LMFDB - Embedded newform 952.1.e.d.237.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [952,1,Mod(237,952)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(952, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("952.237");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 952 = 2^{3} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	952.e (of order $2$, degree $1$, 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:	$0.475109892027$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.2.6571095523328.1

Embedding invariants

Embedding label			237.3
Root			$0.809017 + 0.587785i$ of defining polynomial
Character	$\chi$	$=$	952.237
Dual form			952.1.e.d.237.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.809017 - 0.587785i) q^{2} +1.17557i q^{3} +(0.309017 - 0.951057i) q^{4} -1.90211i q^{5} +(0.690983 + 0.951057i) q^{6} -1.00000 q^{7} +(-0.309017 - 0.951057i) q^{8} -0.381966 q^{9} +O(q^{10})$	\(q+(0.809017 - 0.587785i) q^{2} +1.17557i q^{3} +(0.309017 - 0.951057i) q^{4} -1.90211i q^{5} +(0.690983 + 0.951057i) q^{6} -1.00000 q^{7} +(-0.309017 - 0.951057i) q^{8} -0.381966 q^{9} +(-1.11803 - 1.53884i) q^{10} +(1.11803 + 0.363271i) q^{12} +(-0.809017 + 0.587785i) q^{14} +2.23607 q^{15} +(-0.809017 - 0.587785i) q^{16} +1.00000 q^{17} +(-0.309017 + 0.224514i) q^{18} +(-1.80902 - 0.587785i) q^{20} -1.17557i q^{21} +(1.11803 - 0.363271i) q^{24} -2.61803 q^{25} +0.726543i q^{27} +(-0.309017 + 0.951057i) q^{28} +(1.80902 - 1.31433i) q^{30} +0.618034 q^{31} -1.00000 q^{32} +(0.809017 - 0.587785i) q^{34} +1.90211i q^{35} +(-0.118034 + 0.363271i) q^{36} +(-1.80902 + 0.587785i) q^{40} +1.61803 q^{41} +(-0.690983 - 0.951057i) q^{42} +1.17557i q^{43} +0.726543i q^{45} +(0.690983 - 0.951057i) q^{48} +1.00000 q^{49} +(-2.11803 + 1.53884i) q^{50} +1.17557i q^{51} +1.90211i q^{53} +(0.427051 + 0.587785i) q^{54} +(0.309017 + 0.951057i) q^{56} +(0.690983 - 2.12663i) q^{60} +1.17557i q^{61} +(0.500000 - 0.363271i) q^{62} +0.381966 q^{63} +(-0.809017 + 0.587785i) q^{64} -1.90211i q^{67} +(0.309017 - 0.951057i) q^{68} +(1.11803 + 1.53884i) q^{70} +(0.118034 + 0.363271i) q^{72} -1.61803 q^{73} -3.07768i q^{75} +(-1.11803 + 1.53884i) q^{80} -1.23607 q^{81} +(1.30902 - 0.951057i) q^{82} +(-1.11803 - 0.363271i) q^{84} -1.90211i q^{85} +(0.690983 + 0.951057i) q^{86} +(0.427051 + 0.587785i) q^{90} +0.726543i q^{93} -1.17557i q^{96} -0.618034 q^{97} +(0.809017 - 0.587785i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - q^{4} + 5 q^{6} - 4 q^{7} + q^{8} - 6 q^{9}+O(q^{10}) $ 4 * q + q^2 - q^4 + 5 * q^6 - 4 * q^7 + q^8 - 6 * q^9	$ 4 q + q^{2} - q^{4} + 5 q^{6} - 4 q^{7} + q^{8} - 6 q^{9} - q^{14} - q^{16} + 4 q^{17} + q^{18} - 5 q^{20} - 6 q^{25} + q^{28} + 5 q^{30} - 2 q^{31} - 4 q^{32} + q^{34} + 4 q^{36} - 5 q^{40} + 2 q^{41} - 5 q^{42} + 5 q^{48} + 4 q^{49} - 4 q^{50} - 5 q^{54} - q^{56} + 5 q^{60} + 2 q^{62} + 6 q^{63} - q^{64} - q^{68} - 4 q^{72} - 2 q^{73} + 4 q^{81} + 3 q^{82} + 5 q^{86} - 5 q^{90} + 2 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 - q^4 + 5 * q^6 - 4 * q^7 + q^8 - 6 * q^9 - q^14 - q^16 + 4 * q^17 + q^18 - 5 * q^20 - 6 * q^25 + q^28 + 5 * q^30 - 2 * q^31 - 4 * q^32 + q^34 + 4 * q^36 - 5 * q^40 + 2 * q^41 - 5 * q^42 + 5 * q^48 + 4 * q^49 - 4 * q^50 - 5 * q^54 - q^56 + 5 * q^60 + 2 * q^62 + 6 * q^63 - q^64 - q^68 - 4 * q^72 - 2 * q^73 + 4 * q^81 + 3 * q^82 + 5 * q^86 - 5 * q^90 + 2 * q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/952\mathbb{Z}\right)^\times$.

