LMFDB - Embedded newform 3332.1.cc.b.135.1

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			135.1
Root			$-0.988831 - 0.149042i$ of defining polynomial
Character	$\chi$	$=$	3332.135
Dual form			3332.1.cc.b.543.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.826239 - 0.563320i) q^{2} +(1.07473 - 0.997204i) q^{3} +(0.365341 - 0.930874i) q^{4} +(0.326239 - 1.42935i) q^{6} +(-0.222521 - 0.974928i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(0.0858993 - 1.14625i) q^{9} +(0.123490 + 1.64786i) q^{11} +(-0.535628 - 1.36476i) q^{12} +(-1.48883 - 0.716983i) q^{13} +(-0.733052 - 0.680173i) q^{14} +(-0.733052 - 0.680173i) q^{16} +(-0.988831 - 0.149042i) q^{17} +(-0.574730 - 0.995462i) q^{18} +(-1.21135 - 0.825886i) q^{21} +(1.03030 + 1.29196i) q^{22} +(1.78181 - 0.268565i) q^{23} +(-1.21135 - 0.825886i) q^{24} +(0.826239 + 0.563320i) q^{25} +(-1.63402 + 0.246289i) q^{26} +(-0.136622 - 0.171318i) q^{27} +(-0.988831 - 0.149042i) q^{28} +(0.222521 + 0.385418i) q^{31} +(-0.988831 - 0.149042i) q^{32} +(1.77597 + 1.64786i) q^{33} +(-0.900969 + 0.433884i) q^{34} +(-1.03563 - 0.498732i) q^{36} +(-2.31507 + 0.714104i) q^{39} -1.46610 q^{42} +(1.57906 + 0.487076i) q^{44} +(1.32091 - 1.22563i) q^{46} -1.46610 q^{48} +(-0.900969 + 0.433884i) q^{49} +1.00000 q^{50} +(-1.21135 + 0.825886i) q^{51} +(-1.21135 + 1.12397i) q^{52} +(0.698220 - 1.77904i) q^{53} +(-0.209389 - 0.0645880i) q^{54} +(-0.900969 + 0.433884i) q^{56} +(0.400969 + 0.193096i) q^{62} +(-1.13662 + 0.171318i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(2.39564 + 0.361085i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(1.64715 - 2.06546i) q^{69} +(1.19158 + 1.49419i) q^{71} +(-1.13662 + 0.171318i) q^{72} +(1.44973 - 0.218511i) q^{75} +(1.57906 - 0.487076i) q^{77} +(-1.51053 + 1.89415i) q^{78} +(-0.0747301 + 0.129436i) q^{79} +(0.818951 + 0.123437i) q^{81} +(-1.21135 + 0.825886i) q^{84} +(1.57906 - 0.487076i) q^{88} +(-0.109562 + 1.46200i) q^{89} +(-0.367711 + 1.61105i) q^{91} +(0.400969 - 1.75676i) q^{92} +(0.623490 + 0.192321i) q^{93} +(-1.21135 + 0.825886i) q^{96} +(-0.500000 + 0.866025i) q^{98} +1.89946 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 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}+ \cdots - 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

Character values

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

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$e\left(\frac{16}{21}\right)$	$-1$

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 $

$2$	0.826239	−	0.563320i	0.826239	−	0.563320i
$2$	0.826239	−	0.563320i	0.826239	−	0.563320i
$3$	1.07473	−	0.997204i	1.07473	−	0.997204i	0.0747301	−	0.997204i	$-0.476190\pi$
$3$	1.07473	−	0.997204i	1.07473	−	0.997204i	1.00000			$0$
$4$	0.365341	−	0.930874i	0.365341	−	0.930874i
$5$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$5$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$6$	0.326239	−	1.42935i	0.326239	−	1.42935i
$7$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$7$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$8$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$9$	0.0858993	−	1.14625i	0.0858993	−	1.14625i
$10$		0			0
$11$	0.123490	+	1.64786i	0.123490	+	1.64786i	0.623490	+	0.781831i	$0.285714\pi$
$11$	0.123490	+	1.64786i	0.123490	+	1.64786i	−0.500000	+	0.866025i	$0.666667\pi$
$12$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$13$	−1.48883	−	0.716983i	−1.48883	−	0.716983i	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.48883	−	0.716983i	−1.48883	−	0.716983i	−0.988831	+	0.149042i	$0.952381\pi$
$14$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$15$		0			0
$16$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$17$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$17$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$18$	−0.574730	−	0.995462i	−0.574730	−	0.995462i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	−1.21135	−	0.825886i	−1.21135	−	0.825886i
$22$	1.03030	+	1.29196i	1.03030	+	1.29196i
$23$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.826239	−	0.563320i	$-0.190476\pi$
$23$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.955573	+	0.294755i	$0.0952381\pi$
$24$	−1.21135	−	0.825886i	−1.21135	−	0.825886i
$25$	0.826239	+	0.563320i	0.826239	+	0.563320i
$26$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$27$	−0.136622	−	0.171318i	−0.136622	−	0.171318i
$28$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$29$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$29$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$30$		0			0
$31$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$31$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$32$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$33$	1.77597	+	1.64786i	1.77597	+	1.64786i
$34$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$35$		0			0
$36$	−1.03563	−	0.498732i	−1.03563	−	0.498732i
$37$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$37$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$38$		0			0
