LMFDB - Embedded newform 3332.1.cc.c.2515.1

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			2515.1
Root			$-0.563320 - 0.826239i$ of defining polynomial
Character	$\chi$	$=$	3332.2515
Dual form			3332.1.cc.c.2447.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.365341 - 0.930874i) q^{2} +(-0.149042 - 1.98883i) q^{3} +(-0.733052 + 0.680173i) q^{4} +(-1.79690 + 0.865341i) q^{6} +(-0.433884 + 0.900969i) q^{7} +(0.900969 + 0.433884i) q^{8} +(-2.94440 + 0.443797i) q^{9} +(1.84095 + 0.277479i) q^{11} +(1.46200 + 1.35654i) q^{12} +(-0.455573 + 0.571270i) q^{13} +(0.997204 + 0.0747301i) q^{14} +(0.0747301 - 0.997204i) q^{16} +(0.955573 - 0.294755i) q^{17} +(1.48883 + 2.57873i) q^{18} +(1.85654 + 0.728639i) q^{21} +(-0.414278 - 1.81507i) q^{22} +(1.49419 + 0.460898i) q^{23} +(0.728639 - 1.85654i) q^{24} +(0.365341 - 0.930874i) q^{25} +(0.698220 + 0.215372i) q^{26} +(0.877681 + 3.84537i) q^{27} +(-0.294755 - 0.955573i) q^{28} +(0.433884 + 0.751509i) q^{31} +(-0.955573 + 0.294755i) q^{32} +(0.277479 - 3.70270i) q^{33} +(-0.623490 - 0.781831i) q^{34} +(1.85654 - 2.32803i) q^{36} +(1.20406 + 0.820914i) q^{39} -1.99441i q^{42} +(-1.53825 + 1.04876i) q^{44} +(-0.116853 - 1.55929i) q^{46} -1.99441 q^{48} +(-0.623490 - 0.781831i) q^{49} -1.00000 q^{50} +(-0.728639 - 1.85654i) q^{51} +(-0.0546039 - 0.728639i) q^{52} +(-1.21135 + 1.12397i) q^{53} +(3.25890 - 2.22188i) q^{54} +(-0.781831 + 0.623490i) q^{56} +(0.541044 - 0.678448i) q^{62} +(0.877681 - 2.84537i) q^{63} +(0.623490 + 0.781831i) q^{64} +(-3.54812 + 1.09445i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(0.693950 - 3.04039i) q^{69} +(0.250701 + 1.09839i) q^{71} +(-2.84537 - 0.877681i) q^{72} +(-1.90580 - 0.587862i) q^{75} +(-1.04876 + 1.53825i) q^{77} +(0.324275 - 1.42074i) q^{78} +(0.149042 - 0.258149i) q^{79} +(4.67161 - 1.44100i) q^{81} +(-1.85654 + 0.728639i) q^{84} +(1.53825 + 1.04876i) q^{88} +(0.147791 - 0.0222759i) q^{89} +(-0.317031 - 0.658322i) q^{91} +(-1.40881 + 0.678448i) q^{92} +(1.42996 - 0.974928i) q^{93} +(0.728639 + 1.85654i) q^{96} +(-0.500000 + 0.866025i) q^{98} -5.54365 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

Character values

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

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$e\left(\frac{10}{21}\right)$	$-1$

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 $

$2$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$2$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$3$	−0.149042	−	1.98883i	−0.149042	−	1.98883i	−0.149042	−	0.988831i	$-0.547619\pi$
$3$	−0.149042	−	1.98883i	−0.149042	−	1.98883i		−	1.00000i	$-0.5\pi$
$4$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$5$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$5$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$6$	−1.79690	+	0.865341i	−1.79690	+	0.865341i
$7$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$7$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$8$	0.900969	+	0.433884i	0.900969	+	0.433884i
$9$	−2.94440	+	0.443797i	−2.94440	+	0.443797i
$10$		0			0
$11$	1.84095	+	0.277479i	1.84095	+	0.277479i	0.974928	−	0.222521i	$-0.0714286\pi$
$11$	1.84095	+	0.277479i	1.84095	+	0.277479i	0.866025	+	0.500000i	$0.166667\pi$
$12$	1.46200	+	1.35654i	1.46200	+	1.35654i
$13$	−0.455573	+	0.571270i	−0.455573	+	0.571270i	−0.955573	−	0.294755i	$-0.904762\pi$
$13$	−0.455573	+	0.571270i	−0.455573	+	0.571270i	0.500000	+	0.866025i	$0.333333\pi$
$14$	0.997204	+	0.0747301i	0.997204	+	0.0747301i
$15$		0			0
$16$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$17$	0.955573	−	0.294755i	0.955573	−	0.294755i
$17$	0.955573	−	0.294755i	0.955573	−	0.294755i
$18$	1.48883	+	2.57873i	1.48883	+	2.57873i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	1.85654	+	0.728639i	1.85654	+	0.728639i
$22$	−0.414278	−	1.81507i	−0.414278	−	1.81507i
$23$	1.49419	+	0.460898i	1.49419	+	0.460898i	0.930874	−	0.365341i	$-0.119048\pi$
$23$	1.49419	+	0.460898i	1.49419	+	0.460898i	0.563320	+	0.826239i	$0.309524\pi$
$24$	0.728639	−	1.85654i	0.728639	−	1.85654i
$25$	0.365341	−	0.930874i	0.365341	−	0.930874i
$26$	0.698220	+	0.215372i	0.698220	+	0.215372i
$27$	0.877681	+	3.84537i	0.877681	+	3.84537i
$28$	−0.294755	−	0.955573i	−0.294755	−	0.955573i
$29$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$29$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$30$		0			0
$31$	0.433884	+	0.751509i	0.433884	+	0.751509i	0.997204	−	0.0747301i	$-0.0238095\pi$
$31$	0.433884	+	0.751509i	0.433884	+	0.751509i	−0.563320	+	0.826239i	$0.690476\pi$
$32$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$33$	0.277479	−	3.70270i	0.277479	−	3.70270i
$34$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$35$		0			0
$36$	1.85654	−	2.32803i	1.85654	−	2.32803i
$37$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$37$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$38$		0			0
$39$	1.20406	+	0.820914i	1.20406	+	0.820914i
$40$		0			0
$41$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$41$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$42$		−	1.99441i		−	1.99441i
