LMFDB - Embedded newform 1057.1.bq.a.503.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1057,1,Mod(20,1057)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1057, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 34]))

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

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

chi := DirichletCharacter("1057.20");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1057 = 7 \cdot 151 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1057.bq (of order $50$, degree $20$, 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:	$0.527511718353$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			503.1
Root			$-0.876307 + 0.481754i$ of defining polynomial
Character	$\chi$	$=$	1057.503
Dual form			1057.1.bq.a.601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.535827 + 0.844328i) q^{7} +(-0.837780 + 2.57842i) q^{8} +(0.0627905 - 0.998027i) q^{9} +O(q^{10})$	\(q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.535827 + 0.844328i) q^{7} +(-0.837780 + 2.57842i) q^{8} +(0.0627905 - 0.998027i) q^{9} +(-1.17950 - 1.10762i) q^{11} +(-0.116762 + 1.85588i) q^{14} +(-2.09011 + 1.51855i) q^{16} +(1.18532 - 1.43281i) q^{18} +(-0.563797 - 2.95553i) q^{22} +(-0.500000 - 1.53884i) q^{23} +(-0.992115 - 0.125333i) q^{25} +(-1.56675 + 1.89387i) q^{28} +(0.159566 + 0.836475i) q^{29} -2.09308 q^{32} +(2.38072 - 0.611264i) q^{36} +(0.688925 + 1.46404i) q^{37} +(0.781202 - 1.23098i) q^{43} +(1.69334 - 3.59852i) q^{44} +(0.929776 - 2.86156i) q^{46} +(-0.425779 + 0.904827i) q^{49} +(-1.35556 - 1.27295i) q^{50} +(-1.73879 + 0.955910i) q^{53} +(-2.62594 + 0.674226i) q^{56} +(-0.674229 + 1.43281i) q^{58} +(0.876307 - 0.481754i) q^{63} +(-1.05874 - 0.769217i) q^{64} +(-0.0800484 - 1.27233i) q^{67} +(0.0672897 + 0.106032i) q^{71} +(2.52073 + 0.998027i) q^{72} +(-0.563797 + 2.95553i) q^{74} +(0.303189 - 1.58937i) q^{77} +(1.72897 + 0.684547i) q^{79} +(-0.992115 - 0.125333i) q^{81} +(2.52073 - 0.998027i) q^{86} +(3.84407 - 2.11329i) q^{88} +(3.21748 - 2.33764i) q^{92} +(-1.62954 + 0.895846i) q^{98} +(-1.17950 + 1.10762i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4}+O(q^{10}) $ 20 * q - 5 * q^4	$ 20 q - 5 q^{4} - 5 q^{14} - 5 q^{16} - 5 q^{18} - 10 q^{23} - 5 q^{29} - 10 q^{32} - 10 q^{44} - 5 q^{53} - 5 q^{56} - 5 q^{58} - 5 q^{64} - 5 q^{67} - 5 q^{71} + 20 q^{72} + 20 q^{79} + 20 q^{86} + 15 q^{88} + 15 q^{92}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^14 - 5 * q^16 - 5 * q^18 - 10 * q^23 - 5 * q^29 - 10 * q^32 - 10 * q^44 - 5 * q^53 - 5 * q^56 - 5 * q^58 - 5 * q^64 - 5 * q^67 - 5 * q^71 + 20 * q^72 + 20 * q^79 + 20 * q^86 + 15 * q^88 + 15 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1057\mathbb{Z}\right)^\times$.

$n$	$605$	$610$
$\chi(n)$	$-1$	$e\left(\frac{24}{25}\right)$

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 $

$2$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$2$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$3$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$3$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$4$	0.759544	+	2.33764i	0.759544	+	2.33764i
$5$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$5$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$6$		0			0
$7$	0.535827	+	0.844328i	0.535827	+	0.844328i
$7$	0.535827	+	0.844328i	0.535827	+	0.844328i
$8$	−0.837780	+	2.57842i	−0.837780	+	2.57842i
$9$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$10$		0			0
$11$	−1.17950	−	1.10762i	−1.17950	−	1.10762i	−0.992115	−	0.125333i	$-0.960000\pi$
$11$	−1.17950	−	1.10762i	−1.17950	−	1.10762i	−0.187381	−	0.982287i	$-0.560000\pi$
$12$		0			0
$13$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$13$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$14$	−0.116762	+	1.85588i	−0.116762	+	1.85588i
$15$		0			0
$16$	−2.09011	+	1.51855i	−2.09011	+	1.51855i
$17$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$17$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$18$	1.18532	−	1.43281i	1.18532	−	1.43281i
$19$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$19$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$20$		0			0
$21$		0			0
$22$	−0.563797	−	2.95553i	−0.563797	−	2.95553i
$23$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$23$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$24$		0			0
$25$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$26$		0			0
$27$		0			0
$28$	−1.56675	+	1.89387i	−1.56675	+	1.89387i
$29$	0.159566	+	0.836475i	0.159566	+	0.836475i	0.968583	+	0.248690i	$0.0800000\pi$
$29$	0.159566	+	0.836475i	0.159566	+	0.836475i	−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$31$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$32$	−2.09308			−2.09308
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.38072	−	0.611264i	2.38072	−	0.611264i
$37$	0.688925	+	1.46404i	0.688925	+	1.46404i	0.876307	+	0.481754i	$0.160000\pi$
$37$	0.688925	+	1.46404i	0.688925	+	1.46404i	−0.187381	+	0.982287i	$0.560000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$41$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$42$		0			0
$43$	0.781202	−	1.23098i	0.781202	−	1.23098i	−0.187381	−	0.982287i	$-0.560000\pi$
$43$	0.781202	−	1.23098i	0.781202	−	1.23098i	0.968583	−	0.248690i	$-0.0800000\pi$
