LMFDB - Embedded newform 3332.1.cc.c.135.2

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			135.2
Root			$0.294755 - 0.955573i$ of defining polynomial
Character	$\chi$	$=$	3332.135
Dual form			3332.1.cc.c.543.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.826239 + 0.563320i) q^{2} +(0.997204 - 0.925270i) q^{3} +(0.365341 - 0.930874i) q^{4} +(-0.302705 + 1.32624i) q^{6} +(-0.974928 + 0.222521i) q^{7} +(0.222521 + 0.974928i) q^{8} +(0.0635609 - 0.848162i) q^{9} +(0.0841939 + 1.12349i) q^{11} +(-0.496990 - 1.26631i) q^{12} +(1.48883 + 0.716983i) q^{13} +(0.680173 - 0.733052i) q^{14} +(-0.733052 - 0.680173i) q^{16} +(-0.988831 - 0.149042i) q^{17} +(0.425270 + 0.736589i) q^{18} +(-0.766310 + 1.12397i) q^{21} +(-0.702449 - 0.880843i) q^{22} +(-0.858075 + 0.129334i) q^{23} +(1.12397 + 0.766310i) q^{24} +(0.826239 + 0.563320i) q^{25} +(-1.63402 + 0.246289i) q^{26} +(0.126766 + 0.158960i) q^{27} +(-0.149042 + 0.988831i) q^{28} +(0.974928 + 1.68862i) q^{31} +(0.988831 + 0.149042i) q^{32} +(1.12349 + 1.04245i) q^{33} +(0.900969 - 0.433884i) q^{34} +(-0.766310 - 0.369035i) q^{36} +(2.14807 - 0.662592i) q^{39} -1.36035i q^{42} +(1.07659 + 0.332083i) q^{44} +(0.636119 - 0.590232i) q^{46} -1.36035 q^{48} +(0.900969 - 0.433884i) q^{49} -1.00000 q^{50} +(-1.12397 + 0.766310i) q^{51} +(1.21135 - 1.12397i) q^{52} +(0.698220 - 1.77904i) q^{53} +(-0.194285 - 0.0599289i) q^{54} +(-0.433884 - 0.900969i) q^{56} +(-1.75676 - 0.846011i) q^{62} +(0.126766 + 0.841040i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(-1.51550 - 0.228425i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(-0.736007 + 0.922924i) q^{69} +(0.367554 + 0.460898i) q^{71} +(0.841040 - 0.126766i) q^{72} +(1.34515 - 0.202749i) q^{75} +(-0.332083 - 1.07659i) q^{77} +(-1.40157 + 1.75751i) q^{78} +(-0.997204 + 1.72721i) q^{79} +(1.11453 + 0.167989i) q^{81} +(0.766310 + 1.12397i) q^{84} +(-1.07659 + 0.332083i) q^{88} +(0.109562 - 1.46200i) q^{89} +(-1.61105 - 0.367711i) q^{91} +(-0.193096 + 0.846011i) q^{92} +(2.53464 + 0.781831i) q^{93} +(1.12397 - 0.766310i) q^{96} +(-0.500000 + 0.866025i) q^{98} +0.958252 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$.

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

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
$3$	0.997204	−	0.925270i	0.997204	−	0.925270i		−	1.00000i	$-0.5\pi$
$3$	0.997204	−	0.925270i	0.997204	−	0.925270i	0.997204	+	0.0747301i	$0.0238095\pi$
$4$	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	$0.0952381\pi$
$6$	−0.302705	+	1.32624i	−0.302705	+	1.32624i
$7$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$7$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$8$	0.222521	+	0.974928i	0.222521	+	0.974928i
$9$	0.0635609	−	0.848162i	0.0635609	−	0.848162i
$10$		0			0
$11$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	0.866025	+	0.500000i	$0.166667\pi$
$11$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	−0.781831	+	0.623490i	$0.785714\pi$
$12$	−0.496990	−	1.26631i	−0.496990	−	1.26631i
$13$	1.48883	+	0.716983i	1.48883	+	0.716983i	0.988831	−	0.149042i	$-0.0476190\pi$
$13$	1.48883	+	0.716983i	1.48883	+	0.716983i	0.500000	+	0.866025i	$0.333333\pi$
$14$	0.680173	−	0.733052i	0.680173	−	0.733052i
$15$		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
$18$	0.425270	+	0.736589i	0.425270	+	0.736589i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	−0.766310	+	1.12397i	−0.766310	+	1.12397i
$22$	−0.702449	−	0.880843i	−0.702449	−	0.880843i
$23$	−0.858075	+	0.129334i	−0.858075	+	0.129334i	−0.563320	−	0.826239i	$-0.690476\pi$
$23$	−0.858075	+	0.129334i	−0.858075	+	0.129334i	−0.294755	+	0.955573i	$0.595238\pi$
$24$	1.12397	+	0.766310i	1.12397	+	0.766310i
$25$	0.826239	+	0.563320i	0.826239	+	0.563320i
$26$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$27$	0.126766	+	0.158960i	0.126766	+	0.158960i
$28$	−0.149042	+	0.988831i	−0.149042	+	0.988831i
$29$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$29$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$30$		0			0
$31$	0.974928	+	1.68862i	0.974928	+	1.68862i	0.680173	+	0.733052i	$0.261905\pi$
$31$	0.974928	+	1.68862i	0.974928	+	1.68862i	0.294755	+	0.955573i	$0.404762\pi$
$32$	0.988831	+	0.149042i	0.988831	+	0.149042i
$33$	1.12349	+	1.04245i	1.12349	+	1.04245i
$34$	0.900969	−	0.433884i	0.900969	−	0.433884i
$35$		0			0