$43$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$43$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$44$	−1.53825	+	1.04876i	−1.53825	+	1.04876i
$45$		0			0
$46$	−0.116853	−	1.55929i	−0.116853	−	1.55929i
$47$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$47$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$48$	−1.99441			−1.99441
$49$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$50$	−1.00000			−1.00000
$51$	−0.728639	−	1.85654i	−0.728639	−	1.85654i
$52$	−0.0546039	−	0.728639i	−0.0546039	−	0.728639i
$53$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.222521	+	0.974928i	$0.571429\pi$
$53$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.988831	+	0.149042i	$0.952381\pi$
$54$	3.25890	−	2.22188i	3.25890	−	2.22188i
$55$		0			0
$56$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$57$		0			0
$58$		0			0
$59$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$59$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$60$		0			0
$61$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$61$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$62$	0.541044	−	0.678448i	0.541044	−	0.678448i
$63$	0.877681	−	2.84537i	0.877681	−	2.84537i
$64$	0.623490	+	0.781831i	0.623490	+	0.781831i
$65$		0			0
$66$	−3.54812	+	1.09445i	−3.54812	+	1.09445i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$69$	0.693950	−	3.04039i	0.693950	−	3.04039i
$70$		0			0
$71$	0.250701	+	1.09839i	0.250701	+	1.09839i	0.930874	+	0.365341i	$0.119048\pi$
$71$	0.250701	+	1.09839i	0.250701	+	1.09839i	−0.680173	+	0.733052i	$0.738095\pi$
$72$	−2.84537	−	0.877681i	−2.84537	−	0.877681i
$73$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$73$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$74$		0			0
$75$	−1.90580	−	0.587862i	−1.90580	−	0.587862i
$76$		0			0
$77$	−1.04876	+	1.53825i	−1.04876	+	1.53825i
$78$	0.324275	−	1.42074i	0.324275	−	1.42074i
$79$	0.149042	−	0.258149i	0.149042	−	0.258149i	−0.781831	−	0.623490i	$-0.785714\pi$
$79$	0.149042	−	0.258149i	0.149042	−	0.258149i	0.930874	+	0.365341i	$0.119048\pi$
$80$		0			0
$81$	4.67161	−	1.44100i	4.67161	−	1.44100i
$82$		0			0
$83$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$83$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$84$	−1.85654	+	0.728639i	−1.85654	+	0.728639i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.53825	+	1.04876i	1.53825	+	1.04876i
$89$	0.147791	−	0.0222759i	0.147791	−	0.0222759i	−0.0747301	−	0.997204i	$-0.523810\pi$
$89$	0.147791	−	0.0222759i	0.147791	−	0.0222759i	0.222521	+	0.974928i	$0.428571\pi$
$90$		0			0
$91$	−0.317031	−	0.658322i	−0.317031	−	0.658322i
$92$	−1.40881	+	0.678448i	−1.40881	+	0.678448i
$93$	1.42996	−	0.974928i	1.42996	−	0.974928i
$94$		0			0
$95$		0			0
$96$	0.728639	+	1.85654i	0.728639	+	1.85654i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$99$	−5.54365			−5.54365
$100$	0.365341	+	0.930874i	0.365341	+	0.930874i
$101$	−0.0931869	−	1.24349i	−0.0931869	−	1.24349i	−0.826239	−	0.563320i	$-0.809524\pi$
$101$	−0.0931869	−	1.24349i	−0.0931869	−	1.24349i	0.733052	−	0.680173i	$-0.238095\pi$
$102$	−1.46200	+	1.35654i	−1.46200	+	1.35654i
$103$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$103$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$104$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$105$		0			0
$106$	1.48883	+	0.716983i	1.48883	+	0.716983i
$107$	1.34515	−	0.202749i	1.34515	−	0.202749i	0.563320	−	0.826239i	$-0.309524\pi$
$107$	1.34515	−	0.202749i	1.34515	−	0.202749i	0.781831	+	0.623490i	$0.214286\pi$
$108$	−3.25890	−	2.22188i	−3.25890	−	2.22188i
$109$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$109$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$110$		0			0
$111$		0			0
$112$	0.866025	+	0.500000i	0.866025	+	0.500000i
$113$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$113$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.08786	−	1.88423i	1.08786	−	1.88423i
$118$		0			0
$119$	−0.149042	+	0.988831i	−0.149042	+	0.988831i
$120$		0			0
$121$	2.35654	+	0.726897i	2.35654	+	0.726897i
$122$		0			0
$123$		0			0
$124$	−0.829215	−	0.255779i	−0.829215	−	0.255779i
$125$		0			0
$126$	−2.96934	+	0.222521i	−2.96934	+	0.222521i
$127$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$127$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$	0.116853	−	1.55929i	0.116853	−	1.55929i	−0.563320	−	0.826239i	$-0.690476\pi$
$131$	0.116853	−	1.55929i	0.116853	−	1.55929i	0.680173	−	0.733052i	$-0.261905\pi$
$132$	2.31507	+	2.90301i	2.31507	+	2.90301i
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.988831	+	0.149042i	0.988831	+	0.149042i
$137$	−0.123490	−	0.0841939i	−0.123490	−	0.0841939i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	−0.123490	−	0.0841939i	−0.123490	−	0.0841939i	−0.623490	+	0.781831i	$0.714286\pi$
$138$	−3.08375	+	0.464800i	−3.08375	+	0.464800i
$139$	0.268565	+	0.129334i	0.268565	+	0.129334i	0.563320	−	0.826239i	$-0.309524\pi$
$139$	0.268565	+	0.129334i	0.268565	+	0.129334i	−0.294755	+	0.955573i	$0.595238\pi$
$140$		0			0
$141$		0			0