$44$	1.69334	−	3.59852i	1.69334	−	3.59852i
$45$		0			0
$46$	0.929776	−	2.86156i	0.929776	−	2.86156i
$47$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$47$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$48$		0			0
$49$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$50$	−1.35556	−	1.27295i	−1.35556	−	1.27295i
$51$		0			0
$52$		0			0
$53$	−1.73879	+	0.955910i	−1.73879	+	0.955910i	−0.809017	+	0.587785i	$0.800000\pi$
$53$	−1.73879	+	0.955910i	−1.73879	+	0.955910i	−0.929776	+	0.368125i	$0.880000\pi$
$54$		0			0
$55$		0			0
$56$	−2.62594	+	0.674226i	−2.62594	+	0.674226i
$57$		0			0
$58$	−0.674229	+	1.43281i	−0.674229	+	1.43281i
$59$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$59$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$60$		0			0
$61$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$61$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$62$		0			0
$63$	0.876307	−	0.481754i	0.876307	−	0.481754i
$64$	−1.05874	−	0.769217i	−1.05874	−	0.769217i
$65$		0			0
$66$		0			0
$67$	−0.0800484	−	1.27233i	−0.0800484	−	1.27233i	−0.809017	−	0.587785i	$-0.800000\pi$
$67$	−0.0800484	−	1.27233i	−0.0800484	−	1.27233i	0.728969	−	0.684547i	$-0.240000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.0672897	+	0.106032i	0.0672897	+	0.106032i	0.876307	−	0.481754i	$-0.160000\pi$
$71$	0.0672897	+	0.106032i	0.0672897	+	0.106032i	−0.809017	+	0.587785i	$0.800000\pi$
$72$	2.52073	+	0.998027i	2.52073	+	0.998027i
$73$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$73$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$74$	−0.563797	+	2.95553i	−0.563797	+	2.95553i
$75$		0			0
$76$		0			0
$77$	0.303189	−	1.58937i	0.303189	−	1.58937i
$78$		0			0
$79$	1.72897	+	0.684547i	1.72897	+	0.684547i	1.00000			$0$
$79$	1.72897	+	0.684547i	1.72897	+	0.684547i	0.728969	+	0.684547i	$0.240000\pi$
$80$		0			0
$81$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$82$		0			0
$83$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$83$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$84$		0			0
$85$		0			0
$86$	2.52073	−	0.998027i	2.52073	−	0.998027i
$87$		0			0
$88$	3.84407	−	2.11329i	3.84407	−	2.11329i
$89$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$89$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$90$		0			0
$91$		0			0
$92$	3.21748	−	2.33764i	3.21748	−	2.33764i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$97$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$98$	−1.62954	+	0.895846i	−1.62954	+	0.895846i
$99$	−1.17950	+	1.10762i	−1.17950	+	1.10762i
$100$	−0.460572	−	2.41440i	−0.460572	−	2.41440i
$101$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$101$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$102$		0			0
$103$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$103$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$104$		0			0
$105$		0			0
$106$	−3.66068	−	0.462452i	−3.66068	−	0.462452i
$107$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	0.309017	+	0.951057i	$0.400000\pi$
$107$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	−0.929776	+	0.368125i	$0.880000\pi$
$108$		0			0
$109$	0.812619	+	0.982287i	0.812619	+	0.982287i	1.00000			$0$
$109$	0.812619	+	0.982287i	0.812619	+	0.982287i	−0.187381	+	0.982287i	$0.560000\pi$
$110$		0			0
$111$		0			0
$112$	−2.40209	−	0.951057i	−2.40209	−	0.951057i
$113$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$113$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$114$		0			0
$115$		0			0
$116$	−1.83418	+	1.00835i	−1.83418	+	1.00835i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.101597	+	1.61484i	0.101597	+	1.61484i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.84489	+	0.233064i	1.84489	+	0.233064i
$127$	−0.824805	−	1.75280i	−0.824805	−	1.75280i	−0.637424	−	0.770513i	$-0.720000\pi$
$127$	−0.824805	−	1.75280i	−0.824805	−	1.75280i	−0.187381	−	0.982287i	$-0.560000\pi$
$128$	−0.105209	−	0.323801i	−0.105209	−	0.323801i
$129$		0			0
$130$		0			0
$131$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$131$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$132$		0			0
$133$		0			0
$134$	1.27026	−	2.00160i	1.27026	−	2.00160i
$135$		0			0
$136$		0			0
$137$	0.110048	−	1.74915i	0.110048	−	1.74915i	−0.425779	−	0.904827i	$-0.640000\pi$
$137$	0.110048	−	1.74915i	0.110048	−	1.74915i	0.535827	−	0.844328i	$-0.320000\pi$
$138$		0			0
$139$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$139$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$140$		0			0
$141$		0			0
$142$	−0.0146631	+	0.233064i	−0.0146631	+	0.233064i
$143$		0			0
$144$	1.38432	+	2.18134i	1.38432	+	2.18134i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2.89913	+	2.72246i	−2.89913	+	2.72246i
$149$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$149$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$150$		0			0
$151$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$151$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$152$		0			0
$153$		0			0
$154$	2.19334	−	2.05968i	2.19334	−	2.05968i
$155$		0			0
$156$		0			0
$157$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$157$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$158$	1.85286	+	2.91963i	1.85286	+	2.91963i
$159$		0			0
$160$		0			0
$161$	1.03137	−	1.24672i	1.03137	−	1.24672i
$162$	−1.35556	−	1.27295i	−1.35556	−	1.27295i