$n$	$239$	$409$	$477$	$785$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$	0.809017	−	0.587785i	0.809017	−	0.587785i
$2$	0.809017	−	0.587785i	0.809017	−	0.587785i
$3$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$3$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$5$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$6$	0.690983	+	0.951057i	0.690983	+	0.951057i
$7$	−1.00000			−1.00000
$7$	−1.00000			−1.00000
$8$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$9$	−0.381966			−0.381966
$10$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	1.11803	+	0.363271i	1.11803	+	0.363271i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$15$	2.23607			2.23607
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$	1.00000			1.00000
$17$	1.00000			1.00000
$18$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$21$		−	1.17557i		−	1.17557i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	1.11803	−	0.363271i	1.11803	−	0.363271i
$25$	−2.61803			−2.61803
$26$		0			0
$27$			0.726543i			0.726543i
$28$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	1.80902	−	1.31433i	1.80902	−	1.31433i
$31$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$31$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$	0.809017	−	0.587785i	0.809017	−	0.587785i
$35$			1.90211i			1.90211i
$36$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$41$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$41$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$42$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$43$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$43$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$44$		0			0
$45$			0.726543i			0.726543i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	0.690983	−	0.951057i	0.690983	−	0.951057i
$49$	1.00000			1.00000
$50$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$51$			1.17557i			1.17557i
$52$		0			0
$53$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$53$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$54$	0.427051	+	0.587785i	0.427051	+	0.587785i
$55$		0			0
$56$	0.309017	+	0.951057i	0.309017	+	0.951057i
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	0.690983	−	2.12663i	0.690983	−	2.12663i
$61$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$61$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$62$	0.500000	−	0.363271i	0.500000	−	0.363271i
$63$	0.381966			0.381966
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$67$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$68$	0.309017	−	0.951057i	0.309017	−	0.951057i
$69$		0			0
$70$	1.11803	+	1.53884i	1.11803	+	1.53884i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.118034	+	0.363271i	0.118034	+	0.363271i
$73$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$73$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$		−	3.07768i		−	3.07768i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$81$	−1.23607			−1.23607
$82$	1.30902	−	0.951057i	1.30902	−	0.951057i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$85$		−	1.90211i		−	1.90211i
$86$	0.690983	+	0.951057i	0.690983	+	0.951057i
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.427051	+	0.587785i	0.427051	+	0.587785i
$91$		0			0
$92$		0			0
$93$			0.726543i			0.726543i
$94$		0			0
$95$		0			0
$96$		−	1.17557i		−	1.17557i
$97$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$97$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$98$	0.809017	−	0.587785i	0.809017	−	0.587785i
$99$		0			0
$100$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	0.690983	+	0.951057i	0.690983	+	0.951057i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	−2.23607			−2.23607
$106$	1.11803	+	1.53884i	1.11803	+	1.53884i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	0.690983	+	0.224514i	0.690983	+	0.224514i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	0.809017	+	0.587785i	0.809017	+	0.587785i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.00000			−1.00000
$120$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$121$	−1.00000			−1.00000
$122$	0.690983	+	0.951057i	0.690983	+	0.951057i
$123$			1.90211i			1.90211i
$124$	0.190983	−	0.587785i	0.190983	−	0.587785i
$125$			3.07768i			3.07768i
$126$	0.309017	−	0.224514i	0.309017	−	0.224514i
$127$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$127$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$128$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$129$	−1.38197			−1.38197