$142$	0.930874	−	0.634659i	0.930874	−	0.634659i
$143$	−0.997204	+	0.925270i	−0.997204	+	0.925270i
$144$	0.222521	+	2.96934i	0.222521	+	2.96934i
$145$		0			0
$146$		0			0
$147$	−1.46200	+	1.35654i	−1.46200	+	1.35654i
$148$		0			0
$149$	0.0546039	+	0.139129i	0.0546039	+	0.139129i	0.955573	−	0.294755i	$-0.0952381\pi$
$149$	0.0546039	+	0.139129i	0.0546039	+	0.139129i	−0.900969	+	0.433884i	$0.857143\pi$
$150$	0.149042	+	1.98883i	0.149042	+	1.98883i
$151$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$151$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$152$		0			0
$153$	−2.68278	+	1.29196i	−2.68278	+	1.29196i
$154$	1.81507	+	0.414278i	1.81507	+	0.414278i
$155$		0			0
$156$	−1.44100	+	0.217196i	−1.44100	+	0.217196i
$157$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$157$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$158$	−0.294755	−	0.0444272i	−0.294755	−	0.0444272i
$159$	2.41593	+	2.24165i	2.41593	+	2.24165i
$160$		0			0
$161$	−1.06356	+	1.14625i	−1.06356	+	1.14625i
$162$	−3.04812	−	3.82222i	−3.04812	−	3.82222i
$163$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$163$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.131178	−	0.574730i	0.131178	−	0.574730i	−0.866025	−	0.500000i	$-0.833333\pi$
$167$	0.131178	−	0.574730i	0.131178	−	0.574730i	0.997204	−	0.0747301i	$-0.0238095\pi$
$168$	1.35654	+	1.46200i	1.35654	+	1.46200i
$169$	0.103718	+	0.454418i	0.103718	+	0.454418i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$173$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$174$		0			0
$175$	0.680173	+	0.733052i	0.680173	+	0.733052i
$176$	0.414278	−	1.81507i	0.414278	−	1.81507i
$177$		0			0
$178$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$179$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$179$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$180$		0			0
$181$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$181$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$182$	−0.496990	+	0.535628i	−0.496990	+	0.535628i
$183$		0			0
$184$	1.14625	+	1.06356i	1.14625	+	1.06356i
$185$		0			0
$186$	−1.42996	−	0.974928i	−1.42996	−	0.974928i
$187$	1.84095	−	0.277479i	1.84095	−	0.277479i
$188$		0			0
$189$	−3.84537	−	0.877681i	−3.84537	−	0.877681i
$190$		0			0
$191$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$191$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$192$	1.46200	−	1.35654i	1.46200	−	1.35654i
$193$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$193$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$194$		0			0
$195$		0			0
$196$	0.988831	+	0.149042i	0.988831	+	0.149042i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	2.02532	+	5.16044i	2.02532	+	5.16044i
$199$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	0.866025	+	0.500000i	$0.166667\pi$
$199$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	−0.781831	+	0.623490i	$0.785714\pi$
$200$	0.733052	−	0.680173i	0.733052	−	0.680173i
$201$		0			0
$202$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$203$		0			0
$204$	1.79690	+	0.865341i	1.79690	+	0.865341i
$205$		0			0
$206$		0			0
$207$	−4.60405	−	0.693950i	−4.60405	−	0.693950i
$208$	0.535628	+	0.496990i	0.535628	+	0.496990i
$209$		0			0
$210$		0			0
$211$	1.21572	+	1.52446i	1.21572	+	1.52446i	0.781831	+	0.623490i	$0.214286\pi$
$211$	1.21572	+	1.52446i	1.21572	+	1.52446i	0.433884	+	0.900969i	$0.357143\pi$
$212$	0.123490	−	1.64786i	0.123490	−	1.64786i
$213$	2.14715	−	0.662309i	2.14715	−	0.662309i
$214$	−0.680173	−	1.17809i	−0.680173	−	1.17809i
$215$		0			0
$216$	−0.877681	+	3.84537i	−0.877681	+	3.84537i
$217$	−0.865341	+	0.0648483i	−0.865341	+	0.0648483i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.266948	+	0.680173i	−0.266948	+	0.680173i
$222$		0			0
$223$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$223$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$224$	0.149042	−	0.988831i	0.149042	−	0.988831i
$225$	−0.662592	+	2.90301i	−0.662592	+	2.90301i
$226$		0			0
$227$	−0.563320	−	0.975699i	−0.563320	−	0.975699i	−0.997204	−	0.0747301i	$-0.976190\pi$
$227$	−0.563320	−	0.975699i	−0.563320	−	0.975699i	0.433884	−	0.900969i	$-0.357143\pi$
$228$		0			0
$229$	0.0332580	−	0.443797i	0.0332580	−	0.443797i	−0.955573	−	0.294755i	$-0.904762\pi$
$229$	0.0332580	−	0.443797i	0.0332580	−	0.443797i	0.988831	−	0.149042i	$-0.0476190\pi$
$230$		0			0
$231$	3.21562	+	1.85654i	3.21562	+	1.85654i
$232$		0			0
$233$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$233$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$234$	−2.15142	−	0.324275i	−2.15142	−	0.324275i
$235$		0			0
$236$		0			0
$237$	−0.535628	−	0.257945i	−0.535628	−	0.257945i
$238$	0.974928	−	0.222521i	0.974928	−	0.222521i
$239$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$239$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$240$		0			0
$241$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$241$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$242$	−0.184292	−	2.45921i	−0.184292	−	2.45921i
$243$	−2.12117	−	5.40466i	−2.12117	−	5.40466i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.0648483	+	0.865341i	0.0648483	+	0.865341i