$163$	−1.73879	−	0.955910i	−1.73879	−	0.955910i	−0.929776	−	0.368125i	$-0.880000\pi$
$163$	−1.73879	−	0.955910i	−1.73879	−	0.955910i	−0.809017	−	0.587785i	$-0.800000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$170$		0			0
$171$		0			0
$172$	3.47094	+	0.891185i	3.47094	+	0.891185i
$173$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$173$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$174$		0			0
$175$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$176$	4.14726	+	0.523921i	4.14726	+	0.523921i
$177$		0			0
$178$		0			0
$179$	0.238883	−	0.288760i	0.238883	−	0.288760i	−0.637424	−	0.770513i	$-0.720000\pi$
$179$	0.238883	−	0.288760i	0.238883	−	0.288760i	0.876307	+	0.481754i	$0.160000\pi$
$180$		0			0
$181$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$181$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$182$		0			0
$183$		0			0
$184$	4.38667			4.38667
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.876307	−	0.481754i	$-0.160000\pi$
$191$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$	1.26480	+	1.52888i	1.26480	+	1.52888i	0.728969	+	0.684547i	$0.240000\pi$
$193$	1.26480	+	1.52888i	1.26480	+	1.52888i	0.535827	+	0.844328i	$0.320000\pi$
$194$		0			0
$195$		0			0
$196$	−2.43856	−	0.308061i	−2.43856	−	0.308061i
$197$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$197$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$198$	−2.98509	+	0.377105i	−2.98509	+	0.377105i
$199$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$199$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$200$	1.15434	−	2.45309i	1.15434	−	2.45309i
$201$		0			0
$202$		0			0
$203$	−0.620759	+	0.582932i	−0.620759	+	0.582932i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.56720	+	0.402389i	−1.56720	+	0.402389i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.683098	+	1.07639i	−0.683098	+	1.07639i	0.309017	+	0.951057i	$0.400000\pi$
$211$	−0.683098	+	1.07639i	−0.683098	+	1.07639i	−0.992115	+	0.125333i	$0.960000\pi$
$212$	−3.55526	−	3.33861i	−3.55526	−	3.33861i
$213$		0			0
$214$	−2.37577	+	1.30609i	−2.37577	+	1.30609i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.148854	+	2.36597i	0.148854	+	2.36597i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$223$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$224$	−1.12153	−	1.76724i	−1.12153	−	1.76724i
$225$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$226$	1.61221	−	1.17134i	1.61221	−	1.17134i
$227$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$227$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$228$		0			0
$229$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$229$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$230$		0			0
$231$		0			0
$232$	−2.29047	−	0.289353i	−2.29047	−	0.289353i
$233$	−0.328407	−	0.180543i	−0.328407	−	0.180543i	0.309017	−	0.951057i	$-0.400000\pi$
$233$	−0.328407	−	0.180543i	−0.328407	−	0.180543i	−0.637424	+	0.770513i	$0.720000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.939097	−	0.516273i	0.939097	−	0.516273i	0.0627905	−	0.998027i	$-0.480000\pi$
$239$	0.939097	−	0.516273i	0.939097	−	0.516273i	0.876307	+	0.481754i	$0.160000\pi$
$240$		0			0
$241$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$241$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$242$	−1.61221	+	2.54043i	−1.61221	+	2.54043i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$251$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$252$	1.79176	+	1.68257i	1.79176	+	1.68257i
$253$	−1.11470	+	2.36887i	−1.11470	+	2.36887i
$254$	0.674997	−	3.53846i	0.674997	−	3.53846i
$255$		0			0
$256$	−0.208759	+	0.642495i	−0.208759	+	0.642495i
$257$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$257$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$258$		0			0
$259$	−0.866986	+	1.36615i	−0.866986	+	1.36615i
$260$		0			0
$261$	0.844844	−	0.106729i	0.844844	−	0.106729i
$262$		0			0
$263$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$263$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.809017	+	0.587785i	$0.800000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.91345	−	1.15352i	2.91345	−	1.15352i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$271$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$272$		0			0
$273$		0			0
$274$	2.07741	−	2.51116i	2.07741	−	2.51116i
$275$	1.03137	+	1.24672i	1.03137	+	1.24672i
$276$		0			0
$277$	−1.92189	−	0.242791i	−1.92189	−	0.242791i	−0.929776	−	0.368125i	$-0.880000\pi$
$277$	−1.92189	−	0.242791i	−1.92189	−	0.242791i	−0.992115	+	0.125333i	$0.960000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.598617	+	0.153699i	0.598617	+	0.153699i	0.535827	−	0.844328i	$-0.320000\pi$
$281$	0.598617	+	0.153699i	0.598617	+	0.153699i	0.0627905	+	0.998027i	$0.480000\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$	−0.196754	+	0.237835i	−0.196754	+	0.237835i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.131425	+	2.08895i	−0.131425	+	2.08895i
$289$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$293$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$294$		0			0
$295$		0			0
$296$	−4.35208	+	0.549796i	−4.35208	+	0.549796i
$297$		0			0
$298$	−0.0721631	−	0.222095i	−0.0721631	−	0.222095i
$299$		0			0
$300$		0			0
$301$	1.45794			1.45794