$39$	−2.31507	+	0.714104i	−2.31507	+	0.714104i
$40$		0			0
$41$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$41$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$42$	−1.46610			−1.46610
$43$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$43$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$44$	1.57906	+	0.487076i	1.57906	+	0.487076i
$45$		0			0
$46$	1.32091	−	1.22563i	1.32091	−	1.22563i
$47$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$47$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$48$	−1.46610			−1.46610
$49$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$50$	1.00000			1.00000
$51$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$52$	−1.21135	+	1.12397i	−1.21135	+	1.12397i
$53$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.0747301	−	0.997204i	$-0.476190\pi$
$53$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.623490	−	0.781831i	$-0.285714\pi$
$54$	−0.209389	−	0.0645880i	−0.209389	−	0.0645880i
$55$		0			0
$56$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$57$		0			0
$58$		0			0
$59$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$59$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$60$		0			0
$61$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$61$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$62$	0.400969	+	0.193096i	0.400969	+	0.193096i
$63$	−1.13662	+	0.171318i	−1.13662	+	0.171318i
$64$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$65$		0			0
$66$	2.39564	+	0.361085i	2.39564	+	0.361085i
$67$		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
$69$	1.64715	−	2.06546i	1.64715	−	2.06546i
$70$		0			0
$71$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.826239	+	0.563320i	$0.190476\pi$
$71$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$72$	−1.13662	+	0.171318i	−1.13662	+	0.171318i
$73$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$73$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$74$		0			0
$75$	1.44973	−	0.218511i	1.44973	−	0.218511i
$76$		0			0
$77$	1.57906	−	0.487076i	1.57906	−	0.487076i
$78$	−1.51053	+	1.89415i	−1.51053	+	1.89415i
$79$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$79$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$80$		0			0
$81$	0.818951	+	0.123437i	0.818951	+	0.123437i
$82$		0			0
$83$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$83$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$84$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.57906	−	0.487076i	1.57906	−	0.487076i
$89$	−0.109562	+	1.46200i	−0.109562	+	1.46200i	0.623490	+	0.781831i	$0.285714\pi$
$89$	−0.109562	+	1.46200i	−0.109562	+	1.46200i	−0.733052	+	0.680173i	$0.761905\pi$
$90$		0			0
$91$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$92$	0.400969	−	1.75676i	0.400969	−	1.75676i
$93$	0.623490	+	0.192321i	0.623490	+	0.192321i
$94$		0			0
$95$		0			0
$96$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$99$	1.89946			1.89946
$100$	0.826239	−	0.563320i	0.826239	−	0.563320i
$101$	1.32091	−	1.22563i	1.32091	−	1.22563i	0.365341	−	0.930874i	$-0.380952\pi$
$101$	1.32091	−	1.22563i	1.32091	−	1.22563i	0.955573	−	0.294755i	$-0.0952381\pi$
$102$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$103$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$103$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$104$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$105$		0			0
$106$	−0.425270	−	1.86323i	−0.425270	−	1.86323i
$107$	0.0546039	−	0.728639i	0.0546039	−	0.728639i	−0.900969	−	0.433884i	$-0.857143\pi$
$107$	0.0546039	−	0.728639i	0.0546039	−	0.728639i	0.955573	−	0.294755i	$-0.0952381\pi$
$108$	−0.209389	+	0.0645880i	−0.209389	+	0.0645880i
$109$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$109$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$113$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$113$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.949729	+	1.64498i	−0.949729	+	1.64498i
$118$		0			0
$119$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$120$		0			0
$121$	−1.71135	+	0.257945i	−1.71135	+	0.257945i
$122$		0			0
$123$		0			0
$124$	0.440071	−	0.0663300i	0.440071	−	0.0663300i
$125$		0			0
$126$	−0.842614	+	0.781831i	−0.842614	+	0.781831i
$127$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$127$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$	1.32091	+	1.22563i	1.32091	+	1.22563i	0.955573	+	0.294755i	$0.0952381\pi$
$131$	1.32091	+	1.22563i	1.32091	+	1.22563i	0.365341	+	0.930874i	$0.380952\pi$
$132$	2.18278	−	1.05117i	2.18278	−	1.05117i
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$137$	−1.40097	+	0.432142i	−1.40097	+	0.432142i	−0.900969	−	0.433884i	$-0.857143\pi$
$137$	−1.40097	+	0.432142i	−1.40097	+	0.432142i	−0.500000	+	0.866025i	$0.666667\pi$
$138$	0.197424	−	2.63444i	0.197424	−	2.63444i
$139$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	0.955573	−	0.294755i	$-0.0952381\pi$
$139$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	−0.988831	+	0.149042i	$0.952381\pi$
