LMFDB - Embedded newform 2523.1.h.a.236.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,1,Mod(236,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 1]))

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

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

chi := DirichletCharacter("2523.236");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2523 = 3 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2523.h (of order $14$, degree $6$, not 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:	$1.25914102687$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.87.1
Artin image:	$S_3\times C_{14}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Embedding invariants

Embedding label			236.1
Root			$0.222521 + 0.974928i$ of defining polynomial
Character	$\chi$	$=$	2523.236
Dual form			2523.1.h.a.1037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.222521 + 0.974928i) q^{2} +(0.900969 - 0.433884i) q^{3} +(0.222521 + 0.974928i) q^{6} +(0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(0.623490 - 0.781831i) q^{9} +O(q^{10})$	\(q+(-0.222521 + 0.974928i) q^{2} +(0.900969 - 0.433884i) q^{3} +(0.222521 + 0.974928i) q^{6} +(0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(0.623490 - 0.781831i) q^{9} +(0.623490 + 0.781831i) q^{11} +(-0.623490 - 0.781831i) q^{13} +(0.222521 + 0.974928i) q^{14} +(-0.623490 - 0.781831i) q^{16} +1.00000 q^{17} +(0.623490 + 0.781831i) q^{18} +(0.623490 - 0.781831i) q^{21} +(-0.900969 + 0.433884i) q^{22} +(-0.222521 + 0.974928i) q^{24} +(-0.900969 - 0.433884i) q^{25} +(0.900969 - 0.433884i) q^{26} +(0.222521 - 0.974928i) q^{27} +(0.900969 + 0.433884i) q^{33} +(-0.222521 + 0.974928i) q^{34} +(-0.900969 - 0.433884i) q^{39} -2.00000 q^{41} +(0.623490 + 0.781831i) q^{42} +(0.623490 + 0.781831i) q^{47} +(-0.900969 - 0.433884i) q^{48} +(0.623490 - 0.781831i) q^{50} +(0.900969 - 0.433884i) q^{51} +(0.900969 + 0.433884i) q^{54} +(-0.222521 + 0.974928i) q^{56} +(0.222521 - 0.974928i) q^{63} +(-0.222521 - 0.974928i) q^{64} +(-0.623490 + 0.781831i) q^{66} +(-0.623490 + 0.781831i) q^{67} +(0.222521 + 0.974928i) q^{72} -1.00000 q^{75} +(0.900969 + 0.433884i) q^{77} +(0.623490 - 0.781831i) q^{78} +(-0.222521 - 0.974928i) q^{81} +(0.445042 - 1.94986i) q^{82} -1.00000 q^{88} +(-0.222521 + 0.974928i) q^{89} +(-0.900969 - 0.433884i) q^{91} +(-0.900969 + 0.433884i) q^{94} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + q^{3} + q^{6} + q^{7} + q^{8} - q^{9}+O(q^{10}) $ 6 * q - q^2 + q^3 + q^6 + q^7 + q^8 - q^9	$ 6 q - q^{2} + q^{3} + q^{6} + q^{7} + q^{8} - q^{9} - q^{11} + q^{13} + q^{14} + q^{16} + 6 q^{17} - q^{18} - q^{21} - q^{22} - q^{24} - q^{25} + q^{26} + q^{27} + q^{33} - q^{34} - q^{39} - 12 q^{41} - q^{42} - q^{47} - q^{48} - q^{50} + q^{51} + q^{54} - q^{56} + q^{63} - q^{64} + q^{66} + q^{67} + q^{72} - 6 q^{75} + q^{77} - q^{78} - q^{81} + 2 q^{82} - 6 q^{88} - q^{89} - q^{91} - q^{94} + 6 q^{99}+O(q^{100}) $ 6 * q - q^2 + q^3 + q^6 + q^7 + q^8 - q^9 - q^11 + q^13 + q^14 + q^16 + 6 * q^17 - q^18 - q^21 - q^22 - q^24 - q^25 + q^26 + q^27 + q^33 - q^34 - q^39 - 12 * q^41 - q^42 - q^47 - q^48 - q^50 + q^51 + q^54 - q^56 + q^63 - q^64 + q^66 + q^67 + q^72 - 6 * q^75 + q^77 - q^78 - q^81 + 2 * q^82 - 6 * q^88 - q^89 - q^91 - q^94 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times$.

