LMFDB - Embedded newform 3332.1.cc.b.1563.1

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:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} + \cdots)$

Embedding invariants

Embedding label			1563.1
Root			$-0.733052 + 0.680173i$ of defining polynomial
Character	$\chi$	$=$	3332.1563
Dual form			3332.1.cc.b.1019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.988831 + 0.149042i) q^{2} +(1.36534 + 0.930874i) q^{3} +(0.955573 - 0.294755i) q^{4} +(-1.48883 - 0.716983i) q^{6} +(-0.900969 + 0.433884i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.632289 + 1.61105i) q^{9} +O(q^{10})$	\(q+(-0.988831 + 0.149042i) q^{2} +(1.36534 + 0.930874i) q^{3} +(0.955573 - 0.294755i) q^{4} +(-1.48883 - 0.716983i) q^{6} +(-0.900969 + 0.433884i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.632289 + 1.61105i) q^{9} +(-0.722521 + 1.84095i) q^{11} +(1.57906 + 0.487076i) q^{12} +(-1.23305 - 1.54620i) q^{13} +(0.826239 - 0.563320i) q^{14} +(0.826239 - 0.563320i) q^{16} +(-0.733052 + 0.680173i) q^{17} +(-0.865341 - 1.49881i) q^{18} +(-1.63402 - 0.246289i) q^{21} +(0.440071 - 1.92808i) q^{22} +(-0.914101 - 0.848162i) q^{23} +(-1.63402 - 0.246289i) q^{24} +(-0.988831 - 0.149042i) q^{25} +(1.44973 + 1.34515i) q^{26} +(-0.268680 + 1.17716i) q^{27} +(-0.733052 + 0.680173i) q^{28} +(0.900969 + 1.56052i) q^{31} +(-0.733052 + 0.680173i) q^{32} +(-2.70018 + 1.84095i) q^{33} +(0.623490 - 0.781831i) q^{34} +(1.07906 + 1.35310i) q^{36} +(-0.244221 - 3.25890i) q^{39} +1.65248 q^{42} +(-0.147791 + 1.97213i) q^{44} +(1.03030 + 0.702449i) q^{46} +1.65248 q^{48} +(0.623490 - 0.781831i) q^{49} +1.00000 q^{50} +(-1.63402 + 0.246289i) q^{51} +(-1.63402 - 1.11406i) q^{52} +(0.142820 - 0.0440542i) q^{53} +(0.0902318 - 1.20406i) q^{54} +(0.623490 - 0.781831i) q^{56} +(-1.12349 - 1.40881i) q^{62} +(-1.26868 - 1.17716i) q^{63} +(0.623490 - 0.781831i) q^{64} +(2.39564 - 2.22283i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(-0.458528 - 2.00894i) q^{69} +(-0.0332580 + 0.145713i) q^{71} +(-1.26868 - 1.17716i) q^{72} +(-1.21135 - 1.12397i) q^{75} +(-0.147791 - 1.97213i) q^{77} +(0.727208 + 3.18610i) q^{78} +(-0.365341 + 0.632789i) q^{79} +(-0.193950 + 0.179959i) q^{81} +(-1.63402 + 0.246289i) q^{84} +(-0.147791 - 1.97213i) q^{88} +(0.603718 + 1.53825i) q^{89} +(1.78181 + 0.858075i) q^{91} +(-1.12349 - 0.541044i) q^{92} +(-0.222521 + 2.96934i) q^{93} +(-1.63402 + 0.246289i) q^{96} +(-0.500000 + 0.866025i) q^{98} -3.42270 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) $ 12 * q + q^2 + 13 * q^3 + q^4 - 5 * q^6 - 2 * q^7 - 2 * q^8 + 14 * q^9	$ 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 8 q^{11} - q^{12} - 5 q^{13} + q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + q^{25} - q^{26} + 12 q^{27} + q^{28} + 2 q^{31} + q^{32} - 6 q^{33} - 2 q^{34} - 7 q^{36} - 6 q^{39} + 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} - 2 q^{49} + 12 q^{50} - q^{51} - q^{52} - q^{53} - 6 q^{54} - 2 q^{56} - 4 q^{62} - 2 q^{64} + 15 q^{66} - 6 q^{68} - 3 q^{69} + 2 q^{71} - q^{75} - q^{77} - 9 q^{78} - q^{79} + 13 q^{81} - q^{84} - q^{88} - q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{93} - q^{96} - 6 q^{98} - 14 q^{99}+O(q^{100}) $ 12 * q + q^2 + 13 * q^3 + q^4 - 5 * q^6 - 2 * q^7 - 2 * q^8 + 14 * q^9 - 8 * q^11 - q^12 - 5 * q^13 + q^14 + q^16 + q^17 - 7 * q^18 - q^21 + 2 * q^22 + 2 * q^23 - q^24 + q^25 - q^26 + 12 * q^27 + q^28 + 2 * q^31 + q^32 - 6 * q^33 - 2 * q^34 - 7 * q^36 - 6 * q^39 + 2 * q^42 - q^44 + 2 * q^46 + 2 * q^48 - 2 * q^49 + 12 * q^50 - q^51 - q^52 - q^53 - 6 * q^54 - 2 * q^56 - 4 * q^62 - 2 * q^64 + 15 * q^66 - 6 * q^68 - 3 * q^69 + 2 * q^71 - q^75 - q^77 - 9 * q^78 - q^79 + 13 * q^81 - q^84 - q^88 - q^89 + 2 * q^91 - 4 * q^92 - 2 * q^93 - q^96 - 6 * q^98 - 14 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$e\left(\frac{4}{21}\right)$	$-1$

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 $

$2$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$2$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$3$	1.36534	+	0.930874i	1.36534	+	0.930874i	1.00000			$0$
$3$	1.36534	+	0.930874i	1.36534	+	0.930874i	0.365341	+	0.930874i	$0.380952\pi$
$4$	0.955573	−	0.294755i	0.955573	−	0.294755i
$5$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$5$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$6$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$7$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$7$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$8$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$9$	0.632289	+	1.61105i	0.632289	+	1.61105i
$10$		0			0
$11$	−0.722521	+	1.84095i	−0.722521	+	1.84095i	−0.222521	+	0.974928i	$0.571429\pi$
$11$	−0.722521	+	1.84095i	−0.722521	+	1.84095i	−0.500000	+	0.866025i	$0.666667\pi$
$12$	1.57906	+	0.487076i	1.57906	+	0.487076i
$13$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.733052	−	0.680173i	$-0.761905\pi$
$13$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.500000	−	0.866025i	$-0.666667\pi$
$14$	0.826239	−	0.563320i	0.826239	−	0.563320i
$15$		0			0
$16$	0.826239	−	0.563320i	0.826239	−	0.563320i
$17$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$17$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$18$	−0.865341	−	1.49881i	−0.865341	−	1.49881i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	−1.63402	−	0.246289i	−1.63402	−	0.246289i