$140$		0			0
$141$		0			0
$142$	1.82624	+	0.563320i	1.82624	+	0.563320i
$143$	0.997630	−	2.54192i	0.997630	−	2.54192i
$144$	−0.842614	+	0.781831i	−0.842614	+	0.781831i
$145$		0			0
$146$		0			0
$147$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$148$		0			0
$149$	−1.21135	+	0.825886i	−1.21135	+	0.825886i	−0.988831	−	0.149042i	$-0.952381\pi$
$149$	−1.21135	+	0.825886i	−1.21135	+	0.825886i	−0.222521	+	0.974928i	$0.571429\pi$
$150$	1.07473	−	0.997204i	1.07473	−	0.997204i
$151$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$151$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$152$		0			0
$153$	−0.255779	+	1.12064i	−0.255779	+	1.12064i
$154$	1.03030	−	1.29196i	1.03030	−	1.29196i
$155$		0			0
$156$	−0.181049	+	2.41593i	−0.181049	+	2.41593i
$157$	0.142820	−	0.0440542i	0.142820	−	0.0440542i	−0.222521	−	0.974928i	$-0.571429\pi$
$157$	0.142820	−	0.0440542i	0.142820	−	0.0440542i	0.365341	+	0.930874i	$0.380952\pi$
$158$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$159$	−1.02366	−	2.60825i	−1.02366	−	2.60825i
$160$		0			0
$161$	−0.658322	−	1.67738i	−0.658322	−	1.67738i
$162$	0.746184	−	0.359343i	0.746184	−	0.359343i
$163$	−1.46610	−	1.36035i	−1.46610	−	1.36035i	−0.733052	−	0.680173i	$-0.761905\pi$
$163$	−1.46610	−	1.36035i	−1.46610	−	1.36035i	−0.733052	−	0.680173i	$-0.761905\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.23305	+	1.54620i	−1.23305	+	1.54620i	−0.500000	+	0.866025i	$0.666667\pi$
$167$	−1.23305	+	1.54620i	−1.23305	+	1.54620i	−0.733052	+	0.680173i	$0.761905\pi$
$168$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$169$	1.07906	+	1.35310i	1.07906	+	1.35310i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$173$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$174$		0			0
$175$	0.365341	−	0.930874i	0.365341	−	0.930874i
$176$	1.03030	−	1.29196i	1.03030	−	1.29196i
$177$		0			0
$178$	0.733052	+	1.26968i	0.733052	+	1.26968i
$179$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$179$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$180$		0			0
$181$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$181$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$182$	0.603718	+	1.53825i	0.603718	+	1.53825i
$183$		0			0
$184$	−0.658322	−	1.67738i	−0.658322	−	1.67738i
$185$		0			0
$186$	0.623490	−	0.192321i	0.623490	−	0.192321i
$187$	0.123490	−	1.64786i	0.123490	−	1.64786i
$188$		0			0
$189$	−0.136622	+	0.171318i	−0.136622	+	0.171318i
$190$		0			0
$191$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$191$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$192$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$193$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$193$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$194$		0			0
$195$		0			0
$196$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	1.56941	−	1.07000i	1.56941	−	1.07000i
$199$	−1.40097	+	1.29991i	−1.40097	+	1.29991i	−0.500000	+	0.866025i	$0.666667\pi$
$199$	−1.40097	+	1.29991i	−1.40097	+	1.29991i	−0.900969	+	0.433884i	$0.857143\pi$
$200$	0.365341	−	0.930874i	0.365341	−	0.930874i
$201$		0			0
$202$	0.400969	−	1.75676i	0.400969	−	1.75676i
$203$		0			0
$204$	0.326239	+	1.42935i	0.326239	+	1.42935i
$205$		0			0
$206$		0			0
$207$	−0.154785	−	2.06546i	−0.154785	−	2.06546i
$208$	0.603718	+	1.53825i	0.603718	+	1.53825i
$209$		0			0
$210$		0			0
$211$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$211$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$212$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$213$	2.77064	+	0.417607i	2.77064	+	0.417607i
$214$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$215$		0			0
$216$	−0.136622	+	0.171318i	−0.136622	+	0.171318i
$217$	0.326239	−	0.302705i	0.326239	−	0.302705i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.36534	+	0.930874i	1.36534	+	0.930874i
$222$		0			0
$223$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$223$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$224$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$225$	0.716677	−	0.898684i	0.716677	−	0.898684i
$226$		0			0
$227$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$227$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$228$		0			0
$229$	−0.914101	−	0.848162i	−0.914101	−	0.848162i	0.0747301	−	0.997204i	$-0.476190\pi$
$229$	−0.914101	−	0.848162i	−0.914101	−	0.848162i	−0.988831	+	0.149042i	$0.952381\pi$
$230$		0			0
$231$	1.21135	−	2.09812i	1.21135	−	2.09812i
$232$		0			0
$233$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$233$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$234$	0.141947	+	1.89415i	0.141947	+	1.89415i
$235$		0			0
$236$		0			0
$237$	0.0487597	+	0.213630i	0.0487597	+	0.213630i
$238$	0.623490	+	0.781831i	0.623490	+	0.781831i
$239$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$239$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$240$		0			0
$241$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$241$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$242$	−1.26868	+	1.17716i	−1.26868	+	1.17716i