$n$	$842$	$1684$
$\chi(n)$	$-1$	$e\left(\frac{1}{14}\right)$

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 $

$2$	−0.222521	+	0.974928i	−0.222521	+	0.974928i	0.733052	+	0.680173i	$0.238095\pi$
$2$	−0.222521	+	0.974928i	−0.222521	+	0.974928i	−0.955573	+	0.294755i	$0.904762\pi$
$3$	0.900969	−	0.433884i	0.900969	−	0.433884i
$3$	0.900969	−	0.433884i	0.900969	−	0.433884i
$4$		0			0
$5$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$5$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$6$	0.222521	+	0.974928i	0.222521	+	0.974928i
$7$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$7$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$8$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$9$	0.623490	−	0.781831i	0.623490	−	0.781831i
$10$		0			0
$11$	0.623490	+	0.781831i	0.623490	+	0.781831i	0.988831	−	0.149042i	$-0.0476190\pi$
$11$	0.623490	+	0.781831i	0.623490	+	0.781831i	−0.365341	+	0.930874i	$0.619048\pi$
$12$		0			0
$13$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$13$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$14$	0.222521	+	0.974928i	0.222521	+	0.974928i
$15$		0			0
$16$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$17$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$18$	0.623490	+	0.781831i	0.623490	+	0.781831i
$19$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$19$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$20$		0			0
$21$	0.623490	−	0.781831i	0.623490	−	0.781831i
$22$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$23$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$23$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$24$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$25$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$26$	0.900969	−	0.433884i	0.900969	−	0.433884i
$27$	0.222521	−	0.974928i	0.222521	−	0.974928i
$28$		0			0
$29$		0			0
$29$		0			0
$30$		0			0
$31$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$31$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$32$		0			0
$33$	0.900969	+	0.433884i	0.900969	+	0.433884i
$34$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$35$		0			0
$36$		0			0
$37$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$37$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$38$		0			0
$39$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$40$		0			0
$41$	−2.00000			−2.00000			−1.00000			$\pi$
$41$	−2.00000			−2.00000			−1.00000			$\pi$
$42$	0.623490	+	0.781831i	0.623490	+	0.781831i
$43$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$43$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.623490	+	0.781831i	0.623490	+	0.781831i	0.988831	−	0.149042i	$-0.0476190\pi$
$47$	0.623490	+	0.781831i	0.623490	+	0.781831i	−0.365341	+	0.930874i	$0.619048\pi$
$48$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$49$		0			0
$50$	0.623490	−	0.781831i	0.623490	−	0.781831i
$51$	0.900969	−	0.433884i	0.900969	−	0.433884i
$52$		0			0
$53$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$53$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$54$	0.900969	+	0.433884i	0.900969	+	0.433884i
$55$		0			0
$56$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$61$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$62$		0			0
$63$	0.222521	−	0.974928i	0.222521	−	0.974928i
$64$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$65$		0			0
$66$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$67$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$67$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$71$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$72$	0.222521	+	0.974928i	0.222521	+	0.974928i
$73$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$73$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$74$		0			0
$75$	−1.00000			−1.00000
$76$		0			0
$77$	0.900969	+	0.433884i	0.900969	+	0.433884i
$78$	0.623490	−	0.781831i	0.623490	−	0.781831i
$79$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$79$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$80$		0			0
$81$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$82$	0.445042	−	1.94986i	0.445042	−	1.94986i
$83$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$83$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.00000			−1.00000
$89$	−0.222521	+	0.974928i	−0.222521	+	0.974928i	0.733052	+	0.680173i	$0.238095\pi$
$89$	−0.222521	+	0.974928i	−0.222521	+	0.974928i	−0.955573	+	0.294755i	$0.904762\pi$
$90$		0			0
$91$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$92$		0			0
$93$		0			0
$94$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$95$		0			0
$96$		0			0
$97$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$97$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$98$		0			0
$99$	1.00000			1.00000
$100$		0			0
$101$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	−0.955573	−	0.294755i	$-0.904762\pi$
$101$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	0.733052	−	0.680173i	$-0.238095\pi$
$102$	0.222521	+	0.974928i	0.222521	+	0.974928i