$36$	−0.766310	−	0.369035i	−0.766310	−	0.369035i
$37$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$37$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$38$		0			0
$39$	2.14807	−	0.662592i	2.14807	−	0.662592i
$40$		0			0
$41$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$41$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$42$		−	1.36035i		−	1.36035i
$43$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$43$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$44$	1.07659	+	0.332083i	1.07659	+	0.332083i
$45$		0			0
$46$	0.636119	−	0.590232i	0.636119	−	0.590232i
$47$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$47$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$48$	−1.36035			−1.36035
$49$	0.900969	−	0.433884i	0.900969	−	0.433884i
$50$	−1.00000			−1.00000
$51$	−1.12397	+	0.766310i	−1.12397	+	0.766310i
$52$	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	$-0.285714\pi$
$54$	−0.194285	−	0.0599289i	−0.194285	−	0.0599289i
$55$		0			0
$56$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$57$		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	$0.380952\pi$
$62$	−1.75676	−	0.846011i	−1.75676	−	0.846011i
$63$	0.126766	+	0.841040i	0.126766	+	0.841040i
$64$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$65$		0			0
$66$	−1.51550	−	0.228425i	−1.51550	−	0.228425i
$67$		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
$69$	−0.736007	+	0.922924i	−0.736007	+	0.922924i
$70$		0			0
$71$	0.367554	+	0.460898i	0.367554	+	0.460898i	0.930874	−	0.365341i	$-0.119048\pi$
$71$	0.367554	+	0.460898i	0.367554	+	0.460898i	−0.563320	+	0.826239i	$0.690476\pi$
$72$	0.841040	−	0.126766i	0.841040	−	0.126766i
$73$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$73$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$74$		0			0
$75$	1.34515	−	0.202749i	1.34515	−	0.202749i
$76$		0			0
$77$	−0.332083	−	1.07659i	−0.332083	−	1.07659i
$78$	−1.40157	+	1.75751i	−1.40157	+	1.75751i
$79$	−0.997204	+	1.72721i	−0.997204	+	1.72721i	−0.433884	+	0.900969i	$0.642857\pi$
$79$	−0.997204	+	1.72721i	−0.997204	+	1.72721i	−0.563320	+	0.826239i	$0.690476\pi$
$80$		0			0
$81$	1.11453	+	0.167989i	1.11453	+	0.167989i
$82$		0			0
$83$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$83$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$84$	0.766310	+	1.12397i	0.766310	+	1.12397i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.07659	+	0.332083i	−1.07659	+	0.332083i
$89$	0.109562	−	1.46200i	0.109562	−	1.46200i	−0.623490	−	0.781831i	$-0.714286\pi$
$89$	0.109562	−	1.46200i	0.109562	−	1.46200i	0.733052	−	0.680173i	$-0.238095\pi$
$90$		0			0
$91$	−1.61105	−	0.367711i	−1.61105	−	0.367711i
$92$	−0.193096	+	0.846011i	−0.193096	+	0.846011i
$93$	2.53464	+	0.781831i	2.53464	+	0.781831i
$94$		0			0
$95$		0			0
$96$	1.12397	−	0.766310i	1.12397	−	0.766310i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$99$	0.958252			0.958252
$100$	0.826239	−	0.563320i	0.826239	−	0.563320i
$101$	−1.32091	+	1.22563i	−1.32091	+	1.22563i	−0.365341	+	0.930874i	$0.619048\pi$
$101$	−1.32091	+	1.22563i	−1.32091	+	1.22563i	−0.955573	+	0.294755i	$0.904762\pi$
$102$	0.496990	−	1.26631i	0.496990	−	1.26631i
$103$		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
$107$	0.139129	−	1.85654i	0.139129	−	1.85654i	−0.294755	−	0.955573i	$-0.595238\pi$
$107$	0.139129	−	1.85654i	0.139129	−	1.85654i	0.433884	−	0.900969i	$-0.357143\pi$
$108$	0.194285	−	0.0599289i	0.194285	−	0.0599289i
$109$		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
$112$	0.866025	+	0.500000i	0.866025	+	0.500000i
$113$		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
$117$	0.702749	−	1.21720i	0.702749	−	1.21720i
$118$		0			0
$119$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$120$		0			0
$121$	−0.266310	+	0.0401398i	−0.266310	+	0.0401398i
$122$		0			0
$123$		0			0
$124$	1.92808	−	0.290611i	1.92808	−	0.290611i
$125$		0			0
$126$	−0.578514	−	0.623490i	−0.578514	−	0.623490i
$127$		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
$131$	−0.636119	−	0.590232i	−0.636119	−	0.590232i	0.294755	−	0.955573i	$-0.404762\pi$
$131$	−0.636119	−	0.590232i	−0.636119	−	0.590232i	−0.930874	+	0.365341i	$0.880952\pi$
$132$	1.38084	−	0.664979i	1.38084	−	0.664979i
$133$		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	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.40097	−	0.432142i	1.40097	−	0.432142i	0.900969	+	0.433884i	$0.142857\pi$