$22$	0.440071	−	1.92808i	0.440071	−	1.92808i
$23$	−0.914101	−	0.848162i	−0.914101	−	0.848162i	0.0747301	−	0.997204i	$-0.476190\pi$
$23$	−0.914101	−	0.848162i	−0.914101	−	0.848162i	−0.988831	+	0.149042i	$0.952381\pi$
$24$	−1.63402	−	0.246289i	−1.63402	−	0.246289i
$25$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$26$	1.44973	+	1.34515i	1.44973	+	1.34515i
$27$	−0.268680	+	1.17716i	−0.268680	+	1.17716i
$28$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$29$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$29$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$30$		0			0
$31$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$31$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$32$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$33$	−2.70018	+	1.84095i	−2.70018	+	1.84095i
$34$	0.623490	−	0.781831i	0.623490	−	0.781831i
$35$		0			0
$36$	1.07906	+	1.35310i	1.07906	+	1.35310i
$37$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$37$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$38$		0			0
$39$	−0.244221	−	3.25890i	−0.244221	−	3.25890i
$40$		0			0
$41$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$41$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$42$	1.65248			1.65248
$43$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$43$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$44$	−0.147791	+	1.97213i	−0.147791	+	1.97213i
$45$		0			0
$46$	1.03030	+	0.702449i	1.03030	+	0.702449i
$47$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$47$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$48$	1.65248			1.65248
$49$	0.623490	−	0.781831i	0.623490	−	0.781831i
$50$	1.00000			1.00000
$51$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$52$	−1.63402	−	1.11406i	−1.63402	−	1.11406i
$53$	0.142820	−	0.0440542i	0.142820	−	0.0440542i	−0.222521	−	0.974928i	$-0.571429\pi$
$53$	0.142820	−	0.0440542i	0.142820	−	0.0440542i	0.365341	+	0.930874i	$0.380952\pi$
$54$	0.0902318	−	1.20406i	0.0902318	−	1.20406i
$55$		0			0
$56$	0.623490	−	0.781831i	0.623490	−	0.781831i
$57$		0			0
$58$		0			0
$59$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$59$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$60$		0			0
$61$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$61$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$62$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$63$	−1.26868	−	1.17716i	−1.26868	−	1.17716i
$64$	0.623490	−	0.781831i	0.623490	−	0.781831i
$65$		0			0
$66$	2.39564	−	2.22283i	2.39564	−	2.22283i
$67$		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
$69$	−0.458528	−	2.00894i	−0.458528	−	2.00894i
$70$		0			0
$71$	−0.0332580	+	0.145713i	−0.0332580	+	0.145713i	−0.988831	−	0.149042i	$-0.952381\pi$
$71$	−0.0332580	+	0.145713i	−0.0332580	+	0.145713i	0.955573	+	0.294755i	$0.0952381\pi$
$72$	−1.26868	−	1.17716i	−1.26868	−	1.17716i
$73$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$73$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$74$		0			0
$75$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$76$		0			0
$77$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$78$	0.727208	+	3.18610i	0.727208	+	3.18610i
$79$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$79$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$80$		0			0
$81$	−0.193950	+	0.179959i	−0.193950	+	0.179959i
$82$		0			0
$83$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$83$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$84$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$89$	0.603718	+	1.53825i	0.603718	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$89$	0.603718	+	1.53825i	0.603718	+	1.53825i	−0.222521	+	0.974928i	$0.571429\pi$
$90$		0			0
$91$	1.78181	+	0.858075i	1.78181	+	0.858075i
$92$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$93$	−0.222521	+	2.96934i	−0.222521	+	2.96934i
$94$		0			0
$95$		0			0
$96$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$99$	−3.42270			−3.42270
$100$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$101$	1.03030	+	0.702449i	1.03030	+	0.702449i	0.955573	−	0.294755i	$-0.0952381\pi$
$101$	1.03030	+	0.702449i	1.03030	+	0.702449i	0.0747301	+	0.997204i	$0.476190\pi$
$102$	1.57906	−	0.487076i	1.57906	−	0.487076i
$103$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$103$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$104$	1.78181	+	0.858075i	1.78181	+	0.858075i
$105$		0			0
$106$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i
$107$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.623490	+	0.781831i	$0.285714\pi$
$107$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.0747301	+	0.997204i	$0.476190\pi$
$108$	0.0902318	+	1.20406i	0.0902318	+	1.20406i
$109$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$109$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$113$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$113$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.71135	−	2.96415i	1.71135	−	2.96415i
$118$		0			0
$119$	0.365341	−	0.930874i	0.365341	−	0.930874i
$120$		0			0
$121$	−2.13402	−	1.98008i	−2.13402	−	1.98008i
$122$		0			0
$123$		0			0
$124$	1.32091	+	1.22563i	1.32091	+	1.22563i
$125$		0			0
$126$	1.42996	+	0.974928i	1.42996	+	0.974928i