$103$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$103$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$104$	1.00000			1.00000
$105$		0			0
$106$		0			0
$107$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$107$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$108$		0			0
$109$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$109$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$110$		0			0
$111$		0			0
$112$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$113$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.826239	−	0.563320i	$-0.809524\pi$
$113$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.0747301	+	0.997204i	$0.523810\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.00000			−1.00000
$118$		0			0
$119$	0.900969	−	0.433884i	0.900969	−	0.433884i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$124$		0			0
$125$		0			0
$126$	0.900969	+	0.433884i	0.900969	+	0.433884i
$127$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$127$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$128$	1.00000			1.00000
$129$		0			0
$130$		0			0
$131$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	−0.955573	−	0.294755i	$-0.904762\pi$
$131$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	0.733052	−	0.680173i	$-0.238095\pi$
$132$		0			0
$133$		0			0
$134$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$135$		0			0
$136$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$137$	−1.24698	+	1.56366i	−1.24698	+	1.56366i	−0.623490	+	0.781831i	$0.714286\pi$
$137$	−1.24698	+	1.56366i	−1.24698	+	1.56366i	−0.623490	+	0.781831i	$0.714286\pi$
$138$		0			0
$139$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$139$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$140$		0			0
$141$	0.900969	+	0.433884i	0.900969	+	0.433884i
$142$		0			0
$143$	0.222521	−	0.974928i	0.222521	−	0.974928i
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$149$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$150$	0.222521	−	0.974928i	0.222521	−	0.974928i
$151$	−0.445042	−	1.94986i	−0.445042	−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$151$	−0.445042	−	1.94986i	−0.445042	−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$152$		0			0
$153$	0.623490	−	0.781831i	0.623490	−	0.781831i
$154$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.00000			1.00000
$163$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$163$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$167$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$168$	0.222521	+	0.974928i	0.222521	+	0.974928i
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−1.00000			−1.00000
$176$	0.222521	−	0.974928i	0.222521	−	0.974928i
$177$		0			0
$178$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$179$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$179$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$180$		0			0
$181$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$181$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$182$	0.623490	−	0.781831i	0.623490	−	0.781831i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.623490	+	0.781831i	0.623490	+	0.781831i
$188$		0			0
$189$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$190$		0			0
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$192$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$193$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$193$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$197$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$198$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$199$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$199$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$200$	0.900969	−	0.433884i	0.900969	−	0.433884i
$201$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$202$	1.00000			1.00000
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$207$		0			0
$208$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$209$		0			0
$210$		0			0
$211$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$211$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$	0.623490	+	0.781831i	0.623490	+	0.781831i
$217$		0			0
$218$	0.222521	+	0.974928i	0.222521	+	0.974928i
$219$		0			0
$220$		0			0
$221$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$222$		0			0
$223$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$223$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$224$		0			0
$225$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$226$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$227$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$227$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$228$		0			0
$229$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$229$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$230$		0			0
$231$	1.00000			1.00000
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	0.222521	−	0.974928i	0.222521	−	0.974928i
$235$		0			0
$236$		0			0
$237$		0			0
$238$	0.222521	+	0.974928i	0.222521	+	0.974928i
$239$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$239$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$240$		0			0