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$135$	1.38197			1.38197
$136$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$137$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$137$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$138$		0			0
$139$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$139$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$140$	1.80902	+	0.587785i	1.80902	+	0.587785i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.309017	+	0.224514i	0.309017	+	0.224514i
$145$		0			0
$146$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$147$			1.17557i			1.17557i
$148$		0			0
$149$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$149$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$150$	−1.80902	−	2.48990i	−1.80902	−	2.48990i
$151$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$151$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$152$		0			0
$153$	−0.381966			−0.381966
$154$		0			0
$155$		−	1.17557i		−	1.17557i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$	−2.23607			−2.23607
$160$			1.90211i			1.90211i
$161$		0			0
$162$	−1.00000	+	0.726543i	−1.00000	+	0.726543i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	0.500000	−	1.53884i	0.500000	−	1.53884i
$165$		0			0
$166$		0			0
$167$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$167$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$168$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$169$	−1.00000			−1.00000
$170$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$171$		0			0
$172$	1.11803	+	0.363271i	1.11803	+	0.363271i
$173$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$173$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$174$		0			0
$175$	2.61803			2.61803
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$179$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$180$	0.690983	+	0.224514i	0.690983	+	0.224514i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−1.38197			−1.38197
$184$		0			0
$185$		0			0
$186$	0.427051	+	0.587785i	0.427051	+	0.587785i
$187$		0			0
$188$		0			0
$189$		−	0.726543i		−	0.726543i
$190$		0			0
$191$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$191$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$192$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$195$		0			0
$196$	0.309017	−	0.951057i	0.309017	−	0.951057i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$199$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$200$	0.809017	+	2.48990i	0.809017	+	2.48990i
$201$	2.23607			2.23607
$202$		0			0
$203$		0			0
$204$	1.11803	+	0.363271i	1.11803	+	0.363271i
$205$		−	3.07768i		−	3.07768i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	1.80902	+	0.587785i	1.80902	+	0.587785i
$213$		0			0
$214$		0			0
$215$	2.23607			2.23607
$216$	0.690983	−	0.224514i	0.690983	−	0.224514i
$217$	−0.618034			−0.618034
$218$		0			0
$219$		−	1.90211i		−	1.90211i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	1.00000			1.00000
$225$	1.00000			1.00000
$226$		0			0
$227$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$227$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$239$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$239$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$240$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$241$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$241$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$242$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$243$		−	0.726543i		−	0.726543i
$244$	1.11803	+	0.363271i	1.11803	+	0.363271i
$245$		−	1.90211i		−	1.90211i
$246$	1.11803	+	1.53884i	1.11803	+	1.53884i
$247$		0			0
$248$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$249$		0			0
$250$	1.80902	+	2.48990i	1.80902	+	2.48990i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	0.118034	−	0.363271i	0.118034	−	0.363271i
$253$		0			0
$254$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$255$	2.23607			2.23607
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	−1.11803	+	0.812299i	−1.11803	+	0.812299i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.00000			2.00000			1.00000			$0$
$263$	2.00000			2.00000			1.00000			$0$
$264$		0			0
$265$	3.61803			3.61803
$266$		0			0
$267$		0			0
$268$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	1.11803	−	0.812299i	1.11803	−	0.812299i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$273$		0			0
$274$	1.30902	−	0.951057i	1.30902	−	0.951057i
