LMFDB - Embedded newform 1375.1.bz.a.769.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1375,1,Mod(54,1375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1375, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([11, 25]))

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

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

chi := DirichletCharacter("1375.54");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1375 = 5^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1375.bz (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:	$0.686214392370$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{50}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{50} - \cdots)$

Embedding invariants

Embedding label			769.1
Root			$-0.876307 + 0.481754i$ of defining polynomial
Character	$\chi$	$=$	1375.769
Dual form			1375.1.bz.a.329.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.503997 + 0.536702i) q^{3} +(0.425779 + 0.904827i) q^{4} +(0.809017 - 0.587785i) q^{5} +(0.0287541 - 0.457034i) q^{9} +O(q^{10})$	\(q+(0.503997 + 0.536702i) q^{3} +(0.425779 + 0.904827i) q^{4} +(0.809017 - 0.587785i) q^{5} +(0.0287541 - 0.457034i) q^{9} +(-0.535827 + 0.844328i) q^{11} +(-0.271031 + 0.684547i) q^{12} +(0.723208 + 0.137959i) q^{15} +(-0.637424 + 0.770513i) q^{16} +(0.876307 + 0.481754i) q^{20} +(-0.340480 + 1.32608i) q^{23} +(0.309017 - 0.951057i) q^{25} +(0.827073 - 0.684214i) q^{27} +(0.824805 - 1.75280i) q^{31} +(-0.723208 + 0.137959i) q^{33} +(0.425779 - 0.168578i) q^{36} +(-1.46560 - 1.21245i) q^{37} +(-0.992115 - 0.125333i) q^{44} +(-0.245375 - 0.386649i) q^{45} +(0.147338 + 1.16630i) q^{47} +(-0.734796 + 0.0462295i) q^{48} +(-0.309017 + 0.951057i) q^{49} +(-1.65875 - 0.316423i) q^{53} +(0.0627905 + 0.998027i) q^{55} +(1.18532 + 0.469303i) q^{59} +(0.183098 + 0.713118i) q^{60} +(-0.968583 - 0.248690i) q^{64} +(-0.946441 - 1.72157i) q^{67} +(-0.883312 + 0.485605i) q^{69} +(1.84489 - 0.233064i) q^{71} +(0.666178 - 0.313480i) q^{75} +(-0.0627905 + 0.998027i) q^{80} +(0.329735 + 0.0416553i) q^{81} +(-1.84489 + 0.730444i) q^{89} +(-1.34484 + 0.256543i) q^{92} +(1.35643 - 0.440731i) q^{93} +(0.742395 - 1.35041i) q^{97} +(0.370479 + 0.269169i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{5}+O(q^{10}) $ 20 * q + 5 * q^5	$ 20 q + 5 q^{5} - 20 q^{12} - 5 q^{25} - 5 q^{27} + 5 q^{48} + 5 q^{49} - 5 q^{59} - 5 q^{60} + 5 q^{67} + 5 q^{69} + 5 q^{81} - 5 q^{92} - 5 q^{99}+O(q^{100}) $ 20 * q + 5 * q^5 - 20 * q^12 - 5 * q^25 - 5 * q^27 + 5 * q^48 + 5 * q^49 - 5 * q^59 - 5 * q^60 + 5 * q^67 + 5 * q^69 + 5 * q^81 - 5 * q^92 - 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$.

$n$	$376$	$1002$
$\chi(n)$	$-1$	$e\left(\frac{9}{50}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$2$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$3$	0.503997	+	0.536702i	0.503997	+	0.536702i	0.929776	−	0.368125i	$-0.120000\pi$
$3$	0.503997	+	0.536702i	0.503997	+	0.536702i	−0.425779	+	0.904827i	$0.640000\pi$
$4$	0.425779	+	0.904827i	0.425779	+	0.904827i
$5$	0.809017	−	0.587785i	0.809017	−	0.587785i
$5$	0.809017	−	0.587785i	0.809017	−	0.587785i
$6$		0			0
$7$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$7$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$8$		0			0
$9$	0.0287541	−	0.457034i	0.0287541	−	0.457034i
$10$		0			0
$11$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$11$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$12$	−0.271031	+	0.684547i	−0.271031	+	0.684547i
$13$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$13$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$14$		0			0
$15$	0.723208	+	0.137959i	0.723208	+	0.137959i
$16$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$17$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$17$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$18$		0			0
$19$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$19$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$20$	0.876307	+	0.481754i	0.876307	+	0.481754i
$21$		0			0
$22$		0			0
$23$	−0.340480	+	1.32608i	−0.340480	+	1.32608i	0.535827	+	0.844328i	$0.320000\pi$
$23$	−0.340480	+	1.32608i	−0.340480	+	1.32608i	−0.876307	+	0.481754i	$0.840000\pi$
$24$		0			0
$25$	0.309017	−	0.951057i	0.309017	−	0.951057i
$26$		0			0
$27$	0.827073	−	0.684214i	0.827073	−	0.684214i
$28$		0			0
$29$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$29$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$30$		0			0
$31$	0.824805	−	1.75280i	0.824805	−	1.75280i	0.187381	−	0.982287i	$-0.440000\pi$
$31$	0.824805	−	1.75280i	0.824805	−	1.75280i	0.637424	−	0.770513i	$-0.280000\pi$
$32$		0			0
$33$	−0.723208	+	0.137959i	−0.723208	+	0.137959i
$34$		0			0
$35$		0			0
$36$	0.425779	−	0.168578i	0.425779	−	0.168578i
$37$	−1.46560	−	1.21245i	−1.46560	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$37$	−1.46560	−	1.21245i	−1.46560	−	1.21245i	−0.535827	−	0.844328i	$-0.680000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$41$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$42$		0			0