$243$	1.18429	−	0.807437i	1.18429	−	0.807437i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.326239	−	0.302705i	0.326239	−	0.302705i
$249$		0			0
$250$		0			0
$251$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$251$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$252$	−0.255779	+	1.12064i	−0.255779	+	1.12064i
$253$	0.662592	+	2.90301i	0.662592	+	2.90301i
$254$		0			0
$255$		0			0
$256$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$257$	−0.365341	−	0.930874i	−0.365341	−	0.930874i	−0.988831	−	0.149042i	$-0.952381\pi$
$257$	−0.365341	−	0.930874i	−0.365341	−	0.930874i	0.623490	−	0.781831i	$-0.285714\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.78181	+	0.268565i	1.78181	+	0.268565i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	1.21135	−	2.09812i	1.21135	−	2.09812i
$265$		0			0
$266$		0			0
$267$	1.34017	+	1.68052i	1.34017	+	1.68052i
$268$		0			0
$269$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$269$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$270$		0			0
$271$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$271$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$272$	0.623490	+	0.781831i	0.623490	+	0.781831i
$273$	1.21135	+	2.09812i	1.21135	+	2.09812i
$274$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$275$	−0.826239	+	1.43109i	−0.826239	+	1.43109i
$276$	−1.32091	−	2.28789i	−1.32091	−	2.28789i
$277$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$277$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$278$	−0.109562	−	0.101659i	−0.109562	−	0.101659i
$279$	0.460898	−	0.221957i	0.460898	−	0.221957i
$280$		0			0
$281$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$281$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$282$		0			0
$283$	0.123490	+	1.64786i	0.123490	+	1.64786i	0.623490	+	0.781831i	$0.285714\pi$
$283$	0.123490	+	1.64786i	0.123490	+	1.64786i	−0.500000	+	0.866025i	$0.666667\pi$
$284$	1.82624	−	0.563320i	1.82624	−	0.563320i
$285$		0			0
$286$	−0.607634	−	2.66222i	−0.607634	−	2.66222i
$287$		0			0
$288$	−0.255779	+	1.12064i	−0.255779	+	1.12064i
$289$	0.955573	+	0.294755i	0.955573	+	0.294755i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$293$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$294$	0.326239	+	1.42935i	0.326239	+	1.42935i
$295$		0			0
$296$		0			0
$297$	0.265436	−	0.246289i	0.265436	−	0.246289i
$298$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$299$	−2.84537	−	0.877681i	−2.84537	−	0.877681i
$300$	0.326239	−	1.42935i	0.326239	−	1.42935i
$301$		0			0
$302$		0			0
$303$	0.197424	−	2.63444i	0.197424	−	2.63444i
$304$		0			0
$305$		0			0
$306$	0.419945	+	1.07000i	0.419945	+	1.07000i
$307$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$307$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$308$	0.123490	−	1.64786i	0.123490	−	1.64786i
$309$		0			0
$310$		0			0
$311$	0.988831	+	0.149042i	0.988831	+	0.149042i	0.623490	−	0.781831i	$-0.285714\pi$
$311$	0.988831	+	0.149042i	0.988831	+	0.149042i	0.365341	+	0.930874i	$0.380952\pi$
$312$	1.21135	+	2.09812i	1.21135	+	2.09812i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	0.0931869	−	0.116853i	0.0931869	−	0.116853i
$315$		0			0
$316$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$317$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$317$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$318$	−2.31507	−	1.57839i	−2.31507	−	1.57839i
$319$		0			0
$320$		0			0
$321$	−0.667917	−	0.837541i	−0.667917	−	0.837541i
$322$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$323$		0			0
$324$	0.414101	−	0.717244i	0.414101	−	0.717244i
$325$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$326$	−1.97766	−	0.298085i	−1.97766	−	0.298085i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$331$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$332$		0			0
$333$		0			0
$334$	−0.147791	+	1.97213i	−0.147791	+	1.97213i
$335$		0			0
$336$	0.326239	+	1.42935i	0.326239	+	1.42935i
$337$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$337$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$338$	1.65379	+	0.510127i	1.65379	+	0.510127i
$339$		0			0
$340$		0			0
$341$	−0.607634	+	0.414278i	−0.607634	+	0.414278i
$342$		0			0
$343$	0.623490	+	0.781831i	0.623490	+	0.781831i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$347$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$348$		0			0
$349$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$349$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$350$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$351$	0.0805743	+	0.353019i	0.0805743	+	0.353019i
$352$	0.123490	−	1.64786i	0.123490	−	1.64786i
$353$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.733052	−	0.680173i	$-0.761905\pi$
$353$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.222521	+	0.974928i	$0.571429\pi$
$354$		0			0
$355$		0			0
$356$	1.32091	+	0.636119i	1.32091	+	0.636119i
$357$	1.07473	+	0.997204i	1.07473	+	0.997204i
$358$		0			0