$302$	−1.80113	+	0.462452i	−1.80113	+	0.462452i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$307$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$308$	3.94567	−	0.498454i	3.94567	−	0.498454i
$309$		0			0
$310$		0			0
$311$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$311$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$312$		0			0
$313$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$313$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$314$		0			0
$315$		0			0
$316$	−0.286994	+	4.56165i	−0.286994	+	4.56165i
$317$	0.542804	+	0.656137i	0.542804	+	0.656137i	0.968583	−	0.248690i	$-0.0800000\pi$
$317$	0.542804	+	0.656137i	0.542804	+	0.656137i	−0.425779	+	0.904827i	$0.640000\pi$
$318$		0			0
$319$	0.738289	−	1.16336i	0.738289	−	1.16336i
$320$		0			0
$321$		0			0
$322$	2.91429	−	0.748263i	2.91429	−	0.748263i
$323$		0			0
$324$	−0.460572	−	2.41440i	−0.460572	−	2.41440i
$325$		0			0
$326$	−1.57103	−	3.33861i	−1.57103	−	3.33861i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$331$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	−0.992115	+	0.125333i	$0.960000\pi$
$332$		0			0
$333$	1.50441	−	0.595638i	1.50441	−	0.595638i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.53583	−	0.844328i	1.53583	−	0.844328i	0.535827	−	0.844328i	$-0.320000\pi$
$337$	1.53583	−	0.844328i	1.53583	−	0.844328i	1.00000			$0$
$338$	−1.80113	+	0.462452i	−1.80113	+	0.462452i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$344$	2.51950	+	3.04555i	2.51950	+	3.04555i
$345$		0			0
$346$		0			0
$347$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.728969	−	0.684547i	$-0.240000\pi$
$347$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.535827	+	0.844328i	$0.320000\pi$
$348$		0			0
$349$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$349$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$350$	0.348445	−	1.82662i	0.348445	−	1.82662i
$351$		0			0
$352$	2.46878	+	2.31834i	2.46878	+	2.31834i
$353$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$353$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.674997	−	0.173310i	0.674997	−	0.173310i
$359$	0.110048	+	0.0604991i	0.110048	+	0.0604991i	0.535827	−	0.844328i	$-0.320000\pi$
$359$	0.110048	+	0.0604991i	0.110048	+	0.0604991i	−0.425779	+	0.904827i	$0.640000\pi$
$360$		0			0
$361$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$367$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$368$	3.38187	+	2.45707i	3.38187	+	2.45707i
$369$		0			0
$370$		0			0
$371$	−1.73879	−	0.955910i	−1.73879	−	0.955910i
$372$		0			0
$373$	0.939097	+	1.47978i	0.939097	+	1.47978i	0.876307	+	0.481754i	$0.160000\pi$
$373$	0.939097	+	1.47978i	0.939097	+	1.47978i	0.0627905	+	0.998027i	$0.480000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.362989	+	1.90285i	−0.362989	+	1.90285i	0.0627905	+	0.998027i	$0.480000\pi$
$379$	−0.362989	+	1.90285i	−0.362989	+	1.90285i	−0.425779	+	0.904827i	$0.640000\pi$
$380$		0			0
$381$		0			0
$382$	1.27026	+	2.00160i	1.27026	+	2.00160i
$383$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$383$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$384$		0			0
$385$		0			0
$386$	0.231683	+	3.68250i	0.231683	+	3.68250i
$387$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$388$		0			0
$389$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$389$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$390$		0			0
$391$		0			0
$392$	−1.97632	−	1.85588i	−1.97632	−	1.85588i
$393$		0			0
$394$	−2.79753	+	2.03252i	−2.79753	+	2.03252i
$395$		0			0
$396$	−3.48509	−	1.91595i	−3.48509	−	1.91595i
$397$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$397$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$398$		0			0
$399$		0			0
$400$	2.26395	−	1.24462i	2.26395	−	1.24462i
$401$	1.45794	−	1.36909i	1.45794	−	1.36909i	0.728969	−	0.684547i	$-0.240000\pi$
$401$	1.45794	−	1.36909i	1.45794	−	1.36909i	0.728969	−	0.684547i	$-0.240000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−1.57103	+	0.198467i	−1.57103	+	0.198467i
$407$	0.809017	−	2.48990i	0.809017	−	2.48990i
$408$		0			0
$409$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$409$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−2.79753	−	1.10762i	−2.79753	−	1.10762i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$419$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$420$		0			0
$421$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$421$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$422$	−2.20417	+	0.872693i	−2.20417	+	0.872693i
$423$		0			0
$424$	−1.00801	−	5.28418i	−1.00801	−	5.28418i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−3.55526	−	0.449134i	−3.55526	−	0.449134i
$429$		0			0
$430$		0			0
$431$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	−0.637424	−	0.770513i	$-0.720000\pi$
$431$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	0.309017	−	0.951057i	$-0.400000\pi$
$432$		0			0
$433$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$433$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$434$		0			0
$435$		0			0
$436$	−1.67901	+	2.64570i	−1.67901	+	2.64570i
$437$		0			0
$438$		0			0
$439$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$439$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$440$		0			0
$441$	0.876307	+	0.481754i	0.876307	+	0.481754i
$442$		0			0