$249$		0			0
$250$		0			0
$251$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$251$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$252$	1.29196	+	2.68278i	1.29196	+	2.68278i
$253$	2.62285	+	1.26310i	2.62285	+	1.26310i
$254$		0			0
$255$		0			0
$256$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$257$	−0.733052	−	0.680173i	−0.733052	−	0.680173i	0.222521	−	0.974928i	$-0.428571\pi$
$257$	−0.733052	−	0.680173i	−0.733052	−	0.680173i	−0.955573	+	0.294755i	$0.904762\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.49419	+	0.460898i	−1.49419	+	0.460898i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	1.85654	−	3.21562i	1.85654	−	3.21562i
$265$		0			0
$266$		0			0
$267$	−0.0663300	−	0.290611i	−0.0663300	−	0.290611i
$268$		0			0
$269$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$269$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$270$		0			0
$271$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$271$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$272$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$273$	−1.26204	+	0.728639i	−1.26204	+	0.728639i
$274$	−0.0332580	+	0.145713i	−0.0332580	+	0.145713i
$275$	0.930874	−	1.61232i	0.930874	−	1.61232i
$276$	1.55929	+	2.70077i	1.55929	+	2.70077i
$277$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$277$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$278$	0.0222759	−	0.297251i	0.0222759	−	0.297251i
$279$	−1.61105	−	2.02019i	−1.61105	−	2.02019i
$280$		0			0
$281$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$281$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$282$		0			0
$283$	−1.84095	−	0.277479i	−1.84095	−	0.277479i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	−1.84095	−	0.277479i	−1.84095	−	0.277479i	−0.974928	+	0.222521i	$0.928571\pi$
$284$	−0.930874	−	0.634659i	−0.930874	−	0.634659i
$285$		0			0
$286$	1.22563	+	0.590232i	1.22563	+	0.590232i
$287$		0			0
$288$	2.68278	−	1.29196i	2.68278	−	1.29196i
$289$	0.826239	−	0.563320i	0.826239	−	0.563320i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$293$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$294$	1.79690	+	0.865341i	1.79690	+	0.865341i
$295$		0			0
$296$		0			0
$297$	0.548760	+	7.32269i	0.548760	+	7.32269i
$298$	0.109562	−	0.101659i	0.109562	−	0.101659i
$299$	−0.944011	+	0.643616i	−0.944011	+	0.643616i
$300$	1.79690	−	0.865341i	1.79690	−	0.865341i
$301$		0			0
$302$		0			0
$303$	−2.45921	+	0.370666i	−2.45921	+	0.370666i
$304$		0			0
$305$		0			0
$306$	2.18278	+	2.02532i	2.18278	+	2.02532i
$307$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$307$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$308$	−0.277479	−	1.84095i	−0.277479	−	1.84095i
$309$		0			0
$310$		0			0
$311$	1.65510	−	0.510531i	1.65510	−	0.510531i	0.680173	−	0.733052i	$-0.261905\pi$
$311$	1.65510	−	0.510531i	1.65510	−	0.510531i	0.974928	+	0.222521i	$0.0714286\pi$
$312$	0.728639	+	1.26204i	0.728639	+	1.26204i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−0.440071	+	1.92808i	−0.440071	+	1.92808i
$315$		0			0
$316$	0.0663300	+	0.290611i	0.0663300	+	0.290611i
$317$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$317$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$318$	1.20406	−	3.06789i	1.20406	−	3.06789i
$319$		0			0
$320$		0			0
$321$	−0.603718	−	2.64506i	−0.603718	−	2.64506i
$322$	1.45557	+	0.571270i	1.45557	+	0.571270i
$323$		0			0
$324$	−2.44440	+	4.23383i	−2.44440	+	4.23383i
$325$	0.365341	+	0.632789i	0.365341	+	0.632789i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$331$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$332$		0			0
$333$		0			0
$334$	−0.582926	+	0.0878620i	−0.582926	+	0.0878620i
$335$		0			0
$336$	0.865341	−	1.79690i	0.865341	−	1.79690i
$337$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$337$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$338$	0.385113	−	0.262566i	0.385113	−	0.262566i
$339$		0			0
$340$		0			0
$341$	0.590232	+	1.50389i	0.590232	+	1.50389i
$342$		0			0
$343$	0.974928	−	0.222521i	0.974928	−	0.222521i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.42935	+	1.32624i	−1.42935	+	1.32624i	−0.563320	+	0.826239i	$0.690476\pi$
$347$	−1.42935	+	1.32624i	−1.42935	+	1.32624i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$349$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$350$	0.433884	−	0.900969i	0.433884	−	0.900969i
$351$	−2.59659	−	1.25045i	−2.59659	−	1.25045i
$352$	−1.84095	+	0.277479i	−1.84095	+	0.277479i
$353$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$353$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$354$		0			0
$355$		0			0
$356$	−0.0931869	+	0.116853i	−0.0931869	+	0.116853i
$357$	1.98883	+	0.149042i	1.98883	+	0.149042i
$358$		0			0
$359$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$359$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	1.09445	−	4.79510i	1.09445	−	4.79510i
$364$	0.680173	+	0.266948i	0.680173	+	0.266948i
$365$		0			0
$366$		0			0
$367$	0.496990	−	1.26631i	0.496990	−	1.26631i	−0.433884	−	0.900969i	$-0.642857\pi$