$127$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$127$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$	1.03030	−	0.702449i	1.03030	−	0.702449i	0.0747301	−	0.997204i	$-0.476190\pi$
$131$	1.03030	−	0.702449i	1.03030	−	0.702449i	0.955573	+	0.294755i	$0.0952381\pi$
$132$	−2.03759	+	2.55506i	−2.03759	+	2.55506i
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.365341	−	0.930874i	0.365341	−	0.930874i
$137$	0.123490	+	1.64786i	0.123490	+	1.64786i	0.623490	+	0.781831i	$0.285714\pi$
$137$	0.123490	+	1.64786i	0.123490	+	1.64786i	−0.500000	+	0.866025i	$0.666667\pi$
$138$	0.752824	+	1.91816i	0.752824	+	1.91816i
$139$	−0.658322	+	0.317031i	−0.658322	+	0.317031i	−0.733052	−	0.680173i	$-0.761905\pi$
$139$	−0.658322	+	0.317031i	−0.658322	+	0.317031i	0.0747301	+	0.997204i	$0.476190\pi$
$140$		0			0
$141$		0			0
$142$	0.0111692	−	0.149042i	0.0111692	−	0.149042i
$143$	3.73738	−	1.15283i	3.73738	−	1.15283i
$144$	1.42996	+	0.974928i	1.42996	+	0.974928i
$145$		0			0
$146$		0			0
$147$	1.57906	−	0.487076i	1.57906	−	0.487076i
$148$		0			0
$149$	−1.63402	+	0.246289i	−1.63402	+	0.246289i	−0.900969	−	0.433884i	$-0.857143\pi$
$149$	−1.63402	+	0.246289i	−1.63402	+	0.246289i	−0.733052	+	0.680173i	$0.761905\pi$
$150$	1.36534	+	0.930874i	1.36534	+	0.930874i
$151$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$151$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$152$		0			0
$153$	−1.55929	−	0.750915i	−1.55929	−	0.750915i
$154$	0.440071	+	1.92808i	0.440071	+	1.92808i
$155$		0			0
$156$	−1.19395	−	3.04213i	−1.19395	−	3.04213i
$157$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	0.955573	+	0.294755i	$0.0952381\pi$
$157$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	−0.900969	+	0.433884i	$0.857143\pi$
$158$	0.266948	−	0.680173i	0.266948	−	0.680173i
$159$	0.236007	+	0.0727985i	0.236007	+	0.0727985i
$160$		0			0
$161$	1.19158	+	0.367554i	1.19158	+	0.367554i
$162$	0.164962	−	0.206856i	0.164962	−	0.206856i
$163$	1.65248	−	1.12664i	1.65248	−	1.12664i	0.826239	−	0.563320i	$-0.190476\pi$
$163$	1.65248	−	1.12664i	1.65248	−	1.12664i	0.826239	−	0.563320i	$-0.190476\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.326239	+	1.42935i	0.326239	+	1.42935i	0.826239	+	0.563320i	$0.190476\pi$
$167$	0.326239	+	1.42935i	0.326239	+	1.42935i	−0.500000	+	0.866025i	$0.666667\pi$
$168$	1.57906	−	0.487076i	1.57906	−	0.487076i
$169$	−0.647791	+	2.83816i	−0.647791	+	2.83816i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$173$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$174$		0			0
$175$	0.955573	−	0.294755i	0.955573	−	0.294755i
$176$	0.440071	+	1.92808i	0.440071	+	1.92808i
$177$		0			0
$178$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$179$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$179$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$180$		0			0
$181$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$181$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$182$	−1.88980	−	0.582926i	−1.88980	−	0.582926i
$183$		0			0
$184$	1.19158	+	0.367554i	1.19158	+	0.367554i
$185$		0			0
$186$	−0.222521	−	2.96934i	−0.222521	−	2.96934i
$187$	−0.722521	−	1.84095i	−0.722521	−	1.84095i
$188$		0			0
$189$	−0.268680	−	1.17716i	−0.268680	−	1.17716i
$190$		0			0
$191$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$191$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$192$	1.57906	−	0.487076i	1.57906	−	0.487076i
$193$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$193$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$194$		0			0
$195$		0			0
$196$	0.365341	−	0.930874i	0.365341	−	0.930874i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	3.38447	−	0.510127i	3.38447	−	0.510127i
$199$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	0.623490	−	0.781831i	$-0.285714\pi$
$199$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	−0.500000	+	0.866025i	$0.666667\pi$
$200$	0.955573	−	0.294755i	0.955573	−	0.294755i
$201$		0			0
$202$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$203$		0			0
$204$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$205$		0			0
$206$		0			0
$207$	0.788452	−	2.00894i	0.788452	−	2.00894i
$208$	−1.88980	−	0.582926i	−1.88980	−	0.582926i
$209$		0			0
$210$		0			0
$211$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$211$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$212$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$213$	−0.181049	+	0.167989i	−0.181049	+	0.167989i
$214$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$215$		0			0
$216$	−0.268680	−	1.17716i	−0.268680	−	1.17716i
$217$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.95557	+	0.294755i	1.95557	+	0.294755i
$222$		0			0
$223$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$223$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$224$	0.365341	−	0.930874i	0.365341	−	0.930874i
$225$	−0.385113	−	1.68729i	−0.385113	−	1.68729i
$226$		0			0
$227$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$227$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$228$		0			0
$229$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	−0.733052	−	0.680173i	$-0.761905\pi$
$229$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	0.365341	+	0.930874i	$0.380952\pi$
$230$		0			0
$231$	1.63402	−	2.83021i	1.63402	−	2.83021i