$443$	−0.393950	+	0.476203i	−0.393950	+	0.476203i	−0.929776	−	0.368125i	$-0.880000\pi$
$443$	−0.393950	+	0.476203i	−0.393950	+	0.476203i	0.535827	+	0.844328i	$0.320000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0821721	−	1.30609i	0.0821721	−	1.30609i
$449$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$449$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$450$	−1.35556	+	1.27295i	−1.35556	+	1.27295i
$451$		0			0
$452$	2.63406			2.63406
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$457$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$461$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$462$		0			0
$463$	1.18532	−	1.43281i	1.18532	−	1.43281i	0.309017	−	0.951057i	$-0.400000\pi$
$463$	1.18532	−	1.43281i	1.18532	−	1.43281i	0.876307	−	0.481754i	$-0.160000\pi$
$464$	−1.60374	−	1.50602i	−1.60374	−	1.50602i
$465$		0			0
$466$	−0.296722	−	0.630566i	−0.296722	−	0.630566i
$467$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$467$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$468$		0			0
$469$	1.03137	−	0.749337i	1.03137	−	0.749337i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.28488	+	0.586657i	−2.28488	+	0.586657i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.844844	+	1.79538i	0.844844	+	1.79538i
$478$	1.97708	+	0.249764i	1.97708	+	0.249764i
$479$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$479$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−3.69775	+	1.46404i	−3.69775	+	1.46404i
$485$		0			0
$486$		0			0
$487$	0.791759	−	0.313480i	0.791759	−	0.313480i	0.0627905	−	0.998027i	$-0.480000\pi$
$487$	0.791759	−	0.313480i	0.791759	−	0.313480i	0.728969	+	0.684547i	$0.240000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$491$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.0534698	+	0.113629i	−0.0534698	+	0.113629i
$498$		0			0
$499$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$499$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$503$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$504$	0.508012	+	2.66309i	0.508012	+	2.66309i
$505$		0			0
$506$	−4.26619	+	2.34536i	−4.26619	+	2.34536i
$507$		0			0
$508$	3.47094	−	3.25943i	3.47094	−	3.25943i
$509$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$509$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$510$		0			0
$511$		0			0
$512$	−1.29176	+	0.938518i	−1.29176	+	0.938518i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−2.79753	+	1.10762i	−2.79753	+	1.10762i
$519$		0			0
$520$		0			0
$521$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$521$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$522$	1.38765	+	0.762866i	1.38765	+	0.762866i
$523$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$523$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$524$		0			0
$525$		0			0
$526$	−1.06779	−	1.68257i	−1.06779	−	1.68257i
$527$		0			0
$528$		0			0
$529$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	3.34767	+	0.859536i	3.34767	+	0.859536i
$537$		0			0
$538$		0			0
$539$	1.50441	−	0.595638i	1.50441	−	0.595638i
$540$		0			0
$541$	0.939097	−	0.516273i	0.939097	−	0.516273i	0.0627905	−	0.998027i	$-0.480000\pi$
$541$	0.939097	−	0.516273i	0.939097	−	0.516273i	0.876307	+	0.481754i	$0.160000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.62954	−	0.895846i	−1.62954	−	0.895846i	−0.992115	−	0.125333i	$-0.960000\pi$
$547$	−1.62954	−	0.895846i	−1.62954	−	0.895846i	−0.637424	−	0.770513i	$-0.720000\pi$
$548$	4.17248	−	1.07131i	4.17248	−	1.07131i
$549$		0			0
$550$	0.188925	+	3.00288i	0.188925	+	3.00288i
$551$		0			0
$552$		0			0
$553$	0.348445	+	1.82662i	0.348445	+	1.82662i
$554$	−2.62594	−	2.46592i	−2.62594	−	2.46592i
$555$		0			0
$556$		0			0
$557$	1.96858	−	0.248690i	1.96858	−	0.248690i	0.968583	−	0.248690i	$-0.0800000\pi$
$557$	1.96858	−	0.248690i	1.96858	−	0.248690i	1.00000			$0$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.732570	+	0.885525i	0.732570	+	0.885525i
$563$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$563$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$568$	−0.329768	+	0.0846700i	−0.329768	+	0.0846700i
$569$	−0.328407	+	0.180543i	−0.328407	+	0.180543i	−0.637424	−	0.770513i	$-0.720000\pi$
$569$	−0.328407	+	0.180543i	−0.328407	+	0.180543i	0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$571$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.303189	+	1.58937i	0.303189	+	1.58937i
$576$	−0.834178	+	1.00835i	−0.834178	+	1.00835i
$577$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$577$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$578$	0.348445	−	1.82662i	0.348445	−	1.82662i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.10969	+	0.798431i	3.10969	+	0.798431i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$587$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−3.66316	−	2.01384i	−3.66316	−	2.01384i
$593$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$593$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$594$		0			0
$595$		0			0
$596$	0.0953844	−	0.293563i	0.0953844	−	0.293563i
$597$		0			0
$598$		0			0
$599$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i	0.876307	+	0.481754i	$0.160000\pi$
$599$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i	−0.929776	+	0.368125i	$0.880000\pi$
$600$		0			0