$367$	0.496990	−	1.26631i	0.496990	−	1.26631i	0.930874	−	0.365341i	$-0.119048\pi$
$368$	0.571270	−	1.45557i	0.571270	−	1.45557i
$369$		0			0
$370$		0			0
$371$	−0.487076	−	1.57906i	−0.487076	−	1.57906i
$372$	−0.385113	+	1.68729i	−0.385113	+	1.68729i
$373$	0.365341	−	0.632789i	0.365341	−	0.632789i	−0.623490	−	0.781831i	$-0.714286\pi$
$373$	0.365341	−	0.632789i	0.365341	−	0.632789i	0.988831	+	0.149042i	$0.0476190\pi$
$374$	−0.930874	−	1.61232i	−0.930874	−	1.61232i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	0.587862	+	3.90021i	0.587862	+	3.90021i
$379$	1.07992	−	1.35417i	1.07992	−	1.35417i	0.149042	−	0.988831i	$-0.452381\pi$
$379$	1.07992	−	1.35417i	1.07992	−	1.35417i	0.930874	−	0.365341i	$-0.119048\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$383$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$384$	−1.79690	−	0.865341i	−1.79690	−	0.865341i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.142820	−	1.90580i	−0.142820	−	1.90580i	−0.365341	−	0.930874i	$-0.619048\pi$
$389$	−0.142820	−	1.90580i	−0.142820	−	1.90580i	0.222521	−	0.974928i	$-0.428571\pi$
$390$		0			0
$391$	1.56366			1.56366
$392$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$393$	−3.11858			−3.11858
$394$		0			0
$395$		0			0
$396$	4.06379	−	3.77064i	4.06379	−	3.77064i
$397$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$397$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$398$	1.01507	−	0.488831i	1.01507	−	0.488831i
$399$		0			0
$400$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$401$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$401$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$402$		0			0
$403$	−0.626980	−	0.0945021i	−0.626980	−	0.0945021i
$404$	0.914101	+	0.848162i	0.914101	+	0.848162i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.149042	−	1.98883i	0.149042	−	1.98883i
$409$	1.57906	−	0.487076i	1.57906	−	0.487076i	0.623490	−	0.781831i	$-0.285714\pi$
$409$	1.57906	−	0.487076i	1.57906	−	0.487076i	0.955573	+	0.294755i	$0.0952381\pi$
$410$		0			0
$411$	−0.149042	+	0.258149i	−0.149042	+	0.258149i
$412$		0			0
$413$		0			0
$414$	1.03607	+	4.53932i	1.03607	+	4.53932i
$415$		0			0
$416$	0.266948	−	0.680173i	0.266948	−	0.680173i
$417$	0.217196	−	0.553406i	0.217196	−	0.553406i
$418$		0			0
$419$	0.0663300	+	0.290611i	0.0663300	+	0.290611i	0.997204	−	0.0747301i	$-0.0238095\pi$
$419$	0.0663300	+	0.290611i	0.0663300	+	0.290611i	−0.930874	+	0.365341i	$0.880952\pi$
$420$		0			0
$421$	−0.400969	+	1.75676i	−0.400969	+	1.75676i	0.222521	+	0.974928i	$0.428571\pi$
$421$	−0.400969	+	1.75676i	−0.400969	+	1.75676i	−0.623490	+	0.781831i	$0.714286\pi$
$422$	0.974928	−	1.68862i	0.974928	−	1.68862i
$423$		0			0
$424$	−1.57906	+	0.487076i	−1.57906	+	0.487076i
$425$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$426$	−1.40097	−	1.75676i	−1.40097	−	1.75676i
$427$		0			0
$428$	−0.848162	+	1.06356i	−0.848162	+	1.06356i
$429$	1.98883	+	1.84537i	1.98883	+	1.84537i
$430$		0			0
$431$	1.43109	+	0.975699i	1.43109	+	0.975699i	0.997204	+	0.0747301i	$0.0238095\pi$
$431$	1.43109	+	0.975699i	1.43109	+	0.975699i	0.433884	+	0.900969i	$0.357143\pi$
$432$	3.90021	−	0.587862i	3.90021	−	0.587862i
$433$	1.12349	+	0.541044i	1.12349	+	0.541044i	0.900969	−	0.433884i	$-0.142857\pi$
$433$	1.12349	+	0.541044i	1.12349	+	0.541044i	0.222521	+	0.974928i	$0.428571\pi$
$434$	0.376510	+	0.781831i	0.376510	+	0.781831i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.108903	+	0.277479i	0.108903	+	0.277479i	0.974928	−	0.222521i	$-0.0714286\pi$
$439$	0.108903	+	0.277479i	0.108903	+	0.277479i	−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	2.18278	+	2.02532i	2.18278	+	2.02532i
$442$	0.730682			0.730682
$443$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$443$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	0.268565	−	0.129334i	0.268565	−	0.129334i
$448$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$449$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$449$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$450$	2.94440	−	0.443797i	2.94440	−	0.443797i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.702449	+	0.880843i	−0.702449	+	0.880843i
$455$		0			0
$456$		0			0
$457$	−0.0931869	+	1.24349i	−0.0931869	+	1.24349i	0.733052	+	0.680173i	$0.238095\pi$
$457$	−0.0931869	+	1.24349i	−0.0931869	+	1.24349i	−0.826239	+	0.563320i	$0.809524\pi$
$458$	−0.425270	+	0.131178i	−0.425270	+	0.131178i
$459$	1.97213	+	3.41583i	1.97213	+	3.41583i
$460$		0			0
$461$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	0.365341	+	0.930874i	$0.380952\pi$
$461$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	−0.733052	+	0.680173i	$0.761905\pi$
$462$	0.553406	−	3.67161i	0.553406	−	3.67161i
$463$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$463$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$467$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$468$	0.484144	+	2.12117i	0.484144	+	2.12117i
$469$		0			0
$470$		0			0
$471$	−1.97213	+	3.41583i	−1.97213	+	3.41583i
$472$		0			0
$473$		0			0
$474$	−0.0444272	+	0.592840i	−0.0444272	+	0.592840i