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$279$	−0.236068			−0.236068
$280$	1.80902	−	0.587785i	1.80902	−	0.587785i
$281$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$281$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$282$		0			0
$283$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$283$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.61803			−1.61803
$288$	0.381966			0.381966
$289$	1.00000			1.00000
$290$		0			0
$291$		−	0.726543i		−	0.726543i
$292$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.690983	+	0.951057i	0.690983	+	0.951057i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	0.690983	+	0.951057i	0.690983	+	0.951057i
$299$		0			0
$300$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$301$		−	1.17557i		−	1.17557i
$302$	1.30902	−	0.951057i	1.30902	−	0.951057i
$303$		0			0
$304$		0			0
$305$	2.23607			2.23607
$306$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$311$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$311$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$313$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$314$		0			0
$315$		−	0.726543i		−	0.726543i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$319$		0			0
$320$	1.11803	+	1.53884i	1.11803	+	1.53884i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.381966	+	1.17557i	−0.381966	+	1.17557i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$329$		0			0
$330$		0			0
$331$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$331$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$332$		0			0
$333$		0			0
$334$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$335$	−3.61803			−3.61803
$336$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$339$		0			0
$340$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$	1.11803	−	0.363271i	1.11803	−	0.363271i
$345$		0			0
$346$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	2.11803	−	1.53884i	2.11803	−	1.53884i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	1.17557i		−	1.17557i
$358$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$359$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$359$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$360$	0.690983	−	0.224514i	0.690983	−	0.224514i
$361$	−1.00000			−1.00000
$362$		0			0
$363$		−	1.17557i		−	1.17557i
$364$		0			0
$365$			3.07768i			3.07768i
$366$	−1.11803	+	0.812299i	−1.11803	+	0.812299i
$367$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$367$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$	−0.618034			−0.618034
$370$		0			0
$371$		−	1.90211i		−	1.90211i
$372$	0.690983	+	0.224514i	0.690983	+	0.224514i
$373$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$373$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$374$		0			0
$375$	−3.61803			−3.61803
$376$		0			0
$377$		0			0
$378$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		−	0.726543i		−	0.726543i
$382$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$385$		0			0
$386$		0			0
$387$		−	0.449028i		−	0.449028i
$388$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$389$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$389$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$390$		0			0
$391$		0			0
$392$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$397$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$398$	1.30902	−	0.951057i	1.30902	−	0.951057i
$399$		0			0
$400$	2.11803	+	1.53884i	2.11803	+	1.53884i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	1.80902	−	1.31433i	1.80902	−	1.31433i
$403$		0			0
$404$		0			0
$405$			2.35114i			2.35114i
$406$		0			0
$407$		0			0
$408$	1.11803	−	0.363271i	1.11803	−	0.363271i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	−1.80902	−	2.48990i	−1.80902	−	2.48990i
$411$			1.90211i			1.90211i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.23607			2.23607
$418$		0			0
$419$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$419$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$420$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$421$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$421$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$422$		0			0
$423$		0			0
$424$	1.80902	−	0.587785i	1.80902	−	0.587785i
$425$	−2.61803			−2.61803
$426$		0			0
$427$		−	1.17557i		−	1.17557i
$428$		0			0
$429$		0			0
$430$	1.80902	−	1.31433i	1.80902	−	1.31433i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.427051	−	0.587785i	0.427051	−	0.587785i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$439$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$439$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$440$		0			0