$241$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$241$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$242$		0			0
$243$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$244$		0			0
$245$		0			0
$246$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.0747301	−	0.997204i	$-0.523810\pi$
$251$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.826239	+	0.563320i	$0.809524\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$257$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.00000			1.00000
$263$	0.445042	−	1.94986i	0.445042	−	1.94986i	0.222521	−	0.974928i	$-0.428571\pi$
$263$	0.445042	−	1.94986i	0.445042	−	1.94986i	0.222521	−	0.974928i	$-0.428571\pi$
$264$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$265$		0			0
$266$		0			0
$267$	0.222521	+	0.974928i	0.222521	+	0.974928i
$268$		0			0
$269$	0.623490	−	0.781831i	0.623490	−	0.781831i	−0.365341	−	0.930874i	$-0.619048\pi$
$269$	0.623490	−	0.781831i	0.623490	−	0.781831i	0.988831	+	0.149042i	$0.0476190\pi$
$270$		0			0
$271$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$271$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$272$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$273$	−1.00000			−1.00000
$274$	−1.24698	−	1.56366i	−1.24698	−	1.56366i
$275$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$276$		0			0
$277$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$277$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$278$	−1.00000			−1.00000
$279$		0			0
$280$		0			0
$281$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$281$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$282$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$283$	−1.80194	+	0.867767i	−1.80194	+	0.867767i	−0.900969	+	0.433884i	$0.857143\pi$
$283$	−1.80194	+	0.867767i	−1.80194	+	0.867767i	−0.900969	+	0.433884i	$0.857143\pi$
$284$		0			0
$285$		0			0
$286$	0.900969	+	0.433884i	0.900969	+	0.433884i
$287$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$288$		0			0
$289$		0			0
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.826239	−	0.563320i	$-0.809524\pi$
$293$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.0747301	+	0.997204i	$0.523810\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.900969	−	0.433884i	0.900969	−	0.433884i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	2.00000			2.00000
$303$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$304$		0			0
$305$		0			0
$306$	0.623490	+	0.781831i	0.623490	+	0.781831i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	1.80194	+	0.867767i	1.80194	+	0.867767i
$310$		0			0
$311$	0.623490	−	0.781831i	0.623490	−	0.781831i	−0.365341	−	0.930874i	$-0.619048\pi$
$311$	0.623490	−	0.781831i	0.623490	−	0.781831i	0.988831	+	0.149042i	$0.0476190\pi$
$312$	0.900969	−	0.433884i	0.900969	−	0.433884i
$313$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$313$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.222521	+	0.974928i	−0.222521	+	0.974928i	0.733052	+	0.680173i	$0.238095\pi$
$317$	−0.222521	+	0.974928i	−0.222521	+	0.974928i	−0.955573	+	0.294755i	$0.904762\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.222521	+	0.974928i	0.222521	+	0.974928i
$326$		0			0
$327$	0.623490	−	0.781831i	0.623490	−	0.781831i
$328$	1.24698	−	1.56366i	1.24698	−	1.56366i
$329$	0.900969	+	0.433884i	0.900969	+	0.433884i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−1.00000			−1.00000
$337$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$337$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$338$		0			0
$339$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	2.00000			2.00000			1.00000			$0$
$349$	2.00000			2.00000			1.00000			$0$
$350$	0.222521	−	0.974928i	0.222521	−	0.974928i
$351$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$352$		0			0
$353$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$353$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.623490	−	0.781831i	0.623490	−	0.781831i
$358$		0			0
$359$	−1.24698	−	1.56366i	−1.24698	−	1.56366i	−0.623490	−	0.781831i	$-0.714286\pi$
$359$	−1.24698	−	1.56366i	−1.24698	−	1.56366i	−0.623490	−	0.781831i	$-0.714286\pi$
$360$		0			0
$361$	0.623490	+	0.781831i	0.623490	+	0.781831i
$362$	0.222521	+	0.974928i	0.222521	+	0.974928i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$367$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$368$		0			0
$369$	−1.24698	+	1.56366i	−1.24698	+	1.56366i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$373$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$374$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$375$		0			0
$376$	−1.00000			−1.00000
$377$		0			0
$378$	1.00000			1.00000
$379$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$379$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$380$		0			0
$381$		0			0
$382$	0.445042	−	1.94986i	0.445042	−	1.94986i
$383$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$383$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$384$	0.900969	−	0.433884i	0.900969	−	0.433884i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$397$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$398$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$399$		0			0