$43$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$43$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$44$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$45$	−0.245375	−	0.386649i	−0.245375	−	0.386649i
$46$		0			0
$47$	0.147338	+	1.16630i	0.147338	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$47$	0.147338	+	1.16630i	0.147338	+	1.16630i	−0.728969	+	0.684547i	$0.760000\pi$
$48$	−0.734796	+	0.0462295i	−0.734796	+	0.0462295i
$49$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.65875	−	0.316423i	−1.65875	−	0.316423i	−0.728969	−	0.684547i	$-0.760000\pi$
$53$	−1.65875	−	0.316423i	−1.65875	−	0.316423i	−0.929776	+	0.368125i	$0.880000\pi$
$54$		0			0
$55$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.876307	−	0.481754i	$-0.160000\pi$
$59$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$60$	0.183098	+	0.713118i	0.183098	+	0.713118i
$61$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$61$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$62$		0			0
$63$		0			0
$64$	−0.968583	−	0.248690i	−0.968583	−	0.248690i
$65$		0			0
$66$		0			0
$67$	−0.946441	−	1.72157i	−0.946441	−	1.72157i	−0.637424	−	0.770513i	$-0.720000\pi$
$67$	−0.946441	−	1.72157i	−0.946441	−	1.72157i	−0.309017	−	0.951057i	$-0.600000\pi$
$68$		0			0
$69$	−0.883312	+	0.485605i	−0.883312	+	0.485605i
$70$		0			0
$71$	1.84489	−	0.233064i	1.84489	−	0.233064i	0.876307	−	0.481754i	$-0.160000\pi$
$71$	1.84489	−	0.233064i	1.84489	−	0.233064i	0.968583	+	0.248690i	$0.0800000\pi$
$72$		0			0
$73$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$73$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$74$		0			0
$75$	0.666178	−	0.313480i	0.666178	−	0.313480i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$79$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$80$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$81$	0.329735	+	0.0416553i	0.329735	+	0.0416553i
$82$		0			0
$83$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$83$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.84489	+	0.730444i	−1.84489	+	0.730444i	−0.876307	+	0.481754i	$0.840000\pi$
$89$	−1.84489	+	0.730444i	−1.84489	+	0.730444i	−0.968583	+	0.248690i	$0.920000\pi$
$90$		0			0
$91$		0			0
$92$	−1.34484	+	0.256543i	−1.34484	+	0.256543i
$93$	1.35643	−	0.440731i	1.35643	−	0.440731i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.742395	−	1.35041i	0.742395	−	1.35041i	−0.187381	−	0.982287i	$-0.560000\pi$
$97$	0.742395	−	1.35041i	0.742395	−	1.35041i	0.929776	−	0.368125i	$-0.120000\pi$
$98$		0			0
$99$	0.370479	+	0.269169i	0.370479	+	0.269169i
$100$	0.992115	−	0.125333i	0.992115	−	0.125333i
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−1.77760	+	0.836475i	−1.77760	+	0.836475i	−0.809017	+	0.587785i	$0.800000\pi$
$103$	−1.77760	+	0.836475i	−1.77760	+	0.836475i	−0.968583	+	0.248690i	$0.920000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$107$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$108$	0.971246	+	0.457034i	0.971246	+	0.457034i
$109$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$109$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$110$		0			0
$111$	−0.0879337	−	1.39767i	−0.0879337	−	1.39767i
$112$		0			0
$113$	0.0922765	−	0.233064i	0.0922765	−	0.233064i	−0.876307	−	0.481754i	$-0.840000\pi$
$113$	0.0922765	−	0.233064i	0.0922765	−	0.233064i	0.968583	+	0.248690i	$0.0800000\pi$
$114$		0			0
$115$	0.503997	+	1.27295i	0.503997	+	1.27295i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$122$		0			0
$123$		0			0
$124$	1.93717			1.93717
$125$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$126$		0			0
$127$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$127$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$131$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$132$	−0.432756	−	0.595638i	−0.432756	−	0.595638i
$133$		0			0
$134$		0			0
$135$	0.266945	−	1.03968i	0.266945	−	1.03968i
$136$		0			0
$137$	−0.354691	+	0.895846i	−0.354691	+	0.895846i	0.637424	+	0.770513i	$0.280000\pi$
$137$	−0.354691	+	0.895846i	−0.354691	+	0.895846i	−0.992115	+	0.125333i	$0.960000\pi$
$138$		0			0
$139$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$139$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$140$		0			0
$141$	−0.551699	+	0.666889i	−0.551699	+	0.666889i
$142$		0			0
$143$		0			0
$144$	0.333822	+	0.313480i	0.333822	+	0.313480i
$145$		0			0
$146$		0			0
$147$	−0.666178	+	0.313480i	−0.666178	+	0.313480i
$148$	0.473036	−	1.84235i	0.473036	−	1.84235i
$149$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$149$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$151$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.362989	−	1.90285i	−0.362989	−	1.90285i
$156$		0			0
$157$	1.72108	−	0.559214i	1.72108	−	0.559214i	0.728969	−	0.684547i	$-0.240000\pi$
$157$	1.72108	−	0.559214i	1.72108	−	0.559214i	0.992115	+	0.125333i	$0.0400000\pi$
$158$		0			0
$159$	−0.666178	−	1.04973i	−0.666178	−	1.04973i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.123693	−	0.481754i	−0.123693	−	0.481754i	0.876307	−	0.481754i	$-0.160000\pi$
$163$	−0.123693	−	0.481754i	−0.123693	−	0.481754i	−1.00000			$\pi$
$164$		0			0