$138$	0.0882162	−	1.17716i	0.0882162	−	1.17716i
$139$	−0.443797	−	1.94440i	−0.443797	−	1.94440i	−0.294755	−	0.955573i	$-0.595238\pi$
$139$	−0.443797	−	1.94440i	−0.443797	−	1.94440i	−0.149042	−	0.988831i	$-0.547619\pi$
$140$		0			0
$141$		0			0
$142$	−0.563320	−	0.173761i	−0.563320	−	0.173761i
$143$	−0.680173	+	1.73305i	−0.680173	+	1.73305i
$144$	−0.623490	+	0.578514i	−0.623490	+	0.578514i
$145$		0			0
$146$		0			0
$147$	0.496990	−	1.26631i	0.496990	−	1.26631i
$148$		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	$0.571429\pi$
$150$	−0.997204	+	0.925270i	−0.997204	+	0.925270i
$151$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$151$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$152$		0			0
$153$	−0.189263	+	0.829215i	−0.189263	+	0.829215i
$154$	0.880843	+	0.702449i	0.880843	+	0.702449i
$155$		0			0
$156$	0.167989	−	2.24165i	0.167989	−	2.24165i
$157$	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	$0.380952\pi$
$158$	−0.149042	−	1.98883i	−0.149042	−	1.98883i
$159$	−0.949820	−	2.42010i	−0.949820	−	2.42010i
$160$		0			0
$161$	0.807782	−	0.317031i	0.807782	−	0.317031i
$162$	−1.01550	+	0.489040i	−1.01550	+	0.489040i
$163$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$163$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.185853	+	0.233052i	−0.185853	+	0.233052i	−0.866025	−	0.500000i	$-0.833333\pi$
$167$	−0.185853	+	0.233052i	−0.185853	+	0.233052i	0.680173	+	0.733052i	$0.261905\pi$
$168$	−1.26631	−	0.496990i	−1.26631	−	0.496990i
$169$	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
$175$	−0.930874	−	0.365341i	−0.930874	−	0.365341i
$176$	0.702449	−	0.880843i	0.702449	−	0.880843i
$177$		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	$0.857143\pi$
$182$	1.53825	−	0.603718i	1.53825	−	0.603718i
$183$		0			0
$184$	−0.317031	−	0.807782i	−0.317031	−	0.807782i
$185$		0			0
$186$	−2.53464	+	0.781831i	−2.53464	+	0.781831i
$187$	0.0841939	−	1.12349i	0.0841939	−	1.12349i
$188$		0			0
$189$	−0.158960	−	0.126766i	−0.158960	−	0.126766i
$190$		0			0
$191$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$191$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$192$	−0.496990	+	1.26631i	−0.496990	+	1.26631i
$193$		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			$\pi$
$198$	−0.791745	+	0.539803i	−0.791745	+	0.539803i
$199$	0.432142	−	0.400969i	0.432142	−	0.400969i	−0.433884	−	0.900969i	$-0.642857\pi$
$199$	0.432142	−	0.400969i	0.432142	−	0.400969i	0.866025	+	0.500000i	$0.166667\pi$
$200$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$201$		0			0
$202$	0.400969	−	1.75676i	0.400969	−	1.75676i
$203$		0			0
$204$	0.302705	+	1.32624i	0.302705	+	1.32624i
$205$		0			0
$206$		0			0
$207$	0.0551561	+	0.736007i	0.0551561	+	0.736007i
$208$	−0.603718	−	1.53825i	−0.603718	−	1.53825i
$209$		0			0
$210$		0			0
$211$	1.40881	−	0.678448i	1.40881	−	0.678448i	0.433884	−	0.900969i	$-0.357143\pi$
$211$	1.40881	−	0.678448i	1.40881	−	0.678448i	0.974928	+	0.222521i	$0.0714286\pi$
$212$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$213$	0.792981	+	0.119523i	0.792981	+	0.119523i
$214$	0.930874	+	1.61232i	0.930874	+	1.61232i
$215$		0			0
$216$	−0.126766	+	0.158960i	−0.126766	+	0.158960i
$217$	−1.32624	−	1.42935i	−1.32624	−	1.42935i
$218$		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	$0.285714\pi$
$224$	−0.997204	+	0.0747301i	−0.997204	+	0.0747301i
$225$	0.530303	−	0.664979i	0.530303	−	0.664979i
$226$		0			0
$227$	0.294755	+	0.510531i	0.294755	+	0.510531i	0.974928	−	0.222521i	$-0.0714286\pi$
$227$	0.294755	+	0.510531i	0.294755	+	0.510531i	−0.680173	+	0.733052i	$0.738095\pi$
$228$		0			0
$229$	0.914101	+	0.848162i	0.914101	+	0.848162i	0.988831	−	0.149042i	$-0.0476190\pi$
$229$	0.914101	+	0.848162i	0.914101	+	0.848162i	−0.0747301	+	0.997204i	$0.523810\pi$
$230$		0			0
$231$	−1.32729	−	0.766310i	−1.32729	−	0.766310i
$232$		0			0
$233$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$233$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$234$	0.105033	+	1.40157i	0.105033	+	1.40157i
$235$		0			0
$236$		0			0
$237$	0.603718	+	2.64506i	0.603718	+	2.64506i
$238$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$239$		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	$0.619048\pi$
$242$	0.197424	−	0.183183i	0.197424	−	0.183183i