$475$		0			0
$476$	−0.563320	−	0.826239i	−0.563320	−	0.826239i
$477$	3.06789	−	3.84702i	3.06789	−	3.84702i
$478$		0			0
$479$	1.54620	+	0.233052i	1.54620	+	0.233052i	0.866025	−	0.500000i	$-0.166667\pi$
$479$	1.54620	+	0.233052i	1.54620	+	0.233052i	0.680173	+	0.733052i	$0.261905\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	2.43821	+	1.94440i	2.43821	+	1.94440i
$484$	−2.22188	+	1.07000i	−2.22188	+	1.07000i
$485$		0			0
$486$	−4.25611	+	3.94909i	−4.25611	+	3.94909i
$487$	−0.129436	−	1.72721i	−0.129436	−	1.72721i	−0.563320	−	0.826239i	$-0.690476\pi$
$487$	−0.129436	−	1.72721i	−0.129436	−	1.72721i	0.433884	−	0.900969i	$-0.357143\pi$
$488$		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
$496$	0.781831	−	0.376510i	0.781831	−	0.376510i
$497$	−1.09839	−	0.250701i	−1.09839	−	0.250701i
$498$		0			0
$499$	−0.294755	+	0.0444272i	−0.294755	+	0.0444272i	−0.294755	−	0.955573i	$-0.595238\pi$
$499$	−0.294755	+	0.0444272i	−0.294755	+	0.0444272i			1.00000i	$0.5\pi$
$500$		0			0
$501$	−1.16259	−	0.175233i	−1.16259	−	0.175233i
$502$		0			0
$503$	−0.848162	+	1.06356i	−0.848162	+	1.06356i	0.149042	+	0.988831i	$0.452381\pi$
$503$	−0.848162	+	1.06356i	−0.848162	+	1.06356i	−0.997204	+	0.0747301i	$0.976190\pi$
$504$	2.02532	−	2.18278i	2.02532	−	2.18278i
$505$		0			0
$506$	0.217550	−	2.90301i	0.217550	−	2.90301i
$507$	0.888301	−	0.274005i	0.888301	−	0.274005i
$508$		0			0
$509$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$509$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$510$		0			0
$511$		0			0
$512$	0.222521	+	0.974928i	0.222521	+	0.974928i
$513$		0			0
$514$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$523$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$524$	0.974928	+	1.22252i	0.974928	+	1.22252i
$525$	1.35654	−	1.46200i	1.35654	−	1.46200i
$526$		0			0
$527$	0.636119	+	0.590232i	0.636119	+	0.590232i
$528$	−3.67161	−	0.553406i	−3.67161	−	0.553406i
$529$	1.19395	+	0.814021i	1.19395	+	0.814021i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−0.246289	+	0.167917i	−0.246289	+	0.167917i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.930874	−	1.61232i	−0.930874	−	1.61232i
$540$		0			0
$541$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$541$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$542$		0			0
$543$		0			0
$544$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$545$		0			0
$546$	1.13935	+	0.908598i	1.13935	+	0.908598i
$547$	0.531130	+	0.255779i	0.531130	+	0.255779i	0.680173	−	0.733052i	$-0.261905\pi$
$547$	0.531130	+	0.255779i	0.531130	+	0.255779i	−0.149042	+	0.988831i	$0.547619\pi$
$548$	0.147791	−	0.0222759i	0.147791	−	0.0222759i
$549$		0			0
$550$	−1.84095	−	0.277479i	−1.84095	−	0.277479i
$551$		0			0
$552$	1.94440	−	2.43821i	1.94440	−	2.43821i
$553$	0.167917	+	0.246289i	0.167917	+	0.246289i
$554$		0			0
$555$		0			0
$556$	−0.284841	+	0.0878620i	−0.284841	+	0.0878620i
$557$	0.826239	+	1.43109i	0.826239	+	1.43109i	0.900969	+	0.433884i	$0.142857\pi$
$557$	0.826239	+	1.43109i	0.826239	+	1.43109i	−0.0747301	+	0.997204i	$0.523810\pi$
$558$	−1.29196	+	2.23774i	−1.29196	+	2.23774i
$559$		0			0
$560$		0			0
$561$	−0.826239	−	3.61999i	−0.826239	−	3.61999i
$562$	0.955573	+	0.294755i	0.955573	+	0.294755i
$563$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$563$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$564$		0			0
$565$		0			0
$566$	0.414278	+	1.81507i	0.414278	+	1.81507i
$567$	−0.728639	+	4.83420i	−0.728639	+	4.83420i
$568$	−0.250701	+	1.09839i	−0.250701	+	1.09839i
$569$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$569$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$570$		0			0
$571$	−0.829215	+	0.255779i	−0.829215	+	0.255779i	−0.680173	−	0.733052i	$-0.738095\pi$
$571$	−0.829215	+	0.255779i	−0.829215	+	0.255779i	−0.149042	+	0.988831i	$0.547619\pi$
$572$	0.101659	−	1.35654i	0.101659	−	1.35654i
$573$		0			0
$574$		0			0
$575$	0.974928	−	1.22252i	0.974928	−	1.22252i
$576$	−2.18278	−	2.02532i	−2.18278	−	2.02532i
$577$	−1.44973	−	0.218511i	−1.44973	−	0.218511i	−0.623490	−	0.781831i	$-0.714286\pi$
$577$	−1.44973	−	0.218511i	−1.44973	−	0.218511i	−0.826239	+	0.563320i	$0.809524\pi$
$578$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.54192	+	1.73305i	−2.54192	+	1.73305i
$584$		0			0
$585$		0			0
$586$	0.722521	+	1.84095i	0.722521	+	1.84095i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.149042	−	1.98883i	0.149042	−	1.98883i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.21135	+	0.825886i	−1.21135	+	0.825886i	−0.988831	−	0.149042i	$-0.952381\pi$
$593$	−1.21135	+	0.825886i	−1.21135	+	0.825886i	−0.222521	+	0.974928i	$0.571429\pi$
$594$	6.61601	−	3.18610i	6.61601	−	3.18610i
$595$		0			0
$596$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i
$597$	2.22188	−	0.334895i	2.22188	−	0.334895i
$598$	0.944011	+	0.643616i	0.944011	+	0.643616i
$599$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$599$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$600$	−1.46200	−	1.35654i	−1.46200	−	1.35654i