$165$	−0.503997	+	0.536702i	−0.503997	+	0.536702i
$166$		0			0
$167$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$167$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$168$		0			0
$169$	0.992115	+	0.125333i	0.992115	+	0.125333i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$173$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$174$		0			0
$175$		0			0
$176$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$177$	0.345524	+	0.872693i	0.345524	+	0.872693i
$178$		0			0
$179$	1.73879	−	0.219661i	1.73879	−	0.219661i	0.809017	−	0.587785i	$-0.200000\pi$
$179$	1.73879	−	0.219661i	1.73879	−	0.219661i	0.929776	+	0.368125i	$0.120000\pi$
$180$	0.245375	−	0.386649i	0.245375	−	0.386649i
$181$	−0.541587	+	0.297740i	−0.541587	+	0.297740i	−0.728969	−	0.684547i	$-0.760000\pi$
$181$	−0.541587	+	0.297740i	−0.541587	+	0.297740i	0.187381	+	0.982287i	$0.440000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.89836	−	0.119435i	−1.89836	−	0.119435i
$186$		0			0
$187$		0			0
$188$	−0.992567	+	0.629902i	−0.992567	+	0.629902i
$189$		0			0
$190$		0			0
$191$	−1.84489	−	0.730444i	−1.84489	−	0.730444i	−0.968583	−	0.248690i	$-0.920000\pi$
$191$	−1.84489	−	0.730444i	−1.84489	−	0.730444i	−0.876307	−	0.481754i	$-0.840000\pi$
$192$	−0.354691	−	0.645180i	−0.354691	−	0.645180i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$197$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$197$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$198$		0			0
$199$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$199$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$200$		0			0
$201$	0.446967	−	1.37562i	0.446967	−	1.37562i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.596274	+	0.193741i	0.596274	+	0.193741i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$211$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$212$	−0.419952	−	1.63560i	−0.419952	−	1.63560i
$213$	1.05491	+	0.872693i	1.05491	+	0.872693i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.876307	+	0.481754i	−0.876307	+	0.481754i
$221$		0			0
$222$		0			0
$223$	−1.53799	+	1.27233i	−1.53799	+	1.27233i	−0.728969	+	0.684547i	$0.760000\pi$
$223$	−1.53799	+	1.27233i	−1.53799	+	1.27233i	−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	−0.425779	−	0.168578i	−0.425779	−	0.168578i
$226$		0			0
$227$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$227$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$228$		0			0
$229$	0.159566	−	0.836475i	0.159566	−	0.836475i	−0.809017	−	0.587785i	$-0.800000\pi$
$229$	0.159566	−	0.836475i	0.159566	−	0.836475i	0.968583	−	0.248690i	$-0.0800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$233$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$234$		0			0
$235$	0.804733	+	0.856954i	0.804733	+	0.856954i
$236$	0.0800484	+	1.27233i	0.0800484	+	1.27233i
$237$		0			0
$238$		0			0
$239$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$239$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$240$	−0.567290	+	0.469303i	−0.567290	+	0.469303i
$241$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$241$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$242$		0			0
$243$	−0.487102	−	0.670439i	−0.487102	−	0.670439i
$244$		0			0
$245$	0.309017	+	0.951057i	0.309017	+	0.951057i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$251$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$252$		0			0
$253$	−0.937209	−	0.998027i	−0.937209	−	0.998027i
$254$		0			0
$255$		0			0
$256$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$257$	−0.804733	−	1.10762i	−0.804733	−	1.10762i	−0.992115	−	0.125333i	$-0.960000\pi$
$257$	−0.804733	−	1.10762i	−0.804733	−	1.10762i	0.187381	−	0.982287i	$-0.440000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$263$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$264$		0			0
$265$	−1.52794	+	0.718995i	−1.52794	+	0.718995i
$266$		0			0
$267$	−1.32185	−	0.622015i	−1.32185	−	0.622015i
$268$	1.15475	−	1.58937i	1.15475	−	1.58937i
$269$	−0.620759	−	0.582932i	−0.620759	−	0.582932i	0.309017	−	0.951057i	$-0.400000\pi$
$269$	−0.620759	−	0.582932i	−0.620759	−	0.582932i	−0.929776	+	0.368125i	$0.880000\pi$
$270$		0			0
$271$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$271$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.637424	+	0.770513i	0.637424	+	0.770513i
$276$	−0.815485	−	0.592484i	−0.815485	−	0.592484i
$277$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$277$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$278$		0			0
$279$	−0.777372	−	0.427364i	−0.777372	−	0.427364i
$280$		0			0
$281$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$281$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$282$		0			0
$283$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$283$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$284$	0.996398	+	1.57007i	0.996398	+	1.57007i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.637424	+	0.770513i	0.637424	+	0.770513i
$290$		0			0
$291$	1.09893	−	0.282158i	1.09893	−	0.282158i
$292$		0			0
$293$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$293$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$294$		0			0