$243$	1.09886	−	0.749192i	1.09886	−	0.749192i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−1.42935	+	1.32624i	−1.42935	+	1.32624i
$249$		0			0
$250$		0			0
$251$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$251$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$252$	0.829215	+	0.189263i	0.829215	+	0.189263i
$253$	−0.217550	−	0.953150i	−0.217550	−	0.953150i
$254$		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	$0.0476190\pi$
$257$	0.365341	+	0.930874i	0.365341	+	0.930874i	−0.623490	+	0.781831i	$0.714286\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.858075	+	0.129334i	0.858075	+	0.129334i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	−0.766310	+	1.32729i	−0.766310	+	1.32729i
$265$		0			0
$266$		0			0
$267$	−1.24349	−	1.55929i	−1.24349	−	1.55929i
$268$		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
$273$	−1.94677	+	1.12397i	−1.94677	+	1.12397i
$274$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$275$	−0.563320	+	0.975699i	−0.563320	+	0.975699i
$276$	0.590232	+	1.02231i	0.590232	+	1.02231i
$277$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$277$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$278$	1.46200	+	1.35654i	1.46200	+	1.35654i
$279$	1.49419	−	0.719566i	1.49419	−	0.719566i
$280$		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
$283$	−0.0841939	−	1.12349i	−0.0841939	−	1.12349i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	−0.0841939	−	1.12349i	−0.0841939	−	1.12349i	0.781831	−	0.623490i	$-0.214286\pi$
$284$	0.563320	−	0.173761i	0.563320	−	0.173761i
$285$		0			0
$286$	−0.414278	−	1.81507i	−0.414278	−	1.81507i
$287$		0			0
$288$	0.189263	−	0.829215i	0.189263	−	0.829215i
$289$	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	$0.476190\pi$
$294$	0.302705	+	1.32624i	0.302705	+	1.32624i
$295$		0			0
$296$		0			0
$297$	−0.167917	+	0.155804i	−0.167917	+	0.155804i
$298$	0.535628	−	1.36476i	0.535628	−	1.36476i
$299$	−1.37026	−	0.422669i	−1.37026	−	0.422669i
$300$	0.302705	−	1.32624i	0.302705	−	1.32624i
$301$		0			0
$302$		0			0
$303$	−0.183183	+	2.44440i	−0.183183	+	2.44440i
$304$		0			0
$305$		0			0
$306$	−0.310737	−	0.791745i	−0.310737	−	0.791745i
$307$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$307$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$308$	−1.12349	−	0.0841939i	−1.12349	−	0.0841939i
$309$		0			0
$310$		0			0
$311$	−1.71271	−	0.258149i	−1.71271	−	0.258149i	−0.781831	−	0.623490i	$-0.785714\pi$
$311$	−1.71271	−	0.258149i	−1.71271	−	0.258149i	−0.930874	+	0.365341i	$0.880952\pi$
$312$	1.12397	+	1.94677i	1.12397	+	1.94677i
$313$		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
$316$	1.24349	+	1.55929i	1.24349	+	1.55929i
$317$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$317$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$318$	2.14807	+	1.46453i	2.14807	+	1.46453i
$319$		0			0
$320$		0			0
$321$	−1.57906	−	1.98008i	−1.57906	−	1.98008i
$322$	−0.488831	+	0.716983i	−0.488831	+	0.716983i
$323$		0			0
$324$	0.563561	−	0.976116i	0.563561	−	0.976116i
$325$	0.826239	+	1.43109i	0.826239	+	1.43109i
$326$		0			0
$327$		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
$334$	0.0222759	−	0.297251i	0.0222759	−	0.297251i
$335$		0			0
$336$	1.32624	−	0.302705i	1.32624	−	0.302705i
$337$		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
$341$	−1.81507	+	1.23749i	−1.81507	+	1.23749i
$342$		0			0
$343$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.571270	+	1.45557i	−0.571270	+	1.45557i	0.294755	+	0.955573i	$0.404762\pi$
$347$	−0.571270	+	1.45557i	−0.571270	+	1.45557i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		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	$0.857143\pi$
$350$	0.974928	−	0.222521i	0.974928	−	0.222521i
$351$	0.0747620	+	0.327554i	0.0747620	+	0.327554i
$352$	−0.0841939	+	1.12349i	−0.0841939	+	1.12349i
$353$	−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
$357$	0.925270	−	0.997204i	0.925270	−	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
$363$	−0.228425	+	0.286436i	−0.228425	+	0.286436i
$364$	−0.930874	+	1.36534i	−0.930874	+	1.36534i
$365$		0			0
$366$		0			0
$367$	−1.53825	−	1.04876i	−1.53825	−	1.04876i	−0.974928	−	0.222521i	$-0.928571\pi$
$367$	−1.53825	−	1.04876i	−1.53825	−	1.04876i	−0.563320	−	0.826239i	$-0.690476\pi$
$368$	0.716983	+	0.488831i	0.716983	+	0.488831i