$601$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$601$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	1.24349	+	2.15379i	1.24349	+	2.15379i
$607$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$607$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.08786	−	2.77183i	1.08786	−	2.77183i
$613$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.0747301	−	0.997204i	$-0.476190\pi$
$613$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.623490	−	0.781831i	$-0.285714\pi$
$614$		0			0
$615$		0			0
$616$	−1.61232	+	0.930874i	−1.61232	+	0.930874i
$617$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$617$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$618$		0			0
$619$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$619$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$620$		0			0
$621$	−0.460898	+	6.15025i	−0.460898	+	6.15025i
$622$	−1.07992	−	1.35417i	−1.07992	−	1.35417i
$623$	−0.0440542	+	0.142820i	−0.0440542	+	0.142820i
$624$	0.908598	−	1.13935i	0.908598	−	1.13935i
$625$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$626$		0			0
$627$		0			0
$628$	1.95557	−	0.294755i	1.95557	−	0.294755i
$629$		0			0
$630$		0			0
$631$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$631$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$632$	0.246289	−	0.167917i	0.246289	−	0.167917i
$633$	2.85070	−	2.64506i	2.85070	−	2.64506i
$634$		0			0
$635$		0			0
$636$	−3.29571			−3.29571
$637$	0.730682			0.730682
$638$		0			0
$639$	−1.22563	−	3.12285i	−1.22563	−	3.12285i
$640$		0			0
$641$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$641$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$642$	−2.24165	+	1.52833i	−2.24165	+	1.52833i
$643$	0.531130	−	0.255779i	0.531130	−	0.255779i	−0.149042	−	0.988831i	$-0.547619\pi$
$643$	0.531130	−	0.255779i	0.531130	−	0.255779i	0.680173	+	0.733052i	$0.261905\pi$
$644$		−	1.56366i		−	1.56366i
$645$		0			0
$646$		0			0
$647$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$647$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$648$	4.83420	+	0.728639i	4.83420	+	0.728639i
$649$		0			0
$650$	0.455573	−	0.571270i	0.455573	−	0.571270i
$651$	0.257945	+	1.71135i	0.257945	+	1.71135i
$652$		0			0
$653$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$653$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$659$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$660$		0			0
$661$	−0.658322	+	1.67738i	−0.658322	+	1.67738i	0.0747301	+	0.997204i	$0.476190\pi$
$661$	−0.658322	+	1.67738i	−0.658322	+	1.67738i	−0.733052	+	0.680173i	$0.761905\pi$
$662$		0			0
$663$	1.39254	+	0.429540i	1.39254	+	0.429540i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.294755	+	0.510531i	0.294755	+	0.510531i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−1.98883	−	0.149042i	−1.98883	−	0.149042i
$673$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$673$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$674$		0			0
$675$	3.90021	+	0.587862i	3.90021	+	0.587862i
$676$	−0.385113	−	0.262566i	−0.385113	−	0.262566i
$677$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$677$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.85654	+	1.26577i	−1.85654	+	1.26577i
$682$	1.18429	−	1.09886i	1.18429	−	1.09886i
$683$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	0.866025	+	0.500000i	$0.166667\pi$
$683$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	−0.781831	+	0.623490i	$0.785714\pi$
$684$		0			0
$685$		0			0
$686$	−0.563320	−	0.826239i	−0.563320	−	0.826239i
$687$	−0.887595			−0.887595
$688$		0			0
$689$	−0.0902318	−	1.20406i	−0.0902318	−	1.20406i
$690$		0			0
$691$	0.716983	−	0.488831i	0.716983	−	0.488831i	−0.149042	−	0.988831i	$-0.547619\pi$
$691$	0.716983	−	0.488831i	0.716983	−	0.488831i	0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$	2.40530	−	4.99466i	2.40530	−	4.99466i
$694$	1.75676	+	0.846011i	1.75676	+	0.846011i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.326239	−	0.302705i	−0.326239	−	0.302705i
$699$		0			0
$700$	−0.997204	−	0.0747301i	−0.997204	−	0.0747301i
$701$	−1.03030	−	1.29196i	−1.03030	−	1.29196i	−0.955573	−	0.294755i	$-0.904762\pi$
$701$	−1.03030	−	1.29196i	−1.03030	−	1.29196i	−0.0747301	−	0.997204i	$-0.523810\pi$
$702$	−0.215372	+	2.87394i	−0.215372	+	2.87394i
$703$		0			0
$704$	0.930874	+	1.61232i	0.930874	+	1.61232i
$705$		0			0
$706$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$707$	1.16078	+	0.455573i	1.16078	+	0.455573i
$708$		0			0
$709$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$709$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$710$		0			0
$711$	−0.324275	+	0.826239i	−0.324275	+	0.826239i
$712$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$713$	0.301938	+	1.32288i	0.301938	+	1.32288i
$714$	−0.587862	−	1.90580i	−0.587862	−	1.90580i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.139129	+	1.85654i	−0.139129	+	1.85654i	0.294755	+	0.955573i	$0.404762\pi$
$719$	−0.139129	+	1.85654i	−0.139129	+	1.85654i	−0.433884	+	0.900969i	$0.642857\pi$
$720$		0			0
$721$		0			0
$722$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−4.86348	+	0.733052i	−4.86348	+	0.733052i