$359$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$359$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	−1.58202	+	1.98379i	−1.58202	+	1.98379i
$364$	1.36534	+	0.930874i	1.36534	+	0.930874i
$365$		0			0
$366$		0			0
$367$	0.603718	+	0.411608i	0.603718	+	0.411608i	0.826239	−	0.563320i	$-0.190476\pi$
$367$	0.603718	+	0.411608i	0.603718	+	0.411608i	−0.222521	+	0.974928i	$0.571429\pi$
$368$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$369$		0			0
$370$		0			0
$371$	−1.88980	−	0.284841i	−1.88980	−	0.284841i
$372$	0.406813	−	0.510127i	0.406813	−	0.510127i
$373$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$373$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$374$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	−0.0163752	+	0.218511i	−0.0163752	+	0.218511i
$379$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$379$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$383$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$384$	0.326239	+	1.42935i	0.326239	+	1.42935i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.44973	−	1.34515i	1.44973	−	1.34515i	0.623490	−	0.781831i	$-0.285714\pi$
$389$	1.44973	−	1.34515i	1.44973	−	1.34515i	0.826239	−	0.563320i	$-0.190476\pi$
$390$		0			0
$391$	−1.80194			−1.80194
$392$	0.623490	+	0.781831i	0.623490	+	0.781831i
$393$	2.64183			2.64183
$394$		0			0
$395$		0			0
$396$	0.693950	−	1.76815i	0.693950	−	1.76815i
$397$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$397$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$398$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$399$		0			0
$400$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$401$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$401$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$402$		0			0
$403$	−0.0549581	−	0.733365i	−0.0549581	−	0.733365i
$404$	−0.658322	−	1.67738i	−0.658322	−	1.67738i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.07473	+	0.997204i	1.07473	+	0.997204i
$409$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.900969	−	0.433884i	$-0.857143\pi$
$409$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.988831	+	0.149042i	$0.952381\pi$
$410$		0			0
$411$	−1.07473	+	1.86149i	−1.07473	+	1.86149i
$412$		0			0
$413$		0			0
$414$	−1.29141	−	1.61937i	−1.29141	−	1.61937i
$415$		0			0
$416$	1.36534	+	0.930874i	1.36534	+	0.930874i
$417$	−0.181049	−	0.123437i	−0.181049	−	0.123437i
$418$		0			0
$419$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	0.826239	−	0.563320i	$-0.190476\pi$
$419$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$420$		0			0
$421$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$421$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$422$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$423$		0			0
$424$	−1.88980	−	0.284841i	−1.88980	−	0.284841i
$425$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$426$	2.52446	−	1.21572i	2.52446	−	1.21572i
$427$		0			0
$428$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$429$	−1.46263	−	3.72672i	−1.46263	−	3.72672i
$430$		0			0
$431$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.733052	−	0.680173i	$-0.761905\pi$
$431$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.222521	+	0.974928i	$0.571429\pi$
$432$	−0.0163752	+	0.218511i	−0.0163752	+	0.218511i
$433$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$433$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$434$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.123490	−	0.0841939i	0.123490	−	0.0841939i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	0.123490	−	0.0841939i	0.123490	−	0.0841939i	0.623490	+	0.781831i	$0.285714\pi$
$440$		0			0
$441$	0.419945	+	1.07000i	0.419945	+	1.07000i
$442$	1.65248			1.65248
$443$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$443$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−0.478300	+	2.09557i	−0.478300	+	2.09557i
$448$	0.623490	+	0.781831i	0.623490	+	0.781831i
$449$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$449$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$450$	0.0858993	−	1.14625i	0.0858993	−	1.14625i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−1.72188	−	0.829215i	−1.72188	−	0.829215i
$455$		0			0
$456$		0			0
$457$	1.32091	+	1.22563i	1.32091	+	1.22563i	0.955573	+	0.294755i	$0.0952381\pi$
$457$	1.32091	+	1.22563i	1.32091	+	1.22563i	0.365341	+	0.930874i	$0.380952\pi$
$458$	−1.23305	−	0.185853i	−1.23305	−	0.185853i
$459$	0.109562	+	0.189767i	0.109562	+	0.189767i
$460$		0			0
$461$	1.19158	−	1.49419i	1.19158	−	1.49419i	0.365341	−	0.930874i	$-0.380952\pi$
$461$	1.19158	−	1.49419i	1.19158	−	1.49419i	0.826239	−	0.563320i	$-0.190476\pi$
$462$	−0.181049	−	2.41593i	−0.181049	−	2.41593i
$463$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$463$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$467$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$468$	1.18429	+	1.48506i	1.18429	+	1.48506i
$469$		0			0
$470$		0			0
$471$	0.109562	−	0.189767i	0.109562	−	0.189767i
$472$		0			0
$473$		0			0
$474$	0.160629	+	0.149042i	0.160629	+	0.149042i