$369$		0			0
$370$		0			0
$371$	−0.284841	+	1.88980i	−0.284841	+	1.88980i
$372$	1.65379	−	2.07379i	1.65379	−	2.07379i
$373$	0.826239	−	1.43109i	0.826239	−	1.43109i	−0.0747301	−	0.997204i	$-0.523810\pi$
$373$	0.826239	−	1.43109i	0.826239	−	1.43109i	0.900969	−	0.433884i	$-0.142857\pi$
$374$	0.563320	+	0.975699i	0.563320	+	0.975699i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	0.202749	+	0.0151939i	0.202749	+	0.0151939i
$379$	−1.56052	−	0.751509i	−1.56052	−	0.751509i	−0.563320	−	0.826239i	$-0.690476\pi$
$379$	−1.56052	−	0.751509i	−1.56052	−	0.751509i	−0.997204	+	0.0747301i	$0.976190\pi$
$380$		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	$0.523810\pi$
$384$	−0.302705	−	1.32624i	−0.302705	−	1.32624i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.44973	+	1.34515i	−1.44973	+	1.34515i	−0.623490	+	0.781831i	$0.714286\pi$
$389$	−1.44973	+	1.34515i	−1.44973	+	1.34515i	−0.826239	+	0.563320i	$0.809524\pi$
$390$		0			0
$391$	0.867767			0.867767
$392$	0.623490	+	0.781831i	0.623490	+	0.781831i
$393$	−1.18046			−1.18046
$394$		0			0
$395$		0			0
$396$	0.350089	−	0.892012i	0.350089	−	0.892012i
$397$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$397$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$398$	−0.131178	+	0.574730i	−0.131178	+	0.574730i
$399$		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
$403$	0.240787	+	3.21308i	0.240787	+	3.21308i
$404$	0.658322	+	1.67738i	0.658322	+	1.67738i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.997204	−	0.925270i	−0.997204	−	0.925270i
$409$	−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
$411$	0.997204	−	1.72721i	0.997204	−	1.72721i
$412$		0			0
$413$		0			0
$414$	−0.460180	−	0.577047i	−0.460180	−	0.577047i
$415$		0			0
$416$	1.36534	+	0.930874i	1.36534	+	0.930874i
$417$	−2.24165	−	1.52833i	−2.24165	−	1.52833i
$418$		0			0
$419$	1.24349	+	1.55929i	1.24349	+	1.55929i	0.680173	+	0.733052i	$0.261905\pi$
$419$	1.24349	+	1.55929i	1.24349	+	1.55929i	0.563320	+	0.826239i	$0.309524\pi$
$420$		0			0
$421$	0.277479	−	0.347948i	0.277479	−	0.347948i	−0.623490	−	0.781831i	$-0.714286\pi$
$421$	0.277479	−	0.347948i	0.277479	−	0.347948i	0.900969	+	0.433884i	$0.142857\pi$
$422$	−0.781831	+	1.35417i	−0.781831	+	1.35417i
$423$		0			0
$424$	1.88980	+	0.284841i	1.88980	+	0.284841i
$425$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$426$	−0.722521	+	0.347948i	−0.722521	+	0.347948i
$427$		0			0
$428$	−1.67738	−	0.807782i	−1.67738	−	0.807782i
$429$	0.925270	+	2.35755i	0.925270	+	2.35755i
$430$		0			0
$431$	1.65510	−	0.510531i	1.65510	−	0.510531i	0.680173	−	0.733052i	$-0.261905\pi$
$431$	1.65510	−	0.510531i	1.65510	−	0.510531i	0.974928	+	0.222521i	$0.0714286\pi$
$432$	0.0151939	−	0.202749i	0.0151939	−	0.202749i
$433$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	−0.623490	−	0.781831i	$-0.714286\pi$
$433$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$434$	1.90097	+	0.433884i	1.90097	+	0.433884i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.64786	+	1.12349i	−1.64786	+	1.12349i	−0.781831	+	0.623490i	$0.785714\pi$
$439$	−1.64786	+	1.12349i	−1.64786	+	1.12349i	−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	−0.310737	−	0.791745i	−0.310737	−	0.791745i
$442$	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
$447$	−0.443797	+	1.94440i	−0.443797	+	1.94440i
$448$	0.781831	−	0.623490i	0.781831	−	0.623490i
$449$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$449$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$450$	−0.0635609	+	0.848162i	−0.0635609	+	0.848162i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.531130	−	0.255779i	−0.531130	−	0.255779i
$455$		0			0
$456$		0			0
$457$	−1.32091	−	1.22563i	−1.32091	−	1.22563i	−0.955573	−	0.294755i	$-0.904762\pi$
$457$	−1.32091	−	1.22563i	−1.32091	−	1.22563i	−0.365341	−	0.930874i	$-0.619048\pi$
$458$	−1.23305	−	0.185853i	−1.23305	−	0.185853i
$459$	−0.101659	−	0.176078i	−0.101659	−	0.176078i
$460$		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	$-0.190476\pi$
$462$	1.52833	−	0.114533i	1.52833	−	0.114533i
$463$		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	$0.952381\pi$
$468$	−0.876314	−	1.09886i	−0.876314	−	1.09886i
$469$		0			0
$470$		0			0
$471$	0.101659	−	0.176078i	0.101659	−	0.176078i
$472$		0			0
$473$		0			0
$474$	−1.98883	−	1.84537i	−1.98883	−	1.84537i
$475$		0			0