$295$	1.23480	−	0.317042i	1.23480	−	0.317042i
$296$		0			0
$297$	0.134533	+	1.06494i	0.134533	+	1.06494i
$298$		0			0
$299$		0			0
$300$	0.567290	+	0.469303i	0.567290	+	0.469303i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−1.34484	−	0.532461i	−1.34484	−	0.532461i
$310$		0			0
$311$	−0.362989	−	0.0931997i	−0.362989	−	0.0931997i	0.0627905	−	0.998027i	$-0.480000\pi$
$311$	−0.362989	−	0.0931997i	−0.362989	−	0.0931997i	−0.425779	+	0.904827i	$0.640000\pi$
$312$		0			0
$313$	0.813516	−	0.516273i	0.813516	−	0.516273i	−0.0627905	−	0.998027i	$-0.520000\pi$
$313$	0.813516	−	0.516273i	0.813516	−	0.516273i	0.876307	+	0.481754i	$0.160000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.961606	+	1.74915i	0.961606	+	1.74915i	0.535827	+	0.844328i	$0.320000\pi$
$317$	0.961606	+	1.74915i	0.961606	+	1.74915i	0.425779	+	0.904827i	$0.360000\pi$
$318$		0			0
$319$		0			0
$320$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.102704	+	0.316090i	0.102704	+	0.316090i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.371808	−	0.0469702i	−0.371808	−	0.0469702i	−0.0627905	−	0.998027i	$-0.520000\pi$
$331$	−0.371808	−	0.0469702i	−0.371808	−	0.0469702i	−0.309017	+	0.951057i	$0.600000\pi$
$332$		0			0
$333$	−0.596274	+	0.634967i	−0.596274	+	0.634967i
$334$		0			0
$335$	−1.77760	−	0.836475i	−1.77760	−	0.836475i
$336$		0			0
$337$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$337$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$338$		0			0
$339$	0.171593	−	0.0679385i	0.171593	−	0.0679385i
$340$		0			0
$341$	1.03799	+	1.63560i	1.03799	+	1.63560i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.429183	+	0.912061i	−0.429183	+	0.912061i
$346$		0			0
$347$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$347$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$348$		0			0
$349$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$349$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$353$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$354$		0			0
$355$	1.35556	−	1.27295i	1.35556	−	1.27295i
$356$	−1.44644	−	1.35830i	−1.44644	−	1.35830i
$357$		0			0
$358$		0			0
$359$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$359$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$360$		0			0
$361$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$362$		0			0
$363$	0.271031	−	0.684547i	0.271031	−	0.684547i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.120759	+	0.955910i	−0.120759	+	0.955910i	0.809017	+	0.587785i	$0.200000\pi$
$367$	−0.120759	+	0.955910i	−0.120759	+	0.955910i	−0.929776	+	0.368125i	$0.880000\pi$
$368$	−0.804733	−	1.10762i	−0.804733	−	1.10762i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.976326	+	1.03968i	0.976326	+	1.03968i
$373$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$373$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$374$		0			0
$375$	0.354691	−	0.645180i	0.354691	−	0.645180i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.688925	+	1.46404i	0.688925	+	1.46404i	0.876307	+	0.481754i	$0.160000\pi$
$379$	0.688925	+	1.46404i	0.688925	+	1.46404i	−0.187381	+	0.982287i	$0.560000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.0623382	+	0.493458i	−0.0623382	+	0.493458i	0.929776	+	0.368125i	$0.120000\pi$
$383$	−0.0623382	+	0.493458i	−0.0623382	+	0.493458i	−0.992115	+	0.125333i	$0.960000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.53799	+	0.0967619i	1.53799	+	0.0967619i
$389$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	1.00000			$0$
$389$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	−0.992115	+	0.125333i	$0.960000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.0858089	+	0.449826i	−0.0858089	+	0.449826i
$397$	−0.226810	+	0.106729i	−0.226810	+	0.106729i	−0.535827	−	0.844328i	$-0.680000\pi$
$397$	−0.226810	+	0.106729i	−0.226810	+	0.106729i	0.309017	+	0.951057i	$0.400000\pi$
$398$		0			0
$399$		0			0
$400$	0.535827	+	0.844328i	0.535827	+	0.844328i
$401$	0.866986	+	0.629902i	0.866986	+	0.629902i	0.929776	−	0.368125i	$-0.120000\pi$
$401$	0.866986	+	0.629902i	0.866986	+	0.629902i	−0.0627905	+	0.998027i	$0.520000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.291246	−	0.160114i	0.291246	−	0.160114i
$406$		0			0
$407$	1.80902	−	0.587785i	1.80902	−	0.587785i
$408$		0			0
$409$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$409$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$410$		0			0
$411$	−0.659566	+	0.261141i	−0.659566	+	0.261141i
$412$	−1.51373	−	1.25227i	−1.51373	−	1.25227i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.26480	−	0.159781i	−1.26480	−	0.159781i	−0.535827	−	0.844328i	$-0.680000\pi$
$419$	−1.26480	−	0.159781i	−1.26480	−	0.159781i	−0.728969	+	0.684547i	$0.760000\pi$
$420$		0			0
$421$	1.44644	−	1.35830i	1.44644	−	1.35830i	0.637424	−	0.770513i	$-0.280000\pi$
$421$	1.44644	−	1.35830i	1.44644	−	1.35830i	0.809017	−	0.587785i	$-0.200000\pi$
$422$		0			0
$423$	0.537275	−	0.0338025i	0.537275	−	0.0338025i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$431$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$432$			1.07340i			1.07340i