$601$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$601$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$602$	2.19334	+	1.59355i	2.19334	+	1.59355i
$603$	−1.27485			−1.27485
$604$	−2.28533	−	0.904827i	−2.28533	−	0.904827i
$605$		0			0
$606$		0			0
$607$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$607$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.110048	−	1.74915i	0.110048	−	1.74915i	−0.425779	−	0.904827i	$-0.640000\pi$
$613$	0.110048	−	1.74915i	0.110048	−	1.74915i	0.535827	−	0.844328i	$-0.320000\pi$
$614$		0			0
$615$		0			0
$616$	3.84407	+	2.11329i	3.84407	+	2.11329i
$617$	0.688925	+	1.46404i	0.688925	+	1.46404i	0.876307	+	0.481754i	$0.160000\pi$
$617$	0.688925	+	1.46404i	0.688925	+	1.46404i	−0.187381	+	0.982287i	$0.560000\pi$
$618$		0			0
$619$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$619$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.968583	+	0.248690i	0.968583	+	0.248690i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	0.535827	−	0.844328i	$-0.320000\pi$
$631$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	−0.929776	+	0.368125i	$0.880000\pi$
$632$	−3.21354	+	3.88451i	−3.21354	+	3.88451i
$633$		0			0
$634$	0.0994299	+	1.58039i	0.0994299	+	1.58039i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	2.38226	−	0.943204i	2.38226	−	0.943204i
$639$	0.110048	−	0.0604991i	0.110048	−	0.0604991i
$640$		0			0
$641$	0.542804	+	1.15352i	0.542804	+	1.15352i	0.968583	+	0.248690i	$0.0800000\pi$
$641$	0.542804	+	1.15352i	0.542804	+	1.15352i	−0.425779	+	0.904827i	$0.640000\pi$
$642$		0			0
$643$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$643$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$644$	3.69775	+	1.46404i	3.69775	+	1.46404i
$645$		0			0
$646$		0			0
$647$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$647$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$648$	1.15434	−	2.45309i	1.15434	−	2.45309i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.913880	−	4.79072i	0.913880	−	4.79072i
$653$	0.362576	−	0.770513i	0.362576	−	0.770513i	−0.637424	−	0.770513i	$-0.720000\pi$
$653$	0.362576	−	0.770513i	0.362576	−	0.770513i	1.00000			$0$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	−0.809017	−	0.587785i	$-0.800000\pi$
$659$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	0.535827	+	0.844328i	$0.320000\pi$
$660$		0			0
$661$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$661$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$662$	0.0994299	−	0.211299i	0.0994299	−	0.211299i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	2.91429	+	0.748263i	2.91429	+	0.748263i
$667$	1.20742	−	0.663785i	1.20742	−	0.663785i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.69755	+	0.933237i	1.69755	+	0.933237i	0.968583	+	0.248690i	$0.0800000\pi$
$673$	1.69755	+	0.933237i	1.69755	+	0.933237i	0.728969	+	0.684547i	$0.240000\pi$
$674$	3.23338	+	0.408471i	3.23338	+	0.408471i
$675$		0			0
$676$	−2.28533	−	0.904827i	−2.28533	−	0.904827i
$677$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$677$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.80113	−	0.713118i	−1.80113	−	0.713118i	−0.992115	−	0.125333i	$-0.960000\pi$
$683$	−1.80113	−	0.713118i	−1.80113	−	0.713118i	−0.809017	−	0.587785i	$-0.800000\pi$
$684$		0			0
$685$		0			0
$686$	−1.62954	−	0.895846i	−1.62954	−	0.895846i
$687$		0			0
$688$	0.236507	+	3.75918i	0.236507	+	3.75918i
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$	−1.56720	−	0.402389i	−1.56720	−	0.402389i
$694$	1.72813	+	1.62282i	1.72813	+	1.62282i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.79176	−	1.68257i	1.79176	−	1.68257i
$701$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	1.00000			$0$
$701$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	−0.992115	+	0.125333i	$0.960000\pi$
$702$		0			0
$703$		0			0
$704$	0.396775	+	2.07997i	0.396775	+	2.07997i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$709$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$710$		0			0
$711$	0.791759	−	1.68257i	0.791759	−	1.68257i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.856457	+	0.339095i	0.856457	+	0.339095i
$717$		0			0
$718$	0.0994299	+	0.211299i	0.0994299	+	0.211299i
$719$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$719$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$720$		0			0
$721$		0			0
$722$	−1.85955			−1.85955
$723$		0			0
$724$		0			0
$725$	−0.0534698	−	0.849878i	−0.0534698	−	0.849878i
$726$		0			0
$727$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$727$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$728$		0			0
$729$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$733$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$734$		0			0
$735$		0			0
$736$	1.04654	+	3.22092i	1.04654	+	3.22092i
$737$	−1.31484	+	1.58937i	−1.31484	+	1.58937i
$738$		0			0
$739$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$739$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$740$		0			0
$741$		0			0
$742$	−1.57103	−	3.33861i	−1.57103	−	3.33861i
$743$	1.27760	+	0.702367i	1.27760	+	0.702367i	0.968583	−	0.248690i	$-0.0800000\pi$
$743$	1.27760	+	0.702367i	1.27760	+	0.702367i	0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$		0			0
$746$	−0.204639	+	3.25265i	−0.204639	+	3.25265i
$747$		0			0
$748$		0			0
$749$	−1.44644	+	0.182728i	−1.44644	+	0.182728i
$750$		0			0