$476$	0.294755	−	0.955573i	0.294755	−	0.955573i
$477$	−1.46453	−	0.705280i	−1.46453	−	0.705280i
$478$		0			0
$479$	−0.0648483	−	0.865341i	−0.0648483	−	0.865341i	−0.930874	−	0.365341i	$-0.880952\pi$
$479$	−0.0648483	−	0.865341i	−0.0648483	−	0.865341i	0.866025	−	0.500000i	$-0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	0.512184	−	1.06356i	0.512184	−	1.06356i
$484$	−0.0599289	+	0.262566i	−0.0599289	+	0.262566i
$485$		0			0
$486$	−0.485888	+	1.23802i	−0.485888	+	1.23802i
$487$	1.26968	−	1.17809i	1.26968	−	1.17809i	0.294755	−	0.955573i	$-0.404762\pi$
$487$	1.26968	−	1.17809i	1.26968	−	1.17809i	0.974928	−	0.222521i	$-0.0714286\pi$
$488$		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
$496$	0.433884	−	1.90097i	0.433884	−	1.90097i
$497$	−0.460898	−	0.367554i	−0.460898	−	0.367554i
$498$		0			0
$499$	−0.149042	+	1.98883i	−0.149042	+	1.98883i			1.00000i	$0.5\pi$
$499$	−0.149042	+	1.98883i	−0.149042	+	1.98883i	−0.149042	+	0.988831i	$0.547619\pi$
$500$		0			0
$501$	0.0303029	+	0.404364i	0.0303029	+	0.404364i
$502$		0			0
$503$	−1.67738	−	0.807782i	−1.67738	−	0.807782i	−0.997204	−	0.0747301i	$-0.976190\pi$
$503$	−1.67738	−	0.807782i	−1.67738	−	0.807782i	−0.680173	−	0.733052i	$-0.738095\pi$
$504$	−0.791745	+	0.310737i	−0.791745	+	0.310737i
$505$		0			0
$506$	0.716677	+	0.664979i	0.716677	+	0.664979i
$507$	2.32803	+	0.350894i	2.32803	+	0.350894i
$508$		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	$0.238095\pi$
$524$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$525$	−1.26631	+	0.496990i	−1.26631	+	0.496990i
$526$		0			0
$527$	−0.712362	−	1.81507i	−0.712362	−	1.81507i
$528$	−0.114533	−	1.52833i	−0.114533	−	1.52833i
$529$	−0.236007	+	0.0727985i	−0.236007	+	0.0727985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	1.90580	+	0.587862i	1.90580	+	0.587862i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.563320	+	0.975699i	0.563320	+	0.975699i
$540$		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
$546$	0.975345	−	2.02532i	0.975345	−	2.02532i
$547$	0.0663300	+	0.290611i	0.0663300	+	0.290611i	0.997204	−	0.0747301i	$-0.0238095\pi$
$547$	0.0663300	+	0.290611i	0.0663300	+	0.290611i	−0.930874	+	0.365341i	$0.880952\pi$
$548$	0.109562	−	1.46200i	0.109562	−	1.46200i
$549$		0			0
$550$	−0.0841939	−	1.12349i	−0.0841939	−	1.12349i
$551$		0			0
$552$	−1.06356	−	0.512184i	−1.06356	−	0.512184i
$553$	0.587862	−	1.90580i	0.587862	−	1.90580i
$554$		0			0
$555$		0			0
$556$	−1.97213	−	0.297251i	−1.97213	−	0.297251i
$557$	0.955573	+	1.65510i	0.955573	+	1.65510i	0.733052	+	0.680173i	$0.238095\pi$
$557$	0.955573	+	1.65510i	0.955573	+	1.65510i	0.222521	+	0.974928i	$0.428571\pi$
$558$	−0.829215	+	1.43624i	−0.829215	+	1.43624i
$559$		0			0
$560$		0			0
$561$	−0.955573	−	1.19825i	−0.955573	−	1.19825i
$562$	−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
$566$	0.702449	+	0.880843i	0.702449	+	0.880843i
$567$	−1.12397	+	0.0842299i	−1.12397	+	0.0842299i
$568$	−0.367554	+	0.460898i	−0.367554	+	0.460898i
$569$	−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
$571$	1.92808	+	0.290611i	1.92808	+	0.290611i	0.997204	−	0.0747301i	$-0.0238095\pi$
$571$	1.92808	+	0.290611i	1.92808	+	0.290611i	0.930874	+	0.365341i	$0.119048\pi$
$572$	1.36476	+	1.26631i	1.36476	+	1.26631i
$573$		0			0
$574$		0			0
$575$	−0.781831	−	0.376510i	−0.781831	−	0.376510i
$576$	0.310737	+	0.791745i	0.310737	+	0.791745i
$577$	−0.0546039	−	0.728639i	−0.0546039	−	0.728639i	−0.955573	−	0.294755i	$-0.904762\pi$
$577$	−0.0546039	−	0.728639i	−0.0546039	−	0.728639i	0.900969	−	0.433884i	$-0.142857\pi$
$578$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.05751	+	0.634659i	2.05751	+	0.634659i
$584$		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			$\pi$
$588$	−0.997204	−	0.925270i	−0.997204	−	0.925270i
$589$		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	$0.476190\pi$
$594$	0.0509719	−	0.223322i	0.0509719	−	0.223322i
$595$		0			0
$596$	0.326239	+	1.42935i	0.326239	+	1.42935i
$597$	0.0599289	−	0.799695i	0.0599289	−	0.799695i
$598$	1.37026	−	0.422669i	1.37026	−	0.422669i
$599$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$599$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$600$	0.496990	+	1.26631i	0.496990	+	1.26631i
$601$		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
$606$	−1.22563	−	2.12285i	−1.22563	−	2.12285i