$441$	−0.381966			−0.381966
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.38197			−1.38197
$448$	0.809017	−	0.587785i	0.809017	−	0.587785i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.809017	−	0.587785i	0.809017	−	0.587785i
$451$		0			0
$452$		0			0
$453$			1.90211i			1.90211i
$454$	1.11803	+	1.53884i	1.11803	+	1.53884i
$455$		0			0
$456$		0			0
$457$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$457$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$458$		0			0
$459$			0.726543i			0.726543i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$463$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$464$		0			0
$465$	1.38197			1.38197
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$			1.90211i			1.90211i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$477$		−	0.726543i		−	0.726543i
$478$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$479$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$479$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$480$	−2.23607			−2.23607
$481$		0			0
$482$	0.500000	−	0.363271i	0.500000	−	0.363271i
$483$		0			0
$484$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$485$			1.17557i			1.17557i
$486$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	1.11803	−	0.363271i	1.11803	−	0.363271i
$489$		0			0
$490$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$491$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$491$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$492$	1.80902	+	0.587785i	1.80902	+	0.587785i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	2.92705	+	0.951057i	2.92705	+	0.951057i
$501$		−	1.90211i		−	1.90211i
$502$		0			0
$503$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$503$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$504$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$505$		0			0
$506$		0			0
$507$		−	1.17557i		−	1.17557i
$508$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$	1.80902	−	1.31433i	1.80902	−	1.31433i
$511$	1.61803			1.61803
$512$	0.809017	+	0.587785i	0.809017	+	0.587785i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−0.427051	+	1.31433i	−0.427051	+	1.31433i
$517$		0			0
$518$		0			0
$519$	1.38197			1.38197
$520$		0			0
$521$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$521$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$			3.07768i			3.07768i
$526$	1.61803	−	1.17557i	1.61803	−	1.17557i
$527$	0.618034			0.618034
$528$		0			0
$529$	1.00000			1.00000
$530$	2.92705	−	2.12663i	2.92705	−	2.12663i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$537$	1.38197			1.38197
$538$		0			0
$539$		0			0
$540$	0.427051	−	1.31433i	0.427051	−	1.31433i
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$	−1.00000			−1.00000
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	0.500000	−	1.53884i	0.500000	−	1.53884i
$549$		−	0.449028i		−	0.449028i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	−0.190983	+	0.138757i	−0.190983	+	0.138757i
$559$		0			0
$560$	1.11803	−	1.53884i	1.11803	−	1.53884i
$561$		0			0
$562$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$	1.11803	+	1.53884i	1.11803	+	1.53884i
$567$	1.23607			1.23607
$568$		0			0
$569$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$569$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	0.726543i		−	0.726543i
$574$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$575$		0			0
$576$	0.309017	−	0.224514i	0.309017	−	0.224514i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.809017	−	0.587785i	0.809017	−	0.587785i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$583$		0			0
$584$	0.500000	+	1.53884i	0.500000	+	1.53884i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	1.11803	+	0.363271i	1.11803	+	0.363271i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$			1.90211i			1.90211i
$596$	1.11803	+	0.363271i	1.11803	+	0.363271i
$597$			1.90211i			1.90211i
$598$		0			0
$599$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$599$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$600$	−2.92705	+	0.951057i	−2.92705	+	0.951057i
$601$	−2.00000			−2.00000			−1.00000			$\pi$
$601$	−2.00000			−2.00000			−1.00000			$\pi$
$602$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$603$			0.726543i			0.726543i
$604$	0.500000	−	1.53884i	0.500000	−	1.53884i
$605$			1.90211i			1.90211i
$606$		0			0