$727$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$727$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$728$		−	0.730682i		−	0.730682i
$729$	−6.02815	+	2.90301i	−6.02815	+	2.90301i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.365341	−	0.930874i	−0.365341	−	0.930874i	−0.988831	−	0.149042i	$-0.952381\pi$
$733$	−0.365341	−	0.930874i	−0.365341	−	0.930874i	0.623490	−	0.781831i	$-0.285714\pi$
$734$	−1.36035			−1.36035
$735$		0			0
$736$	−1.56366			−1.56366
$737$		0			0
$738$		0			0
$739$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$739$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$740$		0			0
$741$		0			0
$742$	−1.29196	+	1.03030i	−1.29196	+	1.03030i
$743$	−1.79690	−	0.865341i	−1.79690	−	0.865341i	−0.930874	−	0.365341i	$-0.880952\pi$
$743$	−1.79690	−	0.865341i	−1.79690	−	0.865341i	−0.866025	−	0.500000i	$-0.833333\pi$
$744$	1.71135	−	0.257945i	1.71135	−	0.257945i
$745$		0			0
$746$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$747$		0			0
$748$	−1.16078	+	1.45557i	−1.16078	+	1.45557i
$749$	−0.400969	+	1.29991i	−0.400969	+	1.29991i
$750$		0			0
$751$	0.0222759	−	0.297251i	0.0222759	−	0.297251i	−0.974928	−	0.222521i	$-0.928571\pi$
$751$	0.0222759	−	0.297251i	0.0222759	−	0.297251i	0.997204	−	0.0747301i	$-0.0238095\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	3.41583	−	1.97213i	3.41583	−	1.97213i
$757$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	−0.955573	−	0.294755i	$-0.904762\pi$
$757$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	0.733052	−	0.680173i	$-0.238095\pi$
$758$	−1.65510	−	0.510531i	−1.65510	−	0.510531i
$759$	2.12117	−	5.40466i	2.12117	−	5.40466i
$760$		0			0
$761$	−1.82624	−	0.563320i	−1.82624	−	0.563320i	−0.826239	−	0.563320i	$-0.809524\pi$
$761$	−1.82624	−	0.563320i	−1.82624	−	0.563320i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.149042	+	1.98883i	−0.149042	+	1.98883i
$769$	0.623490	+	0.781831i	0.623490	+	0.781831i	0.988831	−	0.149042i	$-0.0476190\pi$
$769$	0.623490	+	0.781831i	0.623490	+	0.781831i	−0.365341	+	0.930874i	$0.619048\pi$
$770$		0			0
$771$	−1.24349	+	1.55929i	−1.24349	+	1.55929i
$772$		0			0
$773$	1.88980	+	0.284841i	1.88980	+	0.284841i	0.988831	−	0.149042i	$-0.0476190\pi$
$773$	1.88980	+	0.284841i	1.88980	+	0.284841i	0.900969	+	0.433884i	$0.142857\pi$
$774$		0			0
$775$	0.858075	−	0.129334i	0.858075	−	0.129334i
$776$		0			0
$777$		0			0
$778$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$779$		0			0
$780$		0			0
$781$	0.156748	+	2.09165i	0.156748	+	2.09165i
$782$	−0.571270	−	1.45557i	−0.571270	−	1.45557i
$783$		0			0
$784$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$785$		0			0
$786$	1.13935	+	2.90301i	1.13935	+	2.90301i
$787$	0.145713	+	1.94440i	0.145713	+	1.94440i	0.294755	+	0.955573i	$0.404762\pi$
$787$	0.145713	+	1.94440i	0.145713	+	1.94440i	−0.149042	+	0.988831i	$0.547619\pi$
$788$		0

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)

\(f(q)\)	\(=\)	\(q+(-0.365341 - 0.930874i) q^{2} +(-0.149042 - 1.98883i) q^{3} +(-0.733052 + 0.680173i) q^{4} +(-1.79690 + 0.865341i) q^{6} +(-0.433884 + 0.900969i) q^{7} +(0.900969 + 0.433884i) q^{8} +(-2.94440 + 0.443797i) q^{9} +(1.84095 + 0.277479i) q^{11} +(1.46200 + 1.35654i) q^{12} +(-0.455573 + 0.571270i) q^{13} +(0.997204 + 0.0747301i) q^{14} +(0.0747301 - 0.997204i) q^{16} +(0.955573 - 0.294755i) q^{17} +(1.48883 + 2.57873i) q^{18} +(1.85654 + 0.728639i) q^{21} +(-0.414278 - 1.81507i) q^{22} +(1.49419 + 0.460898i) q^{23} +(0.728639 - 1.85654i) q^{24} +(0.365341 - 0.930874i) q^{25} +(0.698220 + 0.215372i) q^{26} +(0.877681 + 3.84537i) q^{27} +(-0.294755 - 0.955573i) q^{28} +(0.433884 + 0.751509i) q^{31} +(-0.955573 + 0.294755i) q^{32} +(0.277479 - 3.70270i) q^{33} +(-0.623490 - 0.781831i) q^{34} +(1.85654 - 2.32803i) q^{36} +(1.20406 + 0.820914i) q^{39} -1.99441i q^{42} +(-1.53825 + 1.04876i) q^{44} +(-0.116853 - 1.55929i) q^{46} -1.99441 q^{48} +(-0.623490 - 0.781831i) q^{49} -1.00000 q^{50} +(-0.728639 - 1.85654i) q^{51} +(-0.0546039 - 0.728639i) q^{52} +(-1.21135 + 1.12397i) q^{53} +(3.25890 - 2.22188i) q^{54} +(-0.781831 + 0.623490i) q^{56} +(0.541044 - 0.678448i) q^{62} +(0.877681 - 2.84537i) q^{63} +(0.623490 + 0.781831i) q^{64} +(-3.54812 + 1.09445i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(0.693950 - 3.04039i) q^{69} +(0.250701 + 1.09839i) q^{71} +(-2.84537 - 0.877681i) q^{72} +(-1.90580 - 0.587862i) q^{75} +(-1.04876 + 1.53825i) q^{77} +(0.324275 - 1.42074i) q^{78} +(0.149042 - 0.258149i) q^{79} +(4.67161 - 1.44100i) q^{81} +(-1.85654 + 0.728639i) q^{84} +(1.53825 + 1.04876i) q^{88} +(0.147791 - 0.0222759i) q^{89} +(-0.317031 - 0.658322i) q^{91} +(-1.40881 + 0.678448i) q^{92} +(1.42996 - 0.974928i) q^{93} +(0.728639 + 1.85654i) q^{96} +(-0.500000 + 0.866025i) q^{98} -5.54365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52}+ \cdots - 12 q^{98}+O(q^{100}) \) 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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Database

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