$433$	0.916350	+	1.66683i	0.916350	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$433$	0.916350	+	1.66683i	0.916350	+	1.66683i	0.187381	+	0.982287i	$0.440000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$439$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$440$		0			0
$441$	0.425779	+	0.168578i	0.425779	+	0.168578i
$442$		0			0
$443$			1.54103i			1.54103i	0.637424	+	0.770513i	$0.280000\pi$
$443$			1.54103i			1.54103i	−0.637424	+	0.770513i	$0.720000\pi$
$444$	1.22721	−	0.674662i	1.22721	−	0.674662i
$445$	−1.06320	+	1.67534i	−1.06320	+	1.67534i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.450527	−	1.38658i	−0.450527	−	1.38658i	−0.876307	−	0.481754i	$-0.840000\pi$
$449$	−0.450527	−	1.38658i	−0.450527	−	1.38658i	0.425779	−	0.904827i	$-0.360000\pi$
$450$		0			0
$451$		0			0
$452$	0.250172	−	0.0157395i	0.250172	−	0.0157395i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$457$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$458$		0			0
$459$		0			0
$460$	−0.937209	+	0.998027i	−0.937209	+	0.998027i
$461$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$461$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$462$		0			0
$463$	0.193142	+	0.159781i	0.193142	+	0.159781i	0.728969	−	0.684547i	$-0.240000\pi$
$463$	0.193142	+	0.159781i	0.193142	+	0.159781i	−0.535827	+	0.844328i	$0.680000\pi$
$464$		0			0
$465$	0.838321	−	1.15385i	0.838321	−	1.15385i
$466$		0			0
$467$	1.15475	−	0.220280i	1.15475	−	0.220280i	0.425779	−	0.904827i	$-0.360000\pi$
$467$	1.15475	−	0.220280i	1.15475	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.16755	+	0.641868i	1.16755	+	0.641868i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.192311	+	0.749004i	−0.192311	+	0.749004i
$478$		0			0
$479$		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
$484$	0.637424	−	0.770513i	0.637424	−	0.770513i
$485$	−0.193142	−	1.52888i	−0.193142	−	1.52888i
$486$		0			0
$487$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$487$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$488$		0			0
$489$	0.196217	−	0.309189i	0.196217	−	0.309189i
$490$		0			0
$491$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$491$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	0.457937			0.457937
$496$	0.824805	+	1.75280i	0.824805	+	1.75280i
$497$		0			0
$498$		0			0
$499$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$499$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$500$	0.728969	−	0.684547i	0.728969	−	0.684547i
$501$		0			0
$502$		0			0
$503$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$503$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.432756	+	0.595638i	0.432756	+	0.595638i
$508$		0			0
$509$	0.110048	−	1.74915i	0.110048	−	1.74915i	−0.425779	−	0.904827i	$-0.640000\pi$
$509$	0.110048	−	1.74915i	0.110048	−	1.74915i	0.535827	−	0.844328i	$-0.320000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.946441	+	1.72157i	−0.946441	+	1.72157i
$516$		0			0
$517$	−1.06369	−	0.500534i	−1.06369	−	0.500534i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.0702235	+	0.368125i	−0.0702235	+	0.368125i	0.929776	+	0.368125i	$0.120000\pi$
$521$	−0.0702235	+	0.368125i	−0.0702235	+	0.368125i	−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$523$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.354691	−	0.645180i	0.354691	−	0.645180i
$529$	−0.766259	−	0.421255i	−0.766259	−	0.421255i
$530$		0			0
$531$	0.248570	−	0.528238i	0.248570	−	0.528238i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.994239	+	0.822506i	0.994239	+	0.822506i
$538$		0			0
$539$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$540$	1.05439	−	0.201136i	1.05439	−	0.201136i
$541$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$541$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$542$		0			0
$543$	−0.432756	−	0.140611i	−0.432756	−	0.140611i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$547$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$548$	−0.961606	+	0.0604991i	−0.961606	+	0.0604991i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.892667	−	1.07905i	−0.892667	−	1.07905i
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$563$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$564$	−0.838321	−	0.215244i	−0.838321	−	0.215244i
$565$	−0.0623382	−	0.242791i	−0.0623382	−	0.242791i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$569$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$570$		0			0
$571$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$571$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$572$		0			0
$573$	−0.537788	−	1.35830i	−0.537788	−	1.35830i
$574$		0			0
$575$	1.15596	+	0.733597i	1.15596	+	0.733597i
$576$	−0.141510	+	0.435524i	−0.141510	+	0.435524i
$577$	1.99211	−	0.125333i	1.99211	−	0.125333i	0.992115	−	0.125333i	$-0.0400000\pi$
$577$	1.99211	−	0.125333i	1.99211	−	0.125333i	1.00000			$0$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.15596	−	1.23098i	1.15596	−	1.23098i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.383238	+	1.49261i	0.383238	+	1.49261i	0.809017	+	0.587785i	$0.200000\pi$