$607$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$607$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$608$		0			0
$609$		0			0
$610$	1.80902	−	1.31433i	1.80902	−	1.31433i
$611$		0			0
$612$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$613$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$613$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$	3.61803			3.61803
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$621$		0			0
$622$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$623$		0			0
$624$		0			0
$625$	3.23607			3.23607
$626$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$631$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$631$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1.17557i			1.17557i
$636$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.80902	+	0.587785i	1.80902	+	0.587785i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$643$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$644$		0			0
$645$			2.62866i			2.62866i
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.381966	+	1.17557i	0.381966	+	1.17557i
$649$		0			0
$650$		0			0
$651$		−	0.726543i		−	0.726543i
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$657$	0.618034			0.618034
$658$		0			0
$659$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$659$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$669$		0			0
$670$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$671$		0			0
$672$			1.17557i			1.17557i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	1.90211i		−	1.90211i
$676$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	0.618034			0.618034
$680$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$681$	−2.23607			−2.23607
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		−	3.07768i		−	3.07768i
$686$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$687$		0			0
$688$	0.690983	−	0.951057i	0.690983	−	0.951057i
$689$		0			0
$690$		0			0
$691$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$691$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$692$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$693$		0			0
$694$		0			0
$695$	−3.61803			−3.61803
$696$		0			0
$697$	1.61803			1.61803
$698$		0			0
$699$		0			0
$700$	0.809017	−	2.48990i	0.809017	−	2.48990i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$715$		0			0
$716$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$717$		−	1.90211i		−	1.90211i
$718$	0.500000	−	0.363271i	0.500000	−	0.363271i
$719$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$719$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$720$	0.427051	−	0.587785i	0.427051	−	0.587785i
$721$		0			0
$722$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$723$			0.726543i			0.726543i
$724$		0			0
$725$		0			0
$726$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−0.381966			−0.381966
$730$	1.80902	+	2.48990i	1.80902	+	2.48990i
$731$			1.17557i			1.17557i
$732$	−0.427051	+	1.31433i	−0.427051	+	1.31433i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$735$	2.23607			2.23607
$736$		0			0
$737$		0			0
$738$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$739$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$739$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$	0.690983	−	0.224514i	0.690983	−	0.224514i
$745$	2.23607			2.23607
$746$	0.690983	+	0.951057i	0.690983	+	0.951057i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	3.07768i		−	3.07768i
$756$	−0.690983	−	0.224514i	−0.690983	−	0.224514i
$757$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$757$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$763$		0			0
$764$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$765$			0.726543i			0.726543i
$766$		0			0
$767$		0			0
$768$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−0.263932	−	0.363271i	−0.263932	−	0.363271i
$775$	−1.61803			−1.61803
$776$	0.190983	+	0.587785i	0.190983	+	0.587785i
$777$		0			0
$778$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$			2.35114i			2.35114i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$795$			4.25325i			4.25325i
$796$	0.500000	−	1.53884i	0.500000	−	1.53884i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	2.61803			2.61803
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.690983	−	2.12663i	0.690983	−	2.12663i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	1.38197	+	1.90211i	1.38197	+	1.90211i