$232$		0			0
$233$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$233$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$234$	−1.25045	+	3.18610i	−1.25045	+	3.18610i
$235$		0			0
$236$		0			0
$237$	−1.08786	+	0.523887i	−1.08786	+	0.523887i
$238$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$239$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$239$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$240$		0			0
$241$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$241$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$242$	2.40530	+	1.63991i	2.40530	+	1.63991i
$243$	0.761623	−	0.114796i	0.761623	−	0.114796i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$249$		0			0
$250$		0			0
$251$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$251$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$252$	−1.55929	−	0.750915i	−1.55929	−	0.750915i
$253$	2.22188	−	1.07000i	2.22188	−	1.07000i
$254$		0			0
$255$		0			0
$256$	0.365341	−	0.930874i	0.365341	−	0.930874i
$257$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.222521	−	0.974928i	$-0.571429\pi$
$257$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.733052	+	0.680173i	$0.761905\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.914101	+	0.848162i	−0.914101	+	0.848162i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	1.63402	−	2.83021i	1.63402	−	2.83021i
$265$		0			0
$266$		0			0
$267$	−0.607634	+	2.66222i	−0.607634	+	2.66222i
$268$		0			0
$269$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$269$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$270$		0			0
$271$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$271$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$272$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$273$	1.63402	+	2.83021i	1.63402	+	2.83021i
$274$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$275$	0.988831	−	1.71271i	0.988831	−	1.71271i
$276$	−1.03030	−	1.78454i	−1.03030	−	1.78454i
$277$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$277$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$278$	0.603718	−	0.411608i	0.603718	−	0.411608i
$279$	−1.94440	+	2.43821i	−1.94440	+	2.43821i
$280$		0			0
$281$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$281$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$282$		0			0
$283$	−0.722521	+	1.84095i	−0.722521	+	1.84095i	−0.222521	+	0.974928i	$0.571429\pi$
$283$	−0.722521	+	1.84095i	−0.722521	+	1.84095i	−0.500000	+	0.866025i	$0.666667\pi$
$284$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$285$		0			0
$286$	−3.52382	+	1.69698i	−3.52382	+	1.69698i
$287$		0			0
$288$	−1.55929	−	0.750915i	−1.55929	−	0.750915i
$289$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$293$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$294$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$295$		0			0
$296$		0			0
$297$	−1.97297	−	1.34515i	−1.97297	−	1.34515i
$298$	1.57906	−	0.487076i	1.57906	−	0.487076i
$299$	−0.184292	+	2.45921i	−0.184292	+	2.45921i
$300$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$301$		0			0
$302$		0			0
$303$	0.752824	+	1.91816i	0.752824	+	1.91816i
$304$		0			0
$305$		0			0
$306$	1.65379	+	0.510127i	1.65379	+	0.510127i
$307$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$307$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$308$	−0.722521	−	1.84095i	−0.722521	−	1.84095i
$309$		0			0
$310$		0			0
$311$	0.733052	−	0.680173i	0.733052	−	0.680173i	−0.222521	−	0.974928i	$-0.571429\pi$
$311$	0.733052	−	0.680173i	0.733052	−	0.680173i	0.955573	+	0.294755i	$0.0952381\pi$
$312$	1.63402	+	2.83021i	1.63402	+	2.83021i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−0.162592	−	0.712362i	−0.162592	−	0.712362i
$315$		0			0
$316$	−0.162592	+	0.712362i	−0.162592	+	0.712362i
$317$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$317$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$318$	−0.244221	−	0.0368104i	−0.244221	−	0.0368104i
$319$		0			0
$320$		0			0
$321$	−0.702749	+	3.07894i	−0.702749	+	3.07894i
$322$	−1.23305	−	0.185853i	−1.23305	−	0.185853i
$323$		0			0
$324$	−0.132289	+	0.229132i	−0.132289	+	0.229132i
$325$	0.988831	+	1.71271i	0.988831	+	1.71271i
$326$	−1.46610	+	1.36035i	−1.46610	+	1.36035i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$331$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$332$		0			0
$333$		0			0
$334$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$335$		0			0
$336$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$337$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$337$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$338$	0.217550	−	2.90301i	0.217550	−	2.90301i
$339$		0			0
$340$		0			0
$341$	−3.52382	+	0.531130i	−3.52382	+	0.531130i
$342$		0			0
$343$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.425270	+	0.131178i	−0.425270	+	0.131178i	−0.500000	−	0.866025i	$-0.666667\pi$
$347$	−0.425270	+	0.131178i	−0.425270	+	0.131178i	0.0747301	+	0.997204i	$0.476190\pi$
$348$		0			0
$349$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$349$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$350$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$351$	2.15142	−	1.03607i	2.15142	−	1.03607i
$352$	−0.722521	−	1.84095i	−0.722521	−	1.84095i