$587$	0.383238	+	1.49261i	0.383238	+	1.49261i	−0.425779	+	0.904827i	$0.640000\pi$
$588$	−0.567290	−	0.469303i	−0.567290	−	0.469303i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.86842	−	0.356420i	1.86842	−	0.356420i
$593$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$593$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.621636	+	1.13075i	−0.621636	+	1.13075i
$598$		0			0
$599$	0.101597	+	0.0738147i	0.101597	+	0.0738147i	0.637424	−	0.770513i	$-0.280000\pi$
$599$	0.101597	+	0.0738147i	0.101597	+	0.0738147i	−0.535827	+	0.844328i	$0.680000\pi$
$600$		0			0
$601$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$601$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$602$		0			0
$603$	−0.814029	+	0.383053i	−0.814029	+	0.383053i
$604$		0			0
$605$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$613$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.250172	+	1.98031i	−0.250172	+	1.98031i	−0.0627905	+	0.998027i	$0.520000\pi$
$617$	−0.250172	+	1.98031i	−0.250172	+	1.98031i	−0.187381	+	0.982287i	$0.560000\pi$
$618$		0			0
$619$	0.362989	+	1.90285i	0.362989	+	1.90285i	0.425779	+	0.904827i	$0.360000\pi$
$619$	0.362989	+	1.90285i	0.362989	+	1.90285i	−0.0627905	+	0.998027i	$0.520000\pi$
$620$	1.56720	−	1.13864i	1.56720	−	1.13864i
$621$	0.625722	+	1.32973i	0.625722	+	1.32973i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$	1.23879	+	1.31918i	1.23879	+	1.31918i
$629$		0			0
$630$		0			0
$631$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	1.00000			$0$
$631$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	−0.929776	+	0.368125i	$0.880000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.666178	−	1.04973i	0.666178	−	1.04973i
$637$		0			0
$638$		0			0
$639$	−0.0534698	−	0.849878i	−0.0534698	−	0.849878i
$640$		0			0
$641$	−1.18532	+	1.43281i	−1.18532	+	1.43281i	−0.309017	+	0.951057i	$0.600000\pi$
$641$	−1.18532	+	1.43281i	−1.18532	+	1.43281i	−0.876307	+	0.481754i	$0.840000\pi$
$642$		0			0
$643$	0.292352	−	0.402389i	0.292352	−	0.402389i	−0.637424	−	0.770513i	$-0.720000\pi$
$643$	0.292352	−	0.402389i	0.292352	−	0.402389i	0.929776	+	0.368125i	$0.120000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.77760	+	0.836475i	−1.77760	+	0.836475i	−0.809017	+	0.587785i	$0.800000\pi$
$647$	−1.77760	+	0.836475i	−1.77760	+	0.836475i	−0.968583	+	0.248690i	$0.920000\pi$
$648$		0			0
$649$	−1.03137	+	0.749337i	−1.03137	+	0.749337i
$650$		0			0
$651$		0			0
$652$	0.383238	−	0.317042i	0.383238	−	0.317042i
$653$	−0.239615	+	0.435857i	−0.239615	+	0.435857i	−0.968583	−	0.248690i	$-0.920000\pi$
$653$	−0.239615	+	0.435857i	−0.239615	+	0.435857i	0.728969	+	0.684547i	$0.240000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$659$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$660$	−0.700215	−	0.227513i	−0.700215	−	0.227513i
$661$	−1.62954	+	0.645180i	−1.62954	+	0.645180i	−0.992115	−	0.125333i	$-0.960000\pi$
$661$	−1.62954	+	0.645180i	−1.62954	+	0.645180i	−0.637424	+	0.770513i	$0.720000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.45800	−	0.184189i	−1.45800	−	0.184189i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$673$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$674$		0			0
$675$	−0.395147	−	0.998027i	−0.395147	−	0.998027i
$676$	0.309017	+	0.951057i	0.309017	+	0.951057i
$677$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$677$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.871808	+	1.58581i	0.871808	+	1.58581i	0.809017	+	0.587785i	$0.200000\pi$
$683$	0.871808	+	1.58581i	0.871808	+	1.58581i	0.0627905	+	0.998027i	$0.480000\pi$
$684$		0			0
$685$	0.239615	+	0.933237i	0.239615	+	0.933237i
$686$		0			0
$687$	0.529359	−	0.335942i	0.529359	−	0.335942i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.35556	−	0.536702i	−1.35556	−	0.536702i	−0.425779	−	0.904827i	$-0.640000\pi$
$691$	−1.35556	−	0.536702i	−1.35556	−	0.536702i	−0.929776	+	0.368125i	$0.880000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$		0			0
$703$		0			0
$704$	0.728969	−	0.684547i	0.728969	−	0.684547i
$705$	−0.0543460	+	0.863805i	−0.0543460	+	0.863805i
$706$		0			0
$707$		0			0
$708$	−0.642520	+	0.684214i	−0.642520	+	0.684214i
$709$	−1.03799	+	0.266509i	−1.03799	+	0.266509i	−0.728969	−	0.684547i	$-0.760000\pi$
$709$	−1.03799	+	0.266509i	−1.03799	+	0.266509i	−0.309017	+	0.951057i	$0.600000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.04353	+	1.69055i	2.04353	+	1.69055i
$714$		0			0
$715$		0			0
$716$	0.939097	+	1.47978i	0.939097	+	1.47978i
$717$		0			0
$718$		0			0
$719$	−0.362576	+	0.770513i	−0.362576	+	0.770513i	0.637424	+	0.770513i	$0.280000\pi$
$719$	−0.362576	+	0.770513i	−0.362576	+	0.770513i	−1.00000			$\pi$
$720$	0.454326	+	0.0573948i	0.454326	+	0.0573948i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$725$		0			0
$726$		0			0
$727$	−0.383238	+	1.49261i	−0.383238	+	1.49261i	0.425779	+	0.904827i	$0.360000\pi$
$727$	−0.383238	+	1.49261i	−0.383238	+	1.49261i	−0.809017	+	0.587785i	$0.800000\pi$
$728$		0			0
$729$	0.176605	−	0.925798i	0.176605	−	0.925798i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$733$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$734$		0			0