$607$	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
$612$	0.702749	+	0.479126i	0.702749	+	0.479126i
$613$	−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
$616$	0.975699	−	0.563320i	0.975699	−	0.563320i
$617$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$617$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$618$		0			0
$619$	−0.866025	−	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
$621$	−0.129334	−	0.120004i	−0.129334	−	0.120004i
$622$	1.56052	−	0.751509i	1.56052	−	0.751509i
$623$	0.218511	+	1.44973i	0.218511	+	1.44973i
$624$	−2.02532	−	0.975345i	−2.02532	−	0.975345i
$625$	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	$0.571429\pi$
$632$	−1.90580	−	0.587862i	−1.90580	−	0.587862i
$633$	0.777125	−	1.98008i	0.777125	−	1.98008i
$634$		0			0
$635$		0			0
$636$	−2.59982			−2.59982
$637$	1.65248			1.65248
$638$		0			0
$639$	0.414278	−	0.282450i	0.414278	−	0.282450i
$640$		0			0
$641$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$641$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$642$	2.42010	+	0.746503i	2.42010	+	0.746503i
$643$	0.0663300	−	0.290611i	0.0663300	−	0.290611i	−0.930874	−	0.365341i	$-0.880952\pi$
$643$	0.0663300	−	0.290611i	0.0663300	−	0.290611i	0.997204	+	0.0747301i	$0.0238095\pi$
$644$		−	0.867767i		−	0.867767i
$645$		0			0
$646$		0			0
$647$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$647$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$648$	0.0842299	+	1.12397i	0.0842299	+	1.12397i
$649$		0			0
$650$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$651$	−2.64506	−	0.198220i	−2.64506	−	0.198220i
$652$		0			0
$653$		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
$663$	−2.22283	+	0.335038i	−2.22283	+	0.335038i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.149042	+	0.258149i	0.149042	+	0.258149i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.925270	+	0.997204i	−0.925270	+	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
$675$	0.0151939	+	0.202749i	0.0151939	+	0.202749i
$676$	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
$681$	0.766310	+	0.236375i	0.766310	+	0.236375i
$682$	0.802576	−	2.04493i	0.802576	−	2.04493i
$683$	0.432142	−	0.400969i	0.432142	−	0.400969i	−0.433884	−	0.900969i	$-0.642857\pi$
$683$	0.432142	−	0.400969i	0.432142	−	0.400969i	0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$	0.294755	−	0.955573i	0.294755	−	0.955573i
$687$	1.69632			1.69632
$688$		0			0
$689$	2.31507	−	2.14807i	2.31507	−	2.14807i
$690$		0			0
$691$	1.86323	+	0.574730i	1.86323	+	0.574730i	0.997204	+	0.0747301i	$0.0238095\pi$
$691$	1.86323	+	0.574730i	1.86323	+	0.574730i	0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$	−0.934227	+	0.213231i	−0.934227	+	0.213231i
$694$	−0.347948	−	1.52446i	−0.347948	−	1.52446i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.455573	−	1.16078i	−0.455573	−	1.16078i
$699$		0			0
$700$	−0.680173	+	0.733052i	−0.680173	+	0.733052i
$701$	1.72188	−	0.829215i	1.72188	−	0.829215i	0.733052	−	0.680173i	$-0.238095\pi$
$701$	1.72188	−	0.829215i	1.72188	−	0.829215i	0.988831	−	0.149042i	$-0.0476190\pi$
$702$	−0.246289	−	0.228523i	−0.246289	−	0.228523i
$703$		0			0
$704$	−0.563320	−	0.975699i	−0.563320	−	0.975699i
$705$		0			0
$706$	0.623490	−	0.781831i	0.623490	−	0.781831i
$707$	1.01507	−	1.48883i	1.01507	−	1.48883i
$708$		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
$711$	1.40157	+	0.955573i	1.40157	+	0.955573i
$712$	1.44973	−	0.218511i	1.44973	−	0.218511i
$713$	−1.05496	−	1.32288i	−1.05496	−	1.32288i
$714$	−0.202749	+	1.34515i	−0.202749	+	1.34515i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.825886	−	0.766310i	−0.825886	−	0.766310i	0.149042	−	0.988831i	$-0.452381\pi$
$719$	−0.825886	−	0.766310i	−0.825886	−	0.766310i	−0.974928	+	0.222521i	$0.928571\pi$
$720$		0			0
$721$		0			0
$722$	0.900969	+	0.433884i	0.900969	+	0.433884i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	0.0273785	−	0.365341i	0.0273785	−	0.365341i
$727$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$727$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$728$		−	1.65248i		−	1.65248i
$729$	0.151777	−	0.664979i	0.151777	−	0.664979i
$730$		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	$0.476190\pi$
$734$	1.86175			1.86175
$735$		0			0
$736$	−0.867767			−0.867767
$737$		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