$353$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	−0.900969	−	0.433884i	$-0.857143\pi$
$353$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	0.826239	−	0.563320i	$-0.190476\pi$
$354$		0			0
$355$		0			0
$356$	1.03030	+	1.29196i	1.03030	+	1.29196i
$357$	1.36534	−	0.930874i	1.36534	−	0.930874i
$358$		0			0
$359$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$359$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	−1.07046	−	4.68999i	−1.07046	−	4.68999i
$364$	1.95557	+	0.294755i	1.95557	+	0.294755i
$365$		0			0
$366$		0			0
$367$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.900969	−	0.433884i	$-0.857143\pi$
$367$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.988831	+	0.149042i	$0.952381\pi$
$368$	−1.23305	−	0.185853i	−1.23305	−	0.185853i
$369$		0			0
$370$		0			0
$371$	−0.109562	+	0.101659i	−0.109562	+	0.101659i
$372$	0.662592	+	2.90301i	0.662592	+	2.90301i
$373$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$373$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$374$	0.988831	+	1.71271i	0.988831	+	1.71271i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	0.441126	+	1.12397i	0.441126	+	1.12397i
$379$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$379$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$383$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$384$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.21135	−	0.825886i	−1.21135	−	0.825886i	−0.222521	−	0.974928i	$-0.571429\pi$
$389$	−1.21135	−	0.825886i	−1.21135	−	0.825886i	−0.988831	+	0.149042i	$0.952381\pi$
$390$		0			0
$391$	1.24698			1.24698
$392$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$393$	2.06061			2.06061
$394$		0			0
$395$		0			0
$396$	−3.27064	+	1.00886i	−3.27064	+	1.00886i
$397$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$397$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$398$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i
$399$		0			0
$400$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$401$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$401$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$402$		0			0
$403$	1.30194	−	3.31728i	1.30194	−	3.31728i
$404$	1.19158	+	0.367554i	1.19158	+	0.367554i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.36534	−	0.930874i	1.36534	−	0.930874i
$409$	−0.109562	+	0.101659i	−0.109562	+	0.101659i	−0.733052	−	0.680173i	$-0.761905\pi$
$409$	−0.109562	+	0.101659i	−0.109562	+	0.101659i	0.623490	+	0.781831i	$0.285714\pi$
$410$		0			0
$411$	−1.36534	+	2.36484i	−1.36534	+	2.36484i
$412$		0			0
$413$		0			0
$414$	−0.480228	+	2.10402i	−0.480228	+	2.10402i
$415$		0			0
$416$	1.95557	+	0.294755i	1.95557	+	0.294755i
$417$	−1.19395	−	0.179959i	−1.19395	−	0.179959i
$418$		0			0
$419$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$419$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$420$		0			0
$421$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$421$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$422$	0.222521	−	0.385418i	0.222521	−	0.385418i
$423$		0			0
$424$	−0.109562	+	0.101659i	−0.109562	+	0.101659i
$425$	0.826239	−	0.563320i	0.826239	−	0.563320i
$426$	0.153989	−	0.193096i	0.153989	−	0.193096i
$427$		0			0
$428$	1.19158	+	1.49419i	1.19158	+	1.49419i
$429$	6.17594	+	1.90503i	6.17594	+	1.90503i
$430$		0			0
$431$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	−0.900969	−	0.433884i	$-0.857143\pi$
$431$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	0.826239	−	0.563320i	$-0.190476\pi$
$432$	0.441126	+	1.12397i	0.441126	+	1.12397i
$433$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$433$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$434$	1.62349	+	0.781831i	1.62349	+	0.781831i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.722521	+	0.108903i	−0.722521	+	0.108903i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−0.722521	+	0.108903i	−0.722521	+	0.108903i	−0.222521	+	0.974928i	$0.571429\pi$
$440$		0			0
$441$	1.65379	+	0.510127i	1.65379	+	0.510127i
$442$	−1.97766			−1.97766
$443$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$443$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.46026	−	1.18480i	−2.46026	−	1.18480i
$448$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$449$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$449$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$450$	0.632289	+	1.61105i	0.632289	+	1.61105i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$455$		0			0
$456$		0			0
$457$	1.03030	−	0.702449i	1.03030	−	0.702449i	0.0747301	−	0.997204i	$-0.476190\pi$
$457$	1.03030	−	0.702449i	1.03030	−	0.702449i	0.955573	+	0.294755i	$0.0952381\pi$
$458$	0.326239	−	0.302705i	0.326239	−	0.302705i
$459$	−0.603718	−	1.04567i	−0.603718	−	1.04567i
$460$		0			0
$461$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	0.955573	−	0.294755i	$-0.0952381\pi$
$461$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	−0.988831	+	0.149042i	$0.952381\pi$
$462$	−1.19395	+	3.04213i	−1.19395	+	3.04213i
$463$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$463$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$467$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$468$	0.761623	−	3.33689i	0.761623	−	3.33689i
$469$		0			0
$470$		0			0
$471$	−0.603718	+	1.04567i	−0.603718	+	1.04567i
$472$		0			0