$475$		0			0
$476$	0.955573	+	0.294755i	0.955573	+	0.294755i
$477$	−1.97924	−	0.953150i	−1.97924	−	0.953150i
$478$		0			0
$479$	−0.134659	−	1.79690i	−0.134659	−	1.79690i	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−0.134659	−	1.79690i	−0.134659	−	1.79690i	0.365341	−	0.930874i	$-0.380952\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−2.38020	−	1.14625i	−2.38020	−	1.14625i
$484$	−0.385113	+	1.68729i	−0.385113	+	1.68729i
$485$		0			0
$486$	0.523663	−	1.33427i	0.523663	−	1.33427i
$487$	0.733052	−	0.680173i	0.733052	−	0.680173i	−0.222521	−	0.974928i	$-0.571429\pi$
$487$	0.733052	−	0.680173i	0.733052	−	0.680173i	0.955573	+	0.294755i	$0.0952381\pi$
$488$		0			0
$489$	−2.93221			−2.93221
$490$		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
$496$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$497$	1.19158	−	1.49419i	1.19158	−	1.49419i
$498$		0			0
$499$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	−0.988831	−	0.149042i	$-0.952381\pi$
$499$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	1.00000			$0$
$500$		0			0
$501$	0.216677	+	2.89135i	0.216677	+	2.89135i
$502$		0			0
$503$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	0.0747301	−	0.997204i	$-0.476190\pi$
$503$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	−0.733052	+	0.680173i	$0.761905\pi$
$504$	0.419945	+	1.07000i	0.419945	+	1.07000i
$505$		0			0
$506$	2.18278	+	2.02532i	2.18278	+	2.02532i
$507$	2.50902	+	0.378174i	2.50902	+	0.378174i
$508$		0			0
$509$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$509$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$510$		0			0
$511$		0			0
$512$	0.623490	+	0.781831i	0.623490	+	0.781831i
$513$		0			0
$514$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$515$		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		−0.733052	−	0.680173i	$-0.761905\pi$
$523$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$524$	1.62349	−	0.781831i	1.62349	−	0.781831i
$525$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$526$		0			0
$527$	−0.162592	−	0.414278i	−0.162592	−	0.414278i
$528$	−0.181049	−	2.41593i	−0.181049	−	2.41593i
$529$	2.14715	−	0.662309i	2.14715	−	0.662309i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	2.05397	+	0.633565i	2.05397	+	0.633565i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$540$		0			0
$541$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$541$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$542$		0			0
$543$		0			0
$544$	0.955573	+	0.294755i	0.955573	+	0.294755i
$545$		0			0
$546$	2.18278	+	1.05117i	2.18278	+	1.05117i
$547$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.365341	+	0.930874i	$0.380952\pi$
$547$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.0747301	+	0.997204i	$0.476190\pi$
$548$	−0.109562	+	1.46200i	−0.109562	+	1.46200i
$549$		0			0
$550$	0.123490	+	1.64786i	0.123490	+	1.64786i
$551$		0			0
$552$	−2.38020	−	1.14625i	−2.38020	−	1.14625i
$553$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$554$		0			0
$555$		0			0
$556$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i
$557$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$557$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$558$	0.255779	−	0.443022i	0.255779	−	0.443022i
$559$		0			0
$560$		0			0
$561$	−1.51053	−	1.89415i	−1.51053	−	1.89415i
$562$	0.988831	−	0.149042i	0.988831	−	0.149042i
$563$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$563$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$564$		0			0
$565$		0			0
$566$	1.03030	+	1.29196i	1.03030	+	1.29196i
$567$	−0.0618916	−	0.825886i	−0.0618916	−	0.825886i
$568$	1.19158	−	1.49419i	1.19158	−	1.49419i
$569$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$569$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$570$		0			0
$571$	0.440071	+	0.0663300i	0.440071	+	0.0663300i	0.365341	−	0.930874i	$-0.380952\pi$
$571$	0.440071	+	0.0663300i	0.440071	+	0.0663300i	0.0747301	+	0.997204i	$0.476190\pi$
$572$	−2.00173	−	1.85734i	−2.00173	−	1.85734i
$573$		0			0
$574$		0			0
$575$	1.62349	+	0.781831i	1.62349	+	0.781831i
$576$	0.419945	+	1.07000i	0.419945	+	1.07000i
$577$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	0.955573	+	0.294755i	$0.0952381\pi$
$577$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	−0.900969	+	0.433884i	$0.857143\pi$
$578$	0.955573	−	0.294755i	0.955573	−	0.294755i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.01782	+	0.930874i	3.01782	+	0.930874i
$584$		0			0
$585$		0			0
$586$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	1.07473	+	0.997204i	1.07473	+	0.997204i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.698220	+	0.215372i	0.698220	+	0.215372i	0.623490	−	0.781831i	$-0.285714\pi$
$593$	0.698220	+	0.215372i	0.698220	+	0.215372i	0.0747301	+	0.997204i	$0.476190\pi$
$594$	0.0805743	−	0.353019i	0.0805743	−	0.353019i
$595$		0			0
$596$	0.326239	+	1.42935i	0.326239	+	1.42935i
$597$	−0.209389	+	2.79410i	−0.209389	+	2.79410i
$598$	−2.84537	+	0.877681i	−2.84537	+	0.877681i