$735$	−0.354691	+	0.645180i	−0.354691	+	0.645180i
$736$		0			0
$737$	1.96070	+	0.123357i	1.96070	+	0.123357i
$738$		0			0
$739$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$739$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$740$	−0.700215	−	1.76854i	−0.700215	−	1.76854i
$741$		0			0
$742$		0			0
$743$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$743$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$751$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$752$	−0.992567	−	0.629902i	−0.992567	−	0.629902i
$753$	0.540110	+	0.575159i	0.540110	+	0.575159i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.905793	−	1.24672i	−0.905793	−	1.24672i	−0.968583	−	0.248690i	$-0.920000\pi$
$757$	−0.905793	−	1.24672i	−0.905793	−	1.24672i	0.0627905	−	0.998027i	$-0.480000\pi$
$758$		0			0
$759$	0.0632925	−	1.00601i	0.0632925	−	1.00601i
$760$		0			0
$761$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$761$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$762$		0			0
$763$		0			0
$764$	−0.124591	−	1.98031i	−0.124591	−	1.98031i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.432756	−	0.595638i	0.432756	−	0.595638i
$769$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$769$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$770$		0			0
$771$	0.188879	−	0.990140i	0.188879	−	0.990140i
$772$		0			0
$773$	0.419952	−	1.63560i	0.419952	−	1.63560i	−0.309017	−	0.951057i	$-0.600000\pi$
$773$	0.419952	−	1.63560i	0.419952	−	1.63560i	0.728969	−	0.684547i	$-0.240000\pi$
$774$		0			0
$775$	−1.41213	−	1.32608i	−1.41213	−	1.32608i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.791759	+	1.68257i	−0.791759	+	1.68257i
$782$		0			0
$783$		0			0
$784$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$785$	1.06369	−	1.46404i	1.06369	−	1.46404i
$786$		0			0
$787$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$787$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−1.15596	−	0.457679i	−1.15596	−	0.457679i
$796$	−1.27760	+	1.19975i	−1.27760	+	1.19975i
$797$	−0.0314168	−	0.248690i	−0.0314168	−	0.248690i	0.968583	−	0.248690i	$-0.0800000\pi$
$797$	−0.0314168	−	0.248690i	−0.0314168	−	0.248690i	−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.280789	+	0.864180i	0.280789	+	0.864180i
$802$		0			0
$803$		0			0
$804$	1.43501	−	0.181284i	1.43501	−	0.181284i
$805$		0			0
$806$		0			0
$807$		−	0.626959i		−	0.626959i
$808$		0			0
$809$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$809$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$810$		0			0
$811$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$811$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.383238	−	0.317042i	−0.383238	−	0.317042i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$821$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$822$		0			0
$823$	−0.621636	−	1.57007i	−0.621636	−	1.57007i	−0.809017	−	0.587785i	$-0.800000\pi$
$823$	−0.621636	−	1.57007i	−0.621636	−	1.57007i	0.187381	−	0.982287i	$-0.440000\pi$
$824$		0			0
$825$	−0.0922765	+	0.730444i	−0.0922765	+	0.730444i
$826$		0			0
$827$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$827$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$828$	0.0785788	+	0.622015i	0.0785788	+	0.622015i
$829$	−0.0915446	+	0.0859661i	−0.0915446	+	0.0859661i	−0.728969	−	0.684547i	$-0.760000\pi$
$829$	−0.0915446	+	0.0859661i	−0.0915446	+	0.0859661i	0.637424	+	0.770513i	$0.280000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.517117	−	2.01404i	−0.517117	−	2.01404i
$838$		0			0
$839$	0.348445	−	0.137959i	0.348445	−	0.137959i	−0.187381	−	0.982287i	$-0.560000\pi$
$839$	0.348445	−	0.137959i	0.348445	−	0.137959i	0.535827	+	0.844328i	$0.320000\pi$
$840$		0			0
$841$	0.535827	+	0.844328i	0.535827	+	0.844328i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.876307	−	0.481754i	0.876307	−	0.481754i
$846$		0			0
$847$		0			0
$848$	1.30113	−	1.07639i	1.30113	−	1.07639i
$849$		0			0
$850$		0			0
$851$	2.10682	−	1.53069i	2.10682	−	1.53069i
$852$	−0.340480	+	1.32608i	−0.340480	+	1.32608i
$853$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$853$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$857$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$858$		0			0
$859$	0.683098	−	0.825723i	0.683098	−	0.825723i	−0.309017	−	0.951057i	$-0.600000\pi$
$859$	0.683098	−	0.825723i	0.683098	−	0.825723i	0.992115	+	0.125333i	$0.0400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.183098	−	0.462452i	0.183098	−	0.462452i	−0.809017	−	0.587785i	$-0.800000\pi$
$863$	0.183098	−	0.462452i	0.183098	−	0.462452i	0.992115	+	0.125333i	$0.0400000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.0922765	+	0.730444i	−0.0922765	+	0.730444i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.595837	−	0.378129i	−0.595837	−	0.378129i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$877$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$878$		0			0
$879$		0			0
$880$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$881$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$881$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	−0.992115	+	0.125333i	$0.960000\pi$