$473$		0			0
$474$	0.997630	−	0.680173i	0.997630	−	0.680173i
$475$		0			0
$476$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$477$	0.161277	+	0.202235i	0.161277	+	0.202235i
$478$		0			0
$479$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	1.28477	+	1.61105i	1.28477	+	1.61105i
$484$	−2.62285	−	1.26310i	−2.62285	−	1.26310i
$485$		0			0
$486$	−0.736007	+	0.227028i	−0.736007	+	0.227028i
$487$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$487$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$488$		0			0
$489$	3.30496			3.30496
$490$		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
$496$	1.62349	+	0.781831i	1.62349	+	0.781831i
$497$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$498$		0			0
$499$	0.266948	+	0.680173i	0.266948	+	0.680173i	1.00000			$0$
$499$	0.266948	+	0.680173i	0.266948	+	0.680173i	−0.733052	+	0.680173i	$0.761905\pi$
$500$		0			0
$501$	−0.885113	+	2.25523i	−0.885113	+	2.25523i
$502$		0			0
$503$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.826239	+	0.563320i	$0.190476\pi$
$503$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$504$	1.65379	+	0.510127i	1.65379	+	0.510127i
$505$		0			0
$506$	−2.03759	+	1.38921i	−2.03759	+	1.38921i
$507$	−3.52642	+	3.27204i	−3.52642	+	3.27204i
$508$		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
$514$	0.988831	+	0.149042i	0.988831	+	0.149042i
$515$		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		0.826239	−	0.563320i	$-0.190476\pi$
$523$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$524$	0.777479	−	0.974928i	0.777479	−	0.974928i
$525$	1.57906	+	0.487076i	1.57906	+	0.487076i
$526$		0			0
$527$	−1.72188	−	0.531130i	−1.72188	−	0.531130i
$528$	−1.19395	+	3.04213i	−1.19395	+	3.04213i
$529$	0.0414721	+	0.553406i	0.0414721	+	0.553406i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	0.204064	−	2.72305i	0.204064	−	2.72305i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.988831	+	1.71271i	0.988831	+	1.71271i
$540$		0			0
$541$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$541$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$542$		0			0
$543$		0			0
$544$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$545$		0			0
$546$	−2.03759	−	2.55506i	−2.03759	−	2.55506i
$547$	1.32091	−	0.636119i	1.32091	−	0.636119i	0.365341	−	0.930874i	$-0.380952\pi$
$547$	1.32091	−	0.636119i	1.32091	−	0.636119i	0.955573	+	0.294755i	$0.0952381\pi$
$548$	0.603718	+	1.53825i	0.603718	+	1.53825i
$549$		0			0
$550$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$551$		0			0
$552$	1.28477	+	1.61105i	1.28477	+	1.61105i
$553$	0.0546039	−	0.728639i	0.0546039	−	0.728639i
$554$		0			0
$555$		0			0
$556$	−0.535628	+	0.496990i	−0.535628	+	0.496990i
$557$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$557$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$558$	1.55929	−	2.70077i	1.55929	−	2.70077i
$559$		0			0
$560$		0			0
$561$	0.727208	−	3.18610i	0.727208	−	3.18610i
$562$	0.733052	+	0.680173i	0.733052	+	0.680173i
$563$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$563$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$564$		0			0
$565$		0			0
$566$	0.440071	−	1.92808i	0.440071	−	1.92808i
$567$	0.0966613	−	0.246289i	0.0966613	−	0.246289i
$568$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$569$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$569$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$570$		0			0
$571$	1.32091	−	1.22563i	1.32091	−	1.22563i	0.365341	−	0.930874i	$-0.380952\pi$
$571$	1.32091	−	1.22563i	1.32091	−	1.22563i	0.955573	−	0.294755i	$-0.0952381\pi$
$572$	3.23154	−	2.20323i	3.23154	−	2.20323i
$573$		0			0
$574$		0			0
$575$	0.777479	+	0.974928i	0.777479	+	0.974928i
$576$	1.65379	+	0.510127i	1.65379	+	0.510127i
$577$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.0747301	−	0.997204i	$-0.476190\pi$
$577$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.623490	−	0.781831i	$-0.285714\pi$
$578$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.0220888	+	0.294755i	−0.0220888	+	0.294755i
$584$		0			0
$585$		0			0
$586$	−0.722521	+	0.108903i	−0.722521	+	0.108903i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	1.36534	−	0.930874i	1.36534	−	0.930874i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.142820	−	1.90580i	0.142820	−	1.90580i	−0.222521	−	0.974928i	$-0.571429\pi$
$593$	0.142820	−	1.90580i	0.142820	−	1.90580i	0.365341	−	0.930874i	$-0.380952\pi$
$594$	2.15142	+	1.03607i	2.15142	+	1.03607i
$595$		0			0
$596$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$597$	0.0902318	+	0.229907i	0.0902318	+	0.229907i
$598$	−0.184292	−	2.45921i	−0.184292	−	2.45921i
$599$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$599$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$600$	1.57906	+	0.487076i	1.57906	+	0.487076i
$601$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$601$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−1.03030	−	1.78454i	−1.03030	−	1.78454i
$607$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$607$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.71135	−	0.257945i	−1.71135	−	0.257945i
$613$	1.44973	+	0.218511i	1.44973	+	0.218511i	0.826239	−	0.563320i	$-0.190476\pi$
$613$	1.44973	+	0.218511i	1.44973	+	0.218511i	0.623490	+	0.781831i	$0.285714\pi$
$614$		0			0
$615$		0			0