$400$	0.222521	+	0.974928i	0.222521	+	0.974928i
$401$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$401$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$402$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$409$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$409$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$410$		0			0
$411$	−0.445042	+	1.94986i	−0.445042	+	1.94986i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.623490	+	0.781831i	0.623490	+	0.781831i
$418$		0			0
$419$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$419$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$420$		0			0
$421$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$421$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$422$		0			0
$423$	1.00000			1.00000
$424$		0			0
$425$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$430$		0			0
$431$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$431$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$432$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$433$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$433$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$439$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$440$		0			0
$441$		0			0
$442$	0.900969	−	0.433884i	0.900969	−	0.433884i
$443$	0.623490	−	0.781831i	0.623490	−	0.781831i	−0.365341	−	0.930874i	$-0.619048\pi$
$443$	0.623490	−	0.781831i	0.623490	−	0.781831i	0.988831	+	0.149042i	$0.0476190\pi$
$444$		0			0
$445$		0			0
$446$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$447$		0			0
$448$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$449$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	−0.955573	−	0.294755i	$-0.904762\pi$
$449$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	0.733052	−	0.680173i	$-0.238095\pi$
$450$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$451$	−1.24698	−	1.56366i	−1.24698	−	1.56366i
$452$		0			0
$453$	−1.24698	−	1.56366i	−1.24698	−	1.56366i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$457$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$458$		0			0
$459$	0.222521	−	0.974928i	0.222521	−	0.974928i
$460$		0			0
$461$	1.80194	−	0.867767i	1.80194	−	0.867767i	0.900969	−	0.433884i	$-0.142857\pi$
$461$	1.80194	−	0.867767i	1.80194	−	0.867767i	0.900969	−	0.433884i	$-0.142857\pi$
$462$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$463$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.80194	−	0.867767i	1.80194	−	0.867767i	0.900969	−	0.433884i	$-0.142857\pi$
$467$	1.80194	−	0.867767i	1.80194	−	0.867767i	0.900969	−	0.433884i	$-0.142857\pi$
$468$		0			0
$469$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.445042	+	1.94986i	0.445042	+	1.94986i	0.222521	+	0.974928i	$0.428571\pi$
$479$	0.445042	+	1.94986i	0.445042	+	1.94986i	0.222521	+	0.974928i	$0.428571\pi$
$480$		0			0
$481$		0			0
$482$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	0.900969	−	0.433884i	0.900969	−	0.433884i
$487$	−0.445042	−	1.94986i	−0.445042	−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$487$	−0.445042	−	1.94986i	−0.445042	−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.445042	−	1.94986i	0.445042	−	1.94986i	0.222521	−	0.974928i	$-0.428571\pi$
$491$	0.445042	−	1.94986i	0.445042	−	1.94986i	0.222521	−	0.974928i	$-0.428571\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$499$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$500$		0			0
$501$		0			0
$502$	0.623490	−	0.781831i	0.623490	−	0.781831i
$503$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.0747301	−	0.997204i	$-0.523810\pi$
$503$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.826239	+	0.563320i	$0.809524\pi$
$504$	0.623490	+	0.781831i	0.623490	+	0.781831i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$509$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$510$		0			0
$511$		0			0
$512$	0.900969	+	0.433884i	0.900969	+	0.433884i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$526$	1.80194	+	0.867767i	1.80194	+	0.867767i
$527$		0			0
$528$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$529$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.24698	+	1.56366i	1.24698	+	1.56366i
$534$	−1.00000			−1.00000
$535$		0			0
$536$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$537$		0			0
$538$	0.623490	+	0.781831i	0.623490	+	0.781831i
$539$		0			0
$540$		0			0
$541$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$541$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$542$		0			0
$543$	0.623490	−	0.781831i	0.623490	−	0.781831i
$544$		0			0
$545$		0			0
$546$	0.222521	−	0.974928i	0.222521	−	0.974928i
$547$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$547$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$548$		0			0
$549$		0			0
$550$	1.00000			1.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.900969	−	0.433884i	0.900969	−	0.433884i
$555$		0			0
$556$		0			0