$811$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$811$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.690983	−	0.951057i	0.690983	−	0.951057i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	1.11803	+	1.53884i	1.11803	+	1.53884i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.00000			1.00000
$834$	1.80902	−	1.31433i	1.80902	−	1.31433i
$835$			3.07768i			3.07768i
$836$		0			0
$837$			0.449028i			0.449028i
$838$	1.11803	+	1.53884i	1.11803	+	1.53884i
$839$	2.00000			2.00000			1.00000			$0$
$839$	2.00000			2.00000			1.00000			$0$
$840$	0.690983	+	2.12663i	0.690983	+	2.12663i
$841$	−1.00000			−1.00000
$842$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$843$		−	0.726543i		−	0.726543i
$844$		0			0
$845$			1.90211i			1.90211i
$846$		0			0
$847$	1.00000			1.00000
$848$	1.11803	−	1.53884i	1.11803	−	1.53884i
$849$	−2.23607			−2.23607
$850$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$855$		0			0
$856$		0			0
$857$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$857$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$	0.690983	−	2.12663i	0.690983	−	2.12663i
$861$		−	1.90211i		−	1.90211i
$862$		0			0
$863$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$863$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$864$		−	0.726543i		−	0.726543i
$865$	−2.23607			−2.23607
$866$		0			0
$867$			1.17557i			1.17557i
$868$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.236068			0.236068
$874$		0			0
$875$		−	3.07768i		−	3.07768i
$876$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$879$		0			0
$880$		0			0
$881$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$881$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$882$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$883$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$883$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$887$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$	0.618034			0.618034
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−1.11803	+	0.812299i	−1.11803	+	0.812299i
$895$	−2.23607			−2.23607
$896$	0.309017	−	0.951057i	0.309017	−	0.951057i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.309017	−	0.951057i	0.309017	−	0.951057i
$901$			1.90211i			1.90211i
$902$		0			0
$903$	1.38197			1.38197
$904$		0			0
$905$		0			0
$906$	1.11803	+	1.53884i	1.11803	+	1.53884i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	1.80902	+	0.587785i	1.80902	+	0.587785i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$915$			2.62866i			2.62866i
$916$		0			0
$917$		0			0
$918$	0.427051	+	0.587785i	0.427051	+	0.587785i
$919$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$919$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$927$		0			0
$928$		0			0
$929$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$929$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$930$	1.11803	−	0.812299i	1.11803	−	0.812299i
$931$		0			0
$932$		0			0
$933$		−	0.726543i		−	0.726543i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	1.11803	+	1.53884i	1.11803	+	1.53884i
$939$		−	0.726543i		−	0.726543i
$940$		0			0
$941$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$941$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−1.38197			−1.38197
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.309017	+	0.951057i	0.309017	+	0.951057i
$953$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$953$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$954$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$955$			1.17557i			1.17557i
$956$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$957$		0			0
$958$	1.30902	−	0.951057i	1.30902	−	0.951057i
$959$	−1.61803			−1.61803
$960$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$961$	−0.618034			−0.618034
$962$		0			0
$963$		0			0
$964$	0.190983	−	0.587785i	0.190983	−	0.587785i
$965$		0			0
$966$		0			0
$967$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$967$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$968$	0.309017	+	0.951057i	0.309017	+	0.951057i
$969$		0			0
$970$	0.690983	+	0.951057i	0.690983	+	0.951057i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−0.690983	−	0.224514i	−0.690983	−	0.224514i
$973$			1.90211i			1.90211i
$974$		0			0
$975$		0			0
$976$	0.690983	−	0.951057i	0.690983	−	0.951057i
$977$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$977$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$978$		0			0
$979$		0			0
$980$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$981$		0			0
$982$	1.11803	+	1.53884i	1.11803	+	1.53884i