$742$	−0.829215	−	1.72188i	−0.829215	−	1.72188i
$743$	−0.302705	−	1.32624i	−0.302705	−	1.32624i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	−0.302705	−	1.32624i	−0.302705	−	1.32624i	0.563320	−	0.826239i	$-0.309524\pi$
$744$	−0.198220	+	2.64506i	−0.198220	+	2.64506i
$745$		0			0
$746$	0.123490	+	1.64786i	0.123490	+	1.64786i
$747$		0			0
$748$	−1.01507	−	0.488831i	−1.01507	−	0.488831i
$749$	0.277479	+	1.84095i	0.277479	+	1.84095i
$750$		0			0
$751$	1.46200	+	1.35654i	1.46200	+	1.35654i	0.781831	+	0.623490i	$0.214286\pi$
$751$	1.46200	+	1.35654i	1.46200	+	1.35654i	0.680173	+	0.733052i	$0.261905\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.176078	+	0.101659i	−0.176078	+	0.101659i
$757$	0.623490	+	0.781831i	0.623490	+	0.781831i	0.988831	−	0.149042i	$-0.0476190\pi$
$757$	0.623490	+	0.781831i	0.623490	+	0.781831i	−0.365341	+	0.930874i	$0.619048\pi$
$758$	1.71271	−	0.258149i	1.71271	−	0.258149i
$759$	−1.09886	−	0.749192i	−1.09886	−	0.749192i
$760$		0			0
$761$	−1.95557	+	0.294755i	−1.95557	+	0.294755i	−0.955573	+	0.294755i	$0.904762\pi$
$761$	−1.95557	+	0.294755i	−1.95557	+	0.294755i	−1.00000			$1.00000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.997204	+	0.925270i	0.997204	+	0.925270i
$769$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.826239	−	0.563320i	$-0.809524\pi$
$769$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.0747301	+	0.997204i	$0.523810\pi$
$770$		0			0
$771$	1.22563	+	0.590232i	1.22563	+	0.590232i
$772$		0			0
$773$	0.147791	+	1.97213i	0.147791	+	1.97213i	0.222521	+	0.974928i	$0.428571\pi$
$773$	0.147791	+	1.97213i	0.147791	+	1.97213i	−0.0747301	+	0.997204i	$0.523810\pi$
$774$		0			0
$775$	−0.145713	+	1.94440i	−0.145713	+	1.94440i
$776$		0			0
$777$		0			0
$778$	0.440071	−	1.92808i	0.440071	−	1.92808i
$779$		0			0
$780$		0			0
$781$	−0.486868	+	0.451748i	−0.486868	+	0.451748i
$782$	−0.716983	+	0.488831i	−0.716983	+	0.488831i
$783$		0			0
$784$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$785$		0			0
$786$	0.975345	−	0.664979i	0.975345	−	0.664979i
$787$	1.14625	−	1.06356i	1.14625	−	1.06356i	0.149042	−	0.988831i	$-0.452381\pi$
$787$	1.14625	−	1.06356i	1.14625	−	1.06356i	0.997204	−	0.0747301i	$-0.0238095\pi$

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)

\(f(q)\)	\(=\)	\(q+(-0.826239 + 0.563320i) q^{2} +(0.997204 - 0.925270i) q^{3} +(0.365341 - 0.930874i) q^{4} +(-0.302705 + 1.32624i) q^{6} +(-0.974928 + 0.222521i) q^{7} +(0.222521 + 0.974928i) q^{8} +(0.0635609 - 0.848162i) q^{9} +(0.0841939 + 1.12349i) q^{11} +(-0.496990 - 1.26631i) q^{12} +(1.48883 + 0.716983i) q^{13} +(0.680173 - 0.733052i) q^{14} +(-0.733052 - 0.680173i) q^{16} +(-0.988831 - 0.149042i) q^{17} +(0.425270 + 0.736589i) q^{18} +(-0.766310 + 1.12397i) q^{21} +(-0.702449 - 0.880843i) q^{22} +(-0.858075 + 0.129334i) q^{23} +(1.12397 + 0.766310i) q^{24} +(0.826239 + 0.563320i) q^{25} +(-1.63402 + 0.246289i) q^{26} +(0.126766 + 0.158960i) q^{27} +(-0.149042 + 0.988831i) q^{28} +(0.974928 + 1.68862i) q^{31} +(0.988831 + 0.149042i) q^{32} +(1.12349 + 1.04245i) q^{33} +(0.900969 - 0.433884i) q^{34} +(-0.766310 - 0.369035i) q^{36} +(2.14807 - 0.662592i) q^{39} -1.36035i q^{42} +(1.07659 + 0.332083i) q^{44} +(0.636119 - 0.590232i) q^{46} -1.36035 q^{48} +(0.900969 - 0.433884i) q^{49} -1.00000 q^{50} +(-1.12397 + 0.766310i) q^{51} +(1.21135 - 1.12397i) q^{52} +(0.698220 - 1.77904i) q^{53} +(-0.194285 - 0.0599289i) q^{54} +(-0.433884 - 0.900969i) q^{56} +(-1.75676 - 0.846011i) q^{62} +(0.126766 + 0.841040i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(-1.51550 - 0.228425i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(-0.736007 + 0.922924i) q^{69} +(0.367554 + 0.460898i) q^{71} +(0.841040 - 0.126766i) q^{72} +(1.34515 - 0.202749i) q^{75} +(-0.332083 - 1.07659i) q^{77} +(-1.40157 + 1.75751i) q^{78} +(-0.997204 + 1.72721i) q^{79} +(1.11453 + 0.167989i) q^{81} +(0.766310 + 1.12397i) q^{84} +(-1.07659 + 0.332083i) q^{88} +(0.109562 - 1.46200i) q^{89} +(-1.61105 - 0.367711i) q^{91} +(-0.193096 + 0.846011i) q^{92} +(2.53464 + 0.781831i) q^{93} +(1.12397 - 0.766310i) q^{96} +(-0.500000 + 0.866025i) q^{98} +0.958252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52}+ \cdots - 12 q^{98}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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