$557$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$557$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.900969	+	0.433884i	0.900969	+	0.433884i
$562$		0			0
$563$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$567$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$568$		0			0
$569$	0.623490	+	0.781831i	0.623490	+	0.781831i	0.988831	−	0.149042i	$-0.0476190\pi$
$569$	0.623490	+	0.781831i	0.623490	+	0.781831i	−0.365341	+	0.930874i	$0.619048\pi$
$570$		0			0
$571$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$571$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$572$		0			0
$573$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$574$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$575$		0			0
$576$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$577$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$577$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$587$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$587$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$593$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$594$	0.222521	+	0.974928i	0.222521	+	0.974928i
$595$		0			0
$596$		0			0
$597$	1.00000			1.00000
$598$		0			0
$599$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.0747301	−	0.997204i	$-0.523810\pi$
$599$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.826239	+	0.563320i	$0.809524\pi$
$600$	0.623490	−	0.781831i	0.623490	−	0.781831i
$601$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$601$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$602$		0			0
$603$	0.222521	+	0.974928i	0.222521	+	0.974928i
$604$		0			0
$605$		0			0
$606$	0.900969	−	0.433884i	0.900969	−	0.433884i
$607$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$607$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.222521	−	0.974928i	0.222521	−	0.974928i
$612$		0			0
$613$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$613$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$614$		0			0
$615$		0			0
$616$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$617$	−1.24698	+	1.56366i	−1.24698	+	1.56366i	−0.623490	+	0.781831i	$0.714286\pi$
$617$	−1.24698	+	1.56366i	−1.24698	+	1.56366i	−0.623490	+	0.781831i	$0.714286\pi$
$618$	−1.24698	+	1.56366i	−1.24698	+	1.56366i
$619$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$619$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$620$		0			0
$621$		0			0
$622$	0.623490	+	0.781831i	0.623490	+	0.781831i
$623$	0.222521	+	0.974928i	0.222521	+	0.974928i
$624$	0.222521	+	0.974928i	0.222521	+	0.974928i
$625$	0.623490	+	0.781831i	0.623490	+	0.781831i
$626$	−1.00000			−1.00000
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$631$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$632$		0			0
$633$		0			0
$634$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.826239	−	0.563320i	$-0.809524\pi$
$641$	−0.900969	+	0.433884i	−0.900969	+	0.433884i	−0.0747301	+	0.997204i	$0.523810\pi$
$642$		0			0
$643$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$643$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$647$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$648$	0.900969	+	0.433884i	0.900969	+	0.433884i
$649$		0			0
$650$	−1.00000			−1.00000
$651$		0			0
$652$		0			0
$653$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	−0.955573	−	0.294755i	$-0.904762\pi$
$653$	−0.222521	−	0.974928i	−0.222521	−	0.974928i	0.733052	−	0.680173i	$-0.238095\pi$
$654$	0.623490	+	0.781831i	0.623490	+	0.781831i
$655$		0			0
$656$	1.24698	+	1.56366i	1.24698	+	1.56366i
$657$		0			0
$658$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$659$	0.623490	−	0.781831i	0.623490	−	0.781831i	−0.365341	−	0.930874i	$-0.619048\pi$
$659$	0.623490	−	0.781831i	0.623490	−	0.781831i	0.988831	+	0.149042i	$0.0476190\pi$
$660$		0			0
$661$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$661$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$662$		0			0
$663$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$673$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$674$		0			0
$675$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$676$		0			0
$677$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.0747301	−	0.997204i	$-0.523810\pi$
$677$	−0.900969	−	0.433884i	−0.900969	−	0.433884i	−0.826239	+	0.563320i	$0.809524\pi$
$678$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$683$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$684$		0			0
$685$		0			0
$686$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$691$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$692$		0			0
$693$	0.900969	−	0.433884i	0.900969	−	0.433884i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.00000			−2.00000
$698$	−0.445042	+	1.94986i	−0.445042	+	1.94986i
$699$		0			0
$700$		0			0
$701$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$701$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$702$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$703$		0			0
$704$	0.623490	−	0.781831i	0.623490	−	0.781831i
$705$		0			0
$706$		0			0
$707$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$708$		0			0