$983$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$983$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$984$	1.80902	−	0.587785i	1.80902	−	0.587785i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.618034			−0.618034
$993$	2.23607			2.23607
$994$		0			0
$995$		−	3.07768i		−	3.07768i
$996$		0			0
$997$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$997$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	952.1.e.d.237.3	yes	4
4.3	odd	2		3808.1.e.d.3569.2		4
7.6	odd	2		952.1.e.c.237.3	✓	4
8.3	odd	2		3808.1.e.d.3569.3		4
8.5	even	2	inner	952.1.e.d.237.4	yes	4
17.16	even	2		952.1.e.c.237.3	✓	4
28.27	even	2		3808.1.e.c.3569.3		4
56.13	odd	2		952.1.e.c.237.4	yes	4
56.27	even	2		3808.1.e.c.3569.2		4
68.67	odd	2		3808.1.e.c.3569.3		4
119.118	odd	2	CM	952.1.e.d.237.3	yes	4
136.67	odd	2		3808.1.e.c.3569.2		4
136.101	even	2		952.1.e.c.237.4	yes	4
476.475	even	2		3808.1.e.d.3569.2		4
952.237	odd	2	inner	952.1.e.d.237.4	yes	4
952.475	even	2		3808.1.e.d.3569.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.1.e.c.237.3	✓	4	7.6	odd	2
952.1.e.c.237.3	✓	4	17.16	even	2
952.1.e.c.237.4	yes	4	56.13	odd	2
952.1.e.c.237.4	yes	4	136.101	even	2
952.1.e.d.237.3	yes	4	1.1	even	1	trivial
952.1.e.d.237.3	yes	4	119.118	odd	2	CM
952.1.e.d.237.4	yes	4	8.5	even	2	inner
952.1.e.d.237.4	yes	4	952.237	odd	2	inner
3808.1.e.c.3569.2		4	56.27	even	2
3808.1.e.c.3569.2		4	136.67	odd	2
3808.1.e.c.3569.3		4	28.27	even	2
3808.1.e.c.3569.3		4	68.67	odd	2
3808.1.e.d.3569.2		4	4.3	odd	2
3808.1.e.d.3569.2		4	476.475	even	2
3808.1.e.d.3569.3		4	8.3	odd	2
3808.1.e.d.3569.3		4	952.475	even	2

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:	\( N \)	\(=\)	\( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	952.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 - 0.587785i) q^{2} +1.17557i q^{3} +(0.309017 - 0.951057i) q^{4} -1.90211i q^{5} +(0.690983 + 0.951057i) q^{6} -1.00000 q^{7} +(-0.309017 - 0.951057i) q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q+(0.809017 - 0.587785i) q^{2} +1.17557i q^{3} +(0.309017 - 0.951057i) q^{4} -1.90211i q^{5} +(0.690983 + 0.951057i) q^{6} -1.00000 q^{7} +(-0.309017 - 0.951057i) q^{8} -0.381966 q^{9} +(-1.11803 - 1.53884i) q^{10} +(1.11803 + 0.363271i) q^{12} +(-0.809017 + 0.587785i) q^{14} +2.23607 q^{15} +(-0.809017 - 0.587785i) q^{16} +1.00000 q^{17} +(-0.309017 + 0.224514i) q^{18} +(-1.80902 - 0.587785i) q^{20} -1.17557i q^{21} +(1.11803 - 0.363271i) q^{24} -2.61803 q^{25} +0.726543i q^{27} +(-0.309017 + 0.951057i) q^{28} +(1.80902 - 1.31433i) q^{30} +0.618034 q^{31} -1.00000 q^{32} +(0.809017 - 0.587785i) q^{34} +1.90211i q^{35} +(-0.118034 + 0.363271i) q^{36} +(-1.80902 + 0.587785i) q^{40} +1.61803 q^{41} +(-0.690983 - 0.951057i) q^{42} +1.17557i q^{43} +0.726543i q^{45} +(0.690983 - 0.951057i) q^{48} +1.00000 q^{49} +(-2.11803 + 1.53884i) q^{50} +1.17557i q^{51} +1.90211i q^{53} +(0.427051 + 0.587785i) q^{54} +(0.309017 + 0.951057i) q^{56} +(0.690983 - 2.12663i) q^{60} +1.17557i q^{61} +(0.500000 - 0.363271i) q^{62} +0.381966 q^{63} +(-0.809017 + 0.587785i) q^{64} -1.90211i q^{67} +(0.309017 - 0.951057i) q^{68} +(1.11803 + 1.53884i) q^{70} +(0.118034 + 0.363271i) q^{72} -1.61803 q^{73} -3.07768i q^{75} +(-1.11803 + 1.53884i) q^{80} -1.23607 q^{81} +(1.30902 - 0.951057i) q^{82} +(-1.11803 - 0.363271i) q^{84} -1.90211i q^{85} +(0.690983 + 0.951057i) q^{86} +(0.427051 + 0.587785i) q^{90} +0.726543i q^{93} -1.17557i q^{96} -0.618034 q^{97} +(0.809017 - 0.587785i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - q^{4} + 5 q^{6} - 4 q^{7} + q^{8} - 6 q^{9}+O(q^{10}) \) 4 * q + q^2 - q^4 + 5 * q^6 - 4 * q^7 + q^8 - 6 * q^9	\( 4 q + q^{2} - q^{4} + 5 q^{6} - 4 q^{7} + q^{8} - 6 q^{9} - q^{14} - q^{16} + 4 q^{17} + q^{18} - 5 q^{20} - 6 q^{25} + q^{28} + 5 q^{30} - 2 q^{31} - 4 q^{32} + q^{34} + 4 q^{36} - 5 q^{40} + 2 q^{41} - 5 q^{42} + 5 q^{48} + 4 q^{49} - 4 q^{50} - 5 q^{54} - q^{56} + 5 q^{60} + 2 q^{62} + 6 q^{63} - q^{64} - q^{68} - 4 q^{72} - 2 q^{73} + 4 q^{81} + 3 q^{82} + 5 q^{86} - 5 q^{90} + 2 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 - q^4 + 5 * q^6 - 4 * q^7 + q^8 - 6 * q^9 - q^14 - q^16 + 4 * q^17 + q^18 - 5 * q^20 - 6 * q^25 + q^28 + 5 * q^30 - 2 * q^31 - 4 * q^32 + q^34 + 4 * q^36 - 5 * q^40 + 2 * q^41 - 5 * q^42 + 5 * q^48 + 4 * q^49 - 4 * q^50 - 5 * q^54 - q^56 + 5 * q^60 + 2 * q^62 + 6 * q^63 - q^64 - q^68 - 4 * q^72 - 2 * q^73 + 4 * q^81 + 3 * q^82 + 5 * q^86 - 5 * q^90 + 2 * q^97 + q^98

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