$751$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$751$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$757$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$758$	−2.62594	+	2.46592i	−2.62594	+	2.46592i
$759$		0			0
$760$		0			0
$761$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$761$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$762$		0			0
$763$	−0.393950	+	1.21245i	−0.393950	+	1.21245i
$764$	−0.196754	+	3.12731i	−0.196754	+	3.12731i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$769$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$770$		0			0
$771$		0			0
$772$	−2.61329	+	4.11788i	−2.61329	+	4.11788i
$773$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$773$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$774$	−0.837780	−	2.57842i	−0.837780	−	2.57842i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.00711	−	3.09957i	−1.00711	−	3.09957i
$779$		0			0
$780$		0			0
$781$	0.0380748	−	0.199595i	0.0380748	−	0.199595i
$782$		0			0
$783$		0			0
$784$	−0.484103	−	2.53776i	−0.484103	−	2.53776i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−4.57066			−4.57066
$789$		0			0
$790$		0			0
$791$	1.03799	−	0.266509i	1.03799	−	0.266509i
$792$	−1.86775	−	3.96918i	−1.86775	−	3.96918i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$797$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$798$		0			0
$799$		0			0
$800$	2.07657	+	0.262332i	2.07657	+	0.262332i
$801$		0			0
$802$	3.68978	−	0.466128i	3.68978	−	0.466128i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.0534698	−	0.849878i	−0.0534698	−	0.849878i	−0.929776	−	0.368125i	$-0.880000\pi$
$809$	−0.0534698	−	0.849878i	−0.0534698	−	0.849878i	0.876307	−	0.481754i	$-0.160000\pi$
$810$		0			0
$811$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$811$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$812$	−1.83418	−	1.00835i	−1.83418	−	1.00835i
$813$		0			0
$814$	3.93860	−	2.86156i	3.93860	−	2.86156i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$821$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$822$		0			0
$823$	0.121636	+	0.0312307i	0.121636	+	0.0312307i	0.309017	−	0.951057i	$-0.400000\pi$
$823$	0.121636	+	0.0312307i	0.121636	+	0.0312307i	−0.187381	+	0.982287i	$0.560000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.968583	+	0.248690i	$0.0800000\pi$
$827$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.876307	+	0.481754i	$0.160000\pi$
$828$	−2.13100	−	3.35791i	−2.13100	−	3.35791i
$829$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$829$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$839$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$840$		0			0
$841$	0.255547	−	0.101178i	0.255547	−	0.101178i
$842$	0.929776	+	0.675522i	0.929776	+	0.675522i
$843$		0			0
$844$	−3.03505	−	0.779269i	−3.03505	−	0.779269i
$845$		0			0
$846$		0			0
$847$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$848$	2.18267	−	4.63841i	2.18267	−	4.63841i
$849$		0			0
$850$		0			0
$851$	1.90846	−	1.79217i	1.90846	−	1.79217i
$852$		0			0
$853$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$853$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$854$		0			0
$855$		0			0
$856$	−2.88134	−	2.70576i	−2.88134	−	2.70576i
$857$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$857$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$858$		0			0
$859$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$859$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$860$		0			0
$861$		0			0
$862$	1.38765	−	2.94890i	1.38765	−	2.94890i
$863$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.187381	+	0.982287i	$0.560000\pi$
$863$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.28109	−	2.72246i	−1.28109	−	2.72246i
$870$		0			0
$871$		0			0
$872$	−3.21354	+	1.27233i	−3.21354	+	1.27233i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$877$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	−0.992115	+	0.125333i	$0.960000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$881$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$882$	0.791759	+	1.68257i	0.791759	+	1.68257i
$883$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$883$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$884$		0			0
$885$		0			0
$886$	−1.11316	+	0.285811i	−1.11316	+	0.285811i
$887$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$887$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$888$		0			0
$889$	1.03799	−	1.63560i	1.03799	−	1.63560i
$890$		0			0
$891$	1.03137	+	1.24672i	1.03137	+	1.24672i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.217020	−	0.262332i	0.217020	−	0.262332i
$897$		0			0
$898$	0.215351	−	0.662783i	0.215351	−	0.662783i
$899$		0			0
$900$	−2.43856	+	0.308061i	−2.43856	+	0.308061i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	2.35050	+	1.70774i	2.35050	+	1.70774i
$905$		0			0
$906$		0			0
$907$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$907$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.124591	+	1.98031i	−0.124591	+	1.98031i	0.0627905	+	0.998027i	$0.480000\pi$
$911$	−0.124591	+	1.98031i	−0.124591	+	1.98031i	−0.187381	+	0.982287i	$0.560000\pi$
$912$		0			0
$913$		0			0
$914$	1.06856	−	3.28869i	1.06856	−	3.28869i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.851559	−	1.80965i	−0.851559	−	1.80965i	−0.425779	−	0.904827i	$-0.640000\pi$