$709$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$709$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$710$		0			0
$711$		0			0
$712$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$713$		0			0
$714$	0.623490	+	0.781831i	0.623490	+	0.781831i
$715$		0			0
$716$		0			0
$717$		0			0
$718$	1.80194	−	0.867767i	1.80194	−	0.867767i
$719$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$719$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$720$		0			0
$721$	1.80194	+	0.867767i	1.80194	+	0.867767i
$722$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$723$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$727$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$728$	0.900969	−	0.433884i	0.900969	−	0.433884i
$729$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$733$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.00000			−1.00000
$738$	−1.24698	−	1.56366i	−1.24698	−	1.56366i
$739$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$739$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.623490	+	0.781831i	0.623490	+	0.781831i	0.988831	−	0.149042i	$-0.0476190\pi$
$743$	0.623490	+	0.781831i	0.623490	+	0.781831i	−0.365341	+	0.930874i	$0.619048\pi$
$744$		0			0
$745$		0			0
$746$	1.24698	−	1.56366i	1.24698	−	1.56366i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$751$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$752$	0.222521	−	0.974928i	0.222521	−	0.974928i
$753$	−1.00000			−1.00000
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$757$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$761$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$762$		0			0
$763$	0.623490	−	0.781831i	0.623490	−	0.781831i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$769$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.80194	+	0.867767i	1.80194	+	0.867767i	0.900969	+	0.433884i	$0.142857\pi$
$773$	1.80194	+	0.867767i	1.80194	+	0.867767i	0.900969	+	0.433884i	$0.142857\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	0.900969	−	0.433884i	0.900969	−	0.433884i
$787$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$787$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$788$		0			0
$789$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$790$		0			0
$791$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$792$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$793$		0			0
$794$	1.24698	+	1.56366i	1.24698	+

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:	\( N \)	\(=\)	\( 2523 = 3 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2523.h (of order \(14\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.222521 + 0.974928i) q^{2} +(0.900969 - 0.433884i) q^{3} +(0.222521 + 0.974928i) q^{6} +(0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(0.623490 - 0.781831i) q^{9} +O(q^{10})\)	\(q+(-0.222521 + 0.974928i) q^{2} +(0.900969 - 0.433884i) q^{3} +(0.222521 + 0.974928i) q^{6} +(0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(0.623490 - 0.781831i) q^{9} +(0.623490 + 0.781831i) q^{11} +(-0.623490 - 0.781831i) q^{13} +(0.222521 + 0.974928i) q^{14} +(-0.623490 - 0.781831i) q^{16} +1.00000 q^{17} +(0.623490 + 0.781831i) q^{18} +(0.623490 - 0.781831i) q^{21} +(-0.900969 + 0.433884i) q^{22} +(-0.222521 + 0.974928i) q^{24} +(-0.900969 - 0.433884i) q^{25} +(0.900969 - 0.433884i) q^{26} +(0.222521 - 0.974928i) q^{27} +(0.900969 + 0.433884i) q^{33} +(-0.222521 + 0.974928i) q^{34} +(-0.900969 - 0.433884i) q^{39} -2.00000 q^{41} +(0.623490 + 0.781831i) q^{42} +(0.623490 + 0.781831i) q^{47} +(-0.900969 - 0.433884i) q^{48} +(0.623490 - 0.781831i) q^{50} +(0.900969 - 0.433884i) q^{51} +(0.900969 + 0.433884i) q^{54} +(-0.222521 + 0.974928i) q^{56} +(0.222521 - 0.974928i) q^{63} +(-0.222521 - 0.974928i) q^{64} +(-0.623490 + 0.781831i) q^{66} +(-0.623490 + 0.781831i) q^{67} +(0.222521 + 0.974928i) q^{72} -1.00000 q^{75} +(0.900969 + 0.433884i) q^{77} +(0.623490 - 0.781831i) q^{78} +(-0.222521 - 0.974928i) q^{81} +(0.445042 - 1.94986i) q^{82} -1.00000 q^{88} +(-0.222521 + 0.974928i) q^{89} +(-0.900969 - 0.433884i) q^{91} +(-0.900969 + 0.433884i) q^{94} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + q^{3} + q^{6} + q^{7} + q^{8} - q^{9}+O(q^{10}) \) 6 * q - q^2 + q^3 + q^6 + q^7 + q^8 - q^9	\( 6 q - q^{2} + q^{3} + q^{6} + q^{7} + q^{8} - q^{9} - q^{11} + q^{13} + q^{14} + q^{16} + 6 q^{17} - q^{18} - q^{21} - q^{22} - q^{24} - q^{25} + q^{26} + q^{27} + q^{33} - q^{34} - q^{39} - 12 q^{41} - q^{42} - q^{47} - q^{48} - q^{50} + q^{51} + q^{54} - q^{56} + q^{63} - q^{64} + q^{66} + q^{67} + q^{72} - 6 q^{75} + q^{77} - q^{78} - q^{81} + 2 q^{82} - 6 q^{88} - q^{89} - q^{91} - q^{94} + 6 q^{99}+O(q^{100}) \) 6 * q - q^2 + q^3 + q^6 + q^7 + q^8 - q^9 - q^11 + q^13 + q^14 + q^16 + 6 * q^17 - q^18 - q^21 - q^22 - q^24 - q^25 + q^26 + q^27 + q^33 - q^34 - q^39 - 12 * q^41 - q^42 - q^47 - q^48 - q^50 + q^51 + q^54 - q^56 + q^63 - q^64 + q^66 + q^67 + q^72 - 6 * q^75 + q^77 - q^78 - q^81 + 2 * q^82 - 6 * q^88 - q^89 - q^91 - q^94 + 6 * q^99

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