$599$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$599$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$600$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$601$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$601$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−1.32091	−	2.28789i	−1.32091	−	2.28789i
$607$	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
$612$	0.949729	+	0.647514i	0.949729	+	0.647514i
$613$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$613$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$614$		0			0
$615$		0			0
$616$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$617$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$617$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$618$		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
$621$	−0.289444	−	0.268565i	−0.289444	−	0.268565i
$622$	0.900969	−	0.433884i	0.900969	−	0.433884i
$623$	1.44973	−	0.218511i	1.44973	−	0.218511i
$624$	2.18278	+	1.05117i	2.18278	+	1.05117i
$625$	0.365341	+	0.930874i	0.365341	+	0.930874i
$626$		0			0
$627$		0			0
$628$	0.0111692	−	0.149042i	0.0111692	−	0.149042i
$629$		0			0
$630$		0			0
$631$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$631$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$632$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$633$	−0.667917	+	1.70182i	−0.667917	+	1.70182i
$634$		0			0
$635$		0			0
$636$	−2.80194			−2.80194
$637$	1.65248			1.65248
$638$		0			0
$639$	1.81507	−	1.23749i	1.81507	−	1.23749i
$640$		0			0
$641$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$641$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$642$	−1.02366	−	0.315758i	−1.02366	−	0.315758i
$643$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.0747301	−	0.997204i	$-0.476190\pi$
$643$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.365341	−	0.930874i	$-0.380952\pi$
$644$	−1.80194			−1.80194
$645$		0			0
$646$		0			0
$647$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$647$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$648$	−0.0618916	−	0.825886i	−0.0618916	−	0.825886i
$649$		0			0
$650$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$651$	0.0487597	−	0.650653i	0.0487597	−	0.650653i
$652$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$653$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$653$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$659$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$660$		0			0
$661$	−0.367711	−	0.250701i	−0.367711	−	0.250701i	0.365341	−	0.930874i	$-0.380952\pi$
$661$	−0.367711	−	0.250701i	−0.367711	−	0.250701i	−0.733052	+	0.680173i	$0.761905\pi$
$662$		0			0
$663$	2.39564	−	0.361085i	2.39564	−	0.361085i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.988831	+	1.71271i	0.988831	+	1.71271i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	1.07473	+	0.997204i	1.07473	+	0.997204i
$673$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$673$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$674$		0			0
$675$	−0.0163752	−	0.218511i	−0.0163752	−	0.218511i
$676$	1.65379	−	0.510127i	1.65379	−	0.510127i
$677$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$677$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−2.67746	−	0.825886i	−2.67746	−	0.825886i
$682$	−0.268680	+	0.684585i	−0.268680	+	0.684585i
$683$	−1.40097	+	1.29991i	−1.40097	+	1.29991i	−0.500000	+	0.866025i	$0.666667\pi$
$683$	−1.40097	+	1.29991i	−1.40097	+	1.29991i	−0.900969	+	0.433884i	$0.857143\pi$
$684$		0			0
$685$		0			0
$686$	0.955573	+	0.294755i	0.955573	+	0.294755i
$687$	−1.82820			−1.82820
$688$		0			0
$689$	−2.31507	+	2.14807i	−2.31507	+	2.14807i
$690$		0			0
$691$	−0.425270	−	0.131178i	−0.425270	−	0.131178i	0.0747301	−	0.997204i	$-0.476190\pi$
$691$	−0.425270	−	0.131178i	−0.425270	−	0.131178i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$	−0.422669	−	1.85183i	−0.422669	−	1.85183i
$694$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.455573	+	1.16078i	0.455573	+	1.16078i
$699$		0			0
$700$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$701$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.733052	+	0.680173i	$0.761905\pi$
$701$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.988831	+	0.149042i	$0.952381\pi$
$702$	0.265436	+	0.246289i	0.265436	+	0.246289i
$703$		0			0
$704$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$705$		0			0
$706$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$707$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$708$		0			0
$709$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$709$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$710$		0			0
$711$	0.141947	+	0.0967776i	0.141947	+	0.0967776i
$712$	1.44973	−	0.218511i	1.44973	−	0.218511i
$713$	0.500000	+	0.626980i	0.500000	+	0.626980i
$714$	1.44973	+	0.218511i	1.44973	+	0.218511i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.21135	−	1.12397i	−1.21135	−	1.12397i	−0.988831	−	0.149042i	$-0.952381\pi$
$719$	−1.21135	−	1.12397i	−1.21135	−	1.12397i	−0.222521	−	0.974928i	$-0.571429\pi$