$919$	−0.851559	−	1.80965i	−0.851559	−	1.80965i	−0.425779	−	0.904827i	$-0.640000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$926$	3.34930	−	0.859954i	3.34930	−	0.859954i
$927$		0			0
$928$	−0.333984	−	1.75081i	−0.333984	−	1.75081i
$929$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$929$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$		0			0
$932$	0.172605	−	0.904827i	0.172605	−	0.904827i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$937$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$938$	2.37065			2.37065
$939$		0			0
$940$		0			0
$941$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$941$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−4.07862	−	1.61484i	−4.07862	−	1.61484i
$947$	−1.92189	+	0.242791i	−1.92189	+	0.242791i	−0.992115	−	0.125333i	$-0.960000\pi$
$947$	−1.92189	+	0.242791i	−1.92189	+	0.242791i	−0.929776	+	0.368125i	$0.880000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.73879	+	0.219661i	−1.73879	+	0.219661i	−0.929776	−	0.368125i	$-0.880000\pi$
$953$	−1.73879	+	0.219661i	−1.73879	+	0.219661i	−0.809017	+	0.587785i	$0.800000\pi$
$954$	−0.691396	+	3.62442i	−0.691396	+	3.62442i
$955$		0			0
$956$	1.92015	+	1.80314i	1.92015	+	1.80314i
$957$		0			0
$958$		0			0
$959$	1.53583	−	0.844328i	1.53583	−	0.844328i
$960$		0			0
$961$	0.728969	−	0.684547i	0.728969	−	0.684547i
$962$		0			0
$963$	1.27760	+	0.702367i	1.27760	+	0.702367i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.273190	−	0.256543i	−0.273190	−	0.256543i	0.535827	−	0.844328i	$-0.320000\pi$
$967$	−0.273190	−	0.256543i	−0.273190	−	0.256543i	−0.809017	+	0.587785i	$0.800000\pi$
$968$	−4.24886	−	1.09092i	−4.24886	−	1.09092i
$969$		0			0
$970$		0			0
$971$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$971$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$972$		0			0
$973$		0			0
$974$	1.53377	+	0.393805i	1.53377	+	0.393805i
$975$		0			0
$976$		0			0
$977$	0.574221	+	0.904827i	0.574221	+	0.904827i	1.00000			$0$
$977$	0.574221	+	0.904827i	0.574221	+	0.904827i	−0.425779	+	0.904827i	$0.640000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.03137	−	0.749337i	1.03137	−	0.749337i
$982$	2.63665	−	1.91564i	2.63665	−	1.91564i
$983$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$983$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.28488	−	0.586657i	−2.28488	−	0.586657i
$990$		0			0
$991$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$991$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$992$		0			0
$993$		0			0
$994$	−0.204639	+	0.112501i	−0.204639	+	0.112501i
$995$		0			0
$996$		0			0
$997$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$997$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$998$	−2.98509	+	2.16880i	−2.98509	+	2.16880i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1057.1.bq.a.503.1	✓	20
7.6	odd	2	CM	1057.1.bq.a.503.1	✓	20
151.148	even	25	inner	1057.1.bq.a.601.1	yes	20
1057.601	odd	50	inner	1057.1.bq.a.601.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1057.1.bq.a.503.1	✓	20	1.1	even	1	trivial
1057.1.bq.a.503.1	✓	20	7.6	odd	2	CM
1057.1.bq.a.601.1	yes	20	151.148	even	25	inner
1057.1.bq.a.601.1	yes	20	1057.601	odd	50	inner

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:	\( N \)	\(=\)	\( 1057 = 7 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1057.bq (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.535827 + 0.844328i) q^{7} +(-0.837780 + 2.57842i) q^{8} +(0.0627905 - 0.998027i) q^{9} +O(q^{10})\)	\(q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.535827 + 0.844328i) q^{7} +(-0.837780 + 2.57842i) q^{8} +(0.0627905 - 0.998027i) q^{9} +(-1.17950 - 1.10762i) q^{11} +(-0.116762 + 1.85588i) q^{14} +(-2.09011 + 1.51855i) q^{16} +(1.18532 - 1.43281i) q^{18} +(-0.563797 - 2.95553i) q^{22} +(-0.500000 - 1.53884i) q^{23} +(-0.992115 - 0.125333i) q^{25} +(-1.56675 + 1.89387i) q^{28} +(0.159566 + 0.836475i) q^{29} -2.09308 q^{32} +(2.38072 - 0.611264i) q^{36} +(0.688925 + 1.46404i) q^{37} +(0.781202 - 1.23098i) q^{43} +(1.69334 - 3.59852i) q^{44} +(0.929776 - 2.86156i) q^{46} +(-0.425779 + 0.904827i) q^{49} +(-1.35556 - 1.27295i) q^{50} +(-1.73879 + 0.955910i) q^{53} +(-2.62594 + 0.674226i) q^{56} +(-0.674229 + 1.43281i) q^{58} +(0.876307 - 0.481754i) q^{63} +(-1.05874 - 0.769217i) q^{64} +(-0.0800484 - 1.27233i) q^{67} +(0.0672897 + 0.106032i) q^{71} +(2.52073 + 0.998027i) q^{72} +(-0.563797 + 2.95553i) q^{74} +(0.303189 - 1.58937i) q^{77} +(1.72897 + 0.684547i) q^{79} +(-0.992115 - 0.125333i) q^{81} +(2.52073 - 0.998027i) q^{86} +(3.84407 - 2.11329i) q^{88} +(3.21748 - 2.33764i) q^{92} +(-1.62954 + 0.895846i) q^{98} +(-1.17950 + 1.10762i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4}+O(q^{10}) \) 20 * q - 5 * q^4	\( 20 q - 5 q^{4} - 5 q^{14} - 5 q^{16} - 5 q^{18} - 10 q^{23} - 5 q^{29} - 10 q^{32} - 10 q^{44} - 5 q^{53} - 5 q^{56} - 5 q^{58} - 5 q^{64} - 5 q^{67} - 5 q^{71} + 20 q^{72} + 20 q^{79} + 20 q^{86} + 15 q^{88} + 15 q^{92}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^14 - 5 * q^16 - 5 * q^18 - 10 * q^23 - 5 * q^29 - 10 * q^32 - 10 * q^44 - 5 * q^53 - 5 * q^56 - 5 * q^58 - 5 * q^64 - 5 * q^67 - 5 * q^71 + 20 * q^72 + 20 * q^79 + 20 * q^86 + 15 * q^88 + 15 * q^92

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