$616$	0.988831	+	1.71271i	0.988831	+	1.71271i
$617$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$617$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$618$		0			0
$619$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$619$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$	1.24402	−	0.848162i	1.24402	−	0.848162i
$622$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$623$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$624$	−2.03759	−	2.55506i	−2.03759	−	2.55506i
$625$	0.955573	+	0.294755i	0.955573	+	0.294755i
$626$		0			0
$627$		0			0
$628$	0.266948	+	0.680173i	0.266948	+	0.680173i
$629$		0			0
$630$		0			0
$631$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$631$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$632$	0.0546039	−	0.728639i	0.0546039	−	0.728639i
$633$	−0.702749	+	0.216769i	−0.702749	+	0.216769i
$634$		0			0
$635$		0			0
$636$	0.246980			0.246980
$637$	−1.97766			−1.97766
$638$		0			0
$639$	−0.255779	+	0.0385525i	−0.255779	+	0.0385525i
$640$		0			0
$641$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$641$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$642$	0.236007	−	3.14929i	0.236007	−	3.14929i
$643$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$643$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$644$	1.24698			1.24698
$645$		0			0
$646$		0			0
$647$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$647$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$648$	0.0966613	−	0.246289i	0.0966613	−	0.246289i
$649$		0			0
$650$	−1.23305	−	1.54620i	−1.23305	−	1.54620i
$651$	−1.08786	−	2.77183i	−1.08786	−	2.77183i
$652$	1.24698	−	1.56366i	1.24698	−	1.56366i
$653$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$653$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$659$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$660$		0			0
$661$	1.78181	+	0.268565i	1.78181	+	0.268565i	0.955573	−	0.294755i	$-0.0952381\pi$
$661$	1.78181	+	0.268565i	1.78181	+	0.268565i	0.826239	+	0.563320i	$0.190476\pi$
$662$		0			0
$663$	2.39564	+	2.22283i	2.39564	+	2.22283i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.733052	+	1.26968i	0.733052	+	1.26968i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	1.36534	−	0.930874i	1.36534	−	0.930874i
$673$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$673$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$674$		0			0
$675$	0.441126	−	1.12397i	0.441126	−	1.12397i
$676$	0.217550	+	2.90301i	0.217550	+	2.90301i
$677$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$677$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.0184568	−	0.246289i	0.0184568	−	0.246289i
$682$	3.40530	−	1.05040i	3.40530	−	1.05040i
$683$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	0.623490	−	0.781831i	$-0.285714\pi$
$683$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$687$	−0.735422			−0.735422
$688$		0			0
$689$	−0.244221	−	0.166507i	−0.244221	−	0.166507i
$690$		0			0
$691$	−0.134659	+	1.79690i	−0.134659	+	1.79690i	0.365341	+	0.930874i	$0.380952\pi$
$691$	−0.134659	+	1.79690i	−0.134659	+	1.79690i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$	3.08375	−	1.48506i	3.08375	−	1.48506i
$694$	0.400969	−	0.193096i	0.400969	−	0.193096i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$699$		0			0
$700$	0.826239	−	0.563320i	0.826239	−	0.563320i
$701$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	−0.733052	−	0.680173i	$-0.761905\pi$
$701$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	0.826239	+	0.563320i	$0.190476\pi$
$702$	−1.97297	+	1.34515i	−1.97297	+	1.34515i
$703$		0			0
$704$	0.988831	+	1.71271i	0.988831	+	1.71271i
$705$		0			0
$706$	0.222521	+	0.974928i	0.222521	+	0.974928i
$707$	−1.23305	−	0.185853i	−1.23305	−	0.185853i
$708$		0			0
$709$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$709$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$710$		0			0
$711$	−1.25045	−	0.188476i	−1.25045	−	0.188476i
$712$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$713$	0.500000	−	2.19064i	0.500000	−	2.19064i
$714$	−1.21135	+	1.12397i	−1.21135	+	1.12397i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.733052	+	0.680173i	$0.761905\pi$
$719$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.900969	+	0.433884i	$0.857143\pi$
$720$		0			0
$721$		0			0
$722$	0.623490	+	0.781831i	0.623490	+	0.781831i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	1.75751	+	4.47806i	1.75751	+	4.47806i
$727$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$727$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$728$	−1.97766			−1.97766
$729$	1.38511	+	0.667035i	1.38511	+	0.667035i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.365341	−	0.930874i	$-0.380952\pi$
$733$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.623490	+	0.781831i	$0.285714\pi$
$734$	1.91115			1.91115
$735$		0			0
$736$	1.24698			1.24698
$737$		0			0
$738$		0			0
$739$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$739$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$740$		0			0
$741$		0			0
$742$	0.0931869	−	0.116853i	0.0931869	−	0.116853i
$743$	−1.48883	+	0.716983i	−1.48883	+	0.716983i	−0.988831	−	0.149042i	$-0.952381\pi$
$743$	−1.48883	+	0.716983i	−1.48883	+	0.716983i	−0.500000	+	0.866025i	$0.666667\pi$
$744$	−1.08786	−	2.77183i	−1.08786	−	2.77183i