$720$		0			0
$721$		0			0
$722$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−0.189617	+	2.53026i	−0.189617	+	2.53026i
$727$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$727$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$728$	1.65248			1.65248
$729$	0.283323	−	1.24132i	0.283323	−	1.24132i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	−0.900969	−	0.433884i	$-0.857143\pi$
$733$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	0.0747301	+	0.997204i	$0.476190\pi$
$734$	0.730682			0.730682
$735$		0			0
$736$	−1.80194			−1.80194
$737$		0			0
$738$		0			0
$739$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$739$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$740$		0			0
$741$		0			0
$742$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$743$	0.326239	+	1.42935i	0.326239	+	1.42935i	0.826239	+	0.563320i	$0.190476\pi$
$743$	0.326239	+	1.42935i	0.326239	+	1.42935i	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0.0487597	−	0.650653i	0.0487597	−	0.650653i
$745$		0			0
$746$	0.123490	+	1.64786i	0.123490	+	1.64786i
$747$		0			0
$748$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$749$	−0.722521	+	0.108903i	−0.722521	+	0.108903i
$750$		0			0
$751$	−0.109562	−	0.101659i	−0.109562	−	0.101659i	0.623490	−	0.781831i	$-0.285714\pi$
$751$	−0.109562	−	0.101659i	−0.109562	−	0.101659i	−0.733052	+	0.680173i	$0.761905\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.109562	+	0.189767i	0.109562	+	0.189767i
$757$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$757$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$758$	0.988831	−	0.149042i	0.988831	−	0.149042i
$759$	3.60700	+	2.45921i	3.60700	+	2.45921i
$760$		0			0
$761$	1.95557	−	0.294755i	1.95557	−	0.294755i	0.955573	−	0.294755i	$-0.0952381\pi$
$761$	1.95557	−	0.294755i	1.95557	−	0.294755i	1.00000			$0$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.07473	+	0.997204i	1.07473	+	0.997204i
$769$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$769$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$770$		0			0
$771$	−1.32091	−	0.636119i	−1.32091	−	0.636119i
$772$		0			0
$773$	−0.147791	−	1.97213i	−0.147791	−	1.97213i	−0.222521	−	0.974928i	$-0.571429\pi$
$773$	−0.147791	−	1.97213i	−0.147791	−	1.97213i	0.0747301	−	0.997204i	$-0.476190\pi$
$774$		0			0
$775$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i
$776$		0			0
$777$		0			0
$778$	0.440071	−	1.92808i	0.440071	−	1.92808i
$779$		0			0
$780$		0			0
$781$	−2.31507	+	2.14807i	−2.31507	+	2.14807i
$782$	−1.48883	+	1.01507i	−1.48883	+	1.01507i
$783$		0			0
$784$	0.955573	+	0.294755i	0.955573	+	0.294755i
$785$		0			0
$786$	2.18278	−	1.48819i	2.18278	−	1.48819i
$787$	−0.914101	+	0.848162i	−0.914101	+	0.848162i	−0.988831	−	0.149042i	$-0.952381\pi$
$787$	−0.914101	+	0.848162i	−0.914101	+	0.848162i	0.0747301	+	0.997204i	$0.476190\pi$
$788$		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)

\(f(q)\)	\(=\)	\(q+(0.826239 - 0.563320i) q^{2} +(1.07473 - 0.997204i) q^{3} +(0.365341 - 0.930874i) q^{4} +(0.326239 - 1.42935i) q^{6} +(-0.222521 - 0.974928i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(0.0858993 - 1.14625i) q^{9} +(0.123490 + 1.64786i) q^{11} +(-0.535628 - 1.36476i) q^{12} +(-1.48883 - 0.716983i) q^{13} +(-0.733052 - 0.680173i) q^{14} +(-0.733052 - 0.680173i) q^{16} +(-0.988831 - 0.149042i) q^{17} +(-0.574730 - 0.995462i) q^{18} +(-1.21135 - 0.825886i) q^{21} +(1.03030 + 1.29196i) q^{22} +(1.78181 - 0.268565i) q^{23} +(-1.21135 - 0.825886i) q^{24} +(0.826239 + 0.563320i) q^{25} +(-1.63402 + 0.246289i) q^{26} +(-0.136622 - 0.171318i) q^{27} +(-0.988831 - 0.149042i) q^{28} +(0.222521 + 0.385418i) q^{31} +(-0.988831 - 0.149042i) q^{32} +(1.77597 + 1.64786i) q^{33} +(-0.900969 + 0.433884i) q^{34} +(-1.03563 - 0.498732i) q^{36} +(-2.31507 + 0.714104i) q^{39} -1.46610 q^{42} +(1.57906 + 0.487076i) q^{44} +(1.32091 - 1.22563i) q^{46} -1.46610 q^{48} +(-0.900969 + 0.433884i) q^{49} +1.00000 q^{50} +(-1.21135 + 0.825886i) q^{51} +(-1.21135 + 1.12397i) q^{52} +(0.698220 - 1.77904i) q^{53} +(-0.209389 - 0.0645880i) q^{54} +(-0.900969 + 0.433884i) q^{56} +(0.400969 + 0.193096i) q^{62} +(-1.13662 + 0.171318i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(2.39564 + 0.361085i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(1.64715 - 2.06546i) q^{69} +(1.19158 + 1.49419i) q^{71} +(-1.13662 + 0.171318i) q^{72} +(1.44973 - 0.218511i) q^{75} +(1.57906 - 0.487076i) q^{77} +(-1.51053 + 1.89415i) q^{78} +(-0.0747301 + 0.129436i) q^{79} +(0.818951 + 0.123437i) q^{81} +(-1.21135 + 0.825886i) q^{84} +(1.57906 - 0.487076i) q^{88} +(-0.109562 + 1.46200i) q^{89} +(-0.367711 + 1.61105i) q^{91} +(0.400969 - 1.75676i) q^{92} +(0.623490 + 0.192321i) q^{93} +(-1.21135 + 0.825886i) q^{96} +(-0.500000 + 0.866025i) q^{98} +1.89946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 8 q^{11} - q^{12} - 5 q^{13} + q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + q^{25} - q^{26}+ \cdots - 14 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more