-0.139129 - 1.85654i) q^{75} +(0.0878620 + 0.582926i) q^{77} +(3.20571 - 1.54379i) q^{78} +(0.680173 + 1.17809i) q^{79} +(0.195461 - 2.60825i) q^{81} +(0.548760 + 1.77904i) q^{84} +(0.582926 - 0.0878620i) q^{88} +(0.535628 - 0.496990i) q^{89} +(-1.49419 + 1.19158i) q^{91} +(-1.21572 - 1.52446i) q^{92} +(-2.87863 - 0.433884i) q^{93} +(1.77904 - 0.548760i) q^{96} +(-0.500000 - 0.866025i) q^{98} +1.45379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9}+O(q^{10}) \) 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9	\( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52} - 2 q^{53} - 4 q^{64} - 22 q^{66} - 12 q^{68} - 14 q^{69} - 4 q^{72} - 6 q^{77} + 30 q^{81} - 6 q^{84} + 2 q^{89} - 12 q^{98}+O(q^{100}) \) 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9 + 10 * q^13 + 2 * q^16 + 2 * q^17 + 10 * q^18 + 6 * q^21 + 2 * q^25 - 2 * q^26 - 2 * q^32 + 8 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^49 - 24 * q^50 + 2 * q^52 - 2 * q^53 - 4 * q^64 - 22 * q^66 - 12 * q^68 - 14 * q^69 - 4 * q^72 - 6 * q^77 + 30 * q^81 - 6 * q^84 + 2 * q^89 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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