$745$		0			0
$746$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$747$		0			0
$748$	−1.23305	−	1.54620i	−1.23305	−	1.54620i
$749$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$750$		0			0
$751$	0.603718	−	0.411608i	0.603718	−	0.411608i	−0.222521	−	0.974928i	$-0.571429\pi$
$751$	0.603718	−	0.411608i	0.603718	−	0.411608i	0.826239	+	0.563320i	$0.190476\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.603718	−	1.04567i	−0.603718	−	1.04567i
$757$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$757$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$758$	0.733052	+	0.680173i	0.733052	+	0.680173i
$759$	4.02966	+	0.607374i	4.02966	+	0.607374i
$760$		0			0
$761$	1.07473	+	0.997204i	1.07473	+	0.997204i	1.00000			$0$
$761$	1.07473	+	0.997204i	1.07473	+	0.997204i	0.0747301	+	0.997204i	$0.476190\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.36534	−	0.930874i	1.36534	−	0.930874i
$769$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$769$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$770$		0			0
$771$	−1.03030	−	1.29196i	−1.03030	−	1.29196i
$772$		0			0
$773$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	0.365341	+	0.930874i	$0.380952\pi$
$773$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	−0.900969	+	0.433884i	$0.857143\pi$
$774$		0			0
$775$	−0.658322	−	1.67738i	−0.658322	−	1.67738i
$776$		0			0
$777$		0			0
$778$	1.32091	+	0.636119i	1.32091	+	0.636119i
$779$		0			0
$780$		0			0
$781$	−0.244221	−	0.166507i	−0.244221	−	0.166507i
$782$	−1.23305	+	0.185853i	−1.23305	+	0.185853i
$783$		0			0
$784$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$785$		0			0
$786$	−2.03759	+	0.307117i	−2.03759	+	0.307117i
$787$	−0.367711

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)

\(f(q)\)	\(=\)	\(q+(-0.988831 + 0.149042i) q^{2} +(1.36534 + 0.930874i) q^{3} +(0.955573 - 0.294755i) q^{4} +(-1.48883 - 0.716983i) q^{6} +(-0.900969 + 0.433884i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.632289 + 1.61105i) q^{9} +O(q^{10})\)	\(q+(-0.988831 + 0.149042i) q^{2} +(1.36534 + 0.930874i) q^{3} +(0.955573 - 0.294755i) q^{4} +(-1.48883 - 0.716983i) q^{6} +(-0.900969 + 0.433884i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.632289 + 1.61105i) q^{9} +(-0.722521 + 1.84095i) q^{11} +(1.57906 + 0.487076i) q^{12} +(-1.23305 - 1.54620i) q^{13} +(0.826239 - 0.563320i) q^{14} +(0.826239 - 0.563320i) q^{16} +(-0.733052 + 0.680173i) q^{17} +(-0.865341 - 1.49881i) q^{18} +(-1.63402 - 0.246289i) q^{21} +(0.440071 - 1.92808i) q^{22} +(-0.914101 - 0.848162i) q^{23} +(-1.63402 - 0.246289i) q^{24} +(-0.988831 - 0.149042i) q^{25} +(1.44973 + 1.34515i) q^{26} +(-0.268680 + 1.17716i) q^{27} +(-0.733052 + 0.680173i) q^{28} +(0.900969 + 1.56052i) q^{31} +(-0.733052 + 0.680173i) q^{32} +(-2.70018 + 1.84095i) q^{33} +(0.623490 - 0.781831i) q^{34} +(1.07906 + 1.35310i) q^{36} +(-0.244221 - 3.25890i) q^{39} +1.65248 q^{42} +(-0.147791 + 1.97213i) q^{44} +(1.03030 + 0.702449i) q^{46} +1.65248 q^{48} +(0.623490 - 0.781831i) q^{49} +1.00000 q^{50} +(-1.63402 + 0.246289i) q^{51} +(-1.63402 - 1.11406i) q^{52} +(0.142820 - 0.0440542i) q^{53} +(0.0902318 - 1.20406i) q^{54} +(0.623490 - 0.781831i) q^{56} +(-1.12349 - 1.40881i) q^{62} +(-1.26868 - 1.17716i) q^{63} +(0.623490 - 0.781831i) q^{64} +(2.39564 - 2.22283i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(-0.458528 - 2.00894i) q^{69} +(-0.0332580 + 0.145713i) q^{71} +(-1.26868 - 1.17716i) q^{72} +(-1.21135 - 1.12397i) q^{75} +(-0.147791 - 1.97213i) q^{77} +(0.727208 + 3.18610i) q^{78} +(-0.365341 + 0.632789i) q^{79} +(-0.193950 + 0.179959i) q^{81} +(-1.63402 + 0.246289i) q^{84} +(-0.147791 - 1.97213i) q^{88} +(0.603718 + 1.53825i) q^{89} +(1.78181 + 0.858075i) q^{91} +(-1.12349 - 0.541044i) q^{92} +(-0.222521 + 2.96934i) q^{93} +(-1.63402 + 0.246289i) q^{96} +(-0.500000 + 0.866025i) q^{98} -3.42270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) \) 12 * q + q^2 + 13 * q^3 + q^4 - 5 * q^6 - 2 * q^7 - 2 * q^8 + 14 * q^9	\( 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 8 q^{11} - q^{12} - 5 q^{13} + q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + q^{25} - q^{26} + 12 q^{27} + q^{28} + 2 q^{31} + q^{32} - 6 q^{33} - 2 q^{34} - 7 q^{36} - 6 q^{39} + 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} - 2 q^{49} + 12 q^{50} - q^{51} - q^{52} - q^{53} - 6 q^{54} - 2 q^{56} - 4 q^{62} - 2 q^{64} + 15 q^{66} - 6 q^{68} - 3 q^{69} + 2 q^{71} - q^{75} - q^{77} - 9 q^{78} - q^{79} + 13 q^{81} - q^{84} - q^{88} - q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{93} - q^{96} - 6 q^{98} - 14 q^{99}+O(q^{100}) \) 12 * q + q^2 + 13 * q^3 + q^4 - 5 * q^6 - 2 * q^7 - 2 * q^8 + 14 * q^9 - 8 * q^11 - q^12 - 5 * q^13 + q^14 + q^16 + q^17 - 7 * q^18 - q^21 + 2 * q^22 + 2 * q^23 - q^24 + q^25 - q^26 + 12 * q^27 + q^28 + 2 * q^31 + q^32 - 6 * q^33 - 2 * q^34 - 7 * q^36 - 6 * q^39 + 2 * q^42 - q^44 + 2 * q^46 + 2 * q^48 - 2 * q^49 + 12 * q^50 - q^51 - q^52 - q^53 - 6 * q^54 - 2 * q^56 - 4 * q^62 - 2 * q^64 + 15 * q^66 - 6 * q^68 - 3 * q^69 + 2 * q^71 - q^75 - q^77 - 9 * q^78 - q^79 + 13 * q^81 - q^84 - q^88 - q^89 + 2 * q^91 - 4 * q^92 - 2 * q^93 - q^96 - 6 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

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