$882$		0			0
$883$	0.0623382	−	0.493458i	0.0623382	−	0.493458i	−0.929776	−	0.368125i	$-0.880000\pi$
$883$	0.0623382	−	0.493458i	0.0623382	−	0.493458i	0.992115	−	0.125333i	$-0.0400000\pi$
$884$		0			0
$885$	0.792491	+	0.502930i	0.792491	+	0.502930i
$886$		0			0
$887$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$887$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.211852	+	0.256085i	−0.211852	+	0.256085i
$892$	−1.80608	−	0.849878i	−1.80608	−	0.849878i
$893$		0			0
$894$		0			0
$895$	1.27760	−	1.19975i	1.27760	−	1.19975i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.0287541	−	0.457034i	−0.0287541	−	0.457034i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.263146	+	0.559214i	−0.263146	+	0.559214i
$906$		0			0
$907$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$907$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.348445	+	0.137959i	−0.348445	+	0.137959i	−0.535827	−	0.844328i	$-0.680000\pi$
$911$	−0.348445	+	0.137959i	−0.348445	+	0.137959i	0.187381	+	0.982287i	$0.440000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.824805	−	0.211774i	0.824805	−	0.211774i
$917$		0			0
$918$		0			0
$919$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$919$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.60601	+	1.01920i	−1.60601	+	1.01920i
$926$		0			0
$927$	0.331184	+	0.836475i	0.331184	+	0.836475i
$928$		0			0
$929$	−1.73879	+	0.219661i	−1.73879	+	0.219661i	−0.929776	−	0.368125i	$-0.880000\pi$
$929$	−1.73879	+	0.219661i	−1.73879	+	0.219661i	−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−0.132925	−	0.241789i	−0.132925	−	0.241789i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$937$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$938$		0			0
$939$	0.687095	+	0.176416i	0.687095	+	0.176416i
$940$	−0.432756	+	1.09302i	−0.432756	+	1.09302i
$941$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$941$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$942$		0			0
$943$		0			0
$944$	−1.11716	+	0.614163i	−1.11716	+	0.614163i
$945$		0			0
$946$		0			0
$947$	1.96070	+	0.374023i	1.96070	+	0.374023i	0.992115	+	0.125333i	$0.0400000\pi$
$947$	1.96070	+	0.374023i	1.96070	+	0.374023i	0.968583	+	0.248690i	$0.0800000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−0.454129	+	1.39767i	−0.454129	+	1.39767i
$952$		0			0
$953$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$953$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$954$		0			0
$955$	−1.92189	+	0.493458i	−1.92189	+	0.493458i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.666178	−	0.313480i	−0.666178	−	0.313480i
$961$	−1.75458	−	2.12093i	−1.75458	−	2.12093i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$967$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.939097	−	0.516273i	−0.939097	−	0.516273i	−0.0627905	−	0.998027i	$-0.520000\pi$
$971$	−0.939097	−	0.516273i	−0.939097	−	0.516273i	−0.876307	+	0.481754i	$0.840000\pi$
$972$	0.399233	−	0.726202i	0.399233	−	0.726202i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.473036	−	1.84235i	0.473036	−	1.84235i	−0.0627905	−	0.998027i	$-0.520000\pi$
$977$	0.473036	−	1.84235i	0.473036	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$978$		0			0
$979$	0.371808	−	1.94908i	0.371808	−	1.94908i
$980$	−0.728969	+	0.684547i	−0.728969	+	0.684547i
$981$		0			0
$982$		0			0
$983$	0.871808	+	0.410241i	0.871808	+	0.410241i	0.809017	−	0.587785i	$-0.200000\pi$
$983$	0.871808	+	0.410241i	0.871808	+	0.410241i	0.0627905	+	0.998027i	$0.480000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.0800484	−	1.27233i	0.0800484	−	1.27233i	−0.728969	−	0.684547i	$-0.760000\pi$
$991$	0.0800484	−	1.27233i	0.0800484	−	1.27233i	0.809017	−	0.587785i	$-0.200000\pi$
$992$		0			0
$993$	−0.162181	−	0.223223i	−0.162181	−	0.223223i
$994$		0			0
$995$	1.41789	+	1.03016i	1.41789	+	1.03016i
$996$		0			0
$997$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$997$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$998$		0			0
$999$	−2.04174			−2.04174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1375.1.bz.a.769.1	yes	20
11.10	odd	2	CM	1375.1.bz.a.769.1	yes	20
125.79	even	50	inner	1375.1.bz.a.329.1	✓	20
1375.329	odd	50	inner	1375.1.bz.a.329.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1375.1.bz.a.329.1	✓	20	125.79	even	50	inner
1375.1.bz.a.329.1	✓	20	1375.329	odd	50	inner
1375.1.bz.a.769.1	yes	20	1.1	even	1	trivial
1375.1.bz.a.769.1	yes	20	11.10	odd	2	CM

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:	\( N \)	\(=\)	\( 1375 = 5^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1375.bz (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.503997 + 0.536702i) q^{3} +(0.425779 + 0.904827i) q^{4} +(0.809017 - 0.587785i) q^{5} +(0.0287541 - 0.457034i) q^{9} +O(q^{10})\)	\(q+(0.503997 + 0.536702i) q^{3} +(0.425779 + 0.904827i) q^{4} +(0.809017 - 0.587785i) q^{5} +(0.0287541 - 0.457034i) q^{9} +(-0.535827 + 0.844328i) q^{11} +(-0.271031 + 0.684547i) q^{12} +(0.723208 + 0.137959i) q^{15} +(-0.637424 + 0.770513i) q^{16} +(0.876307 + 0.481754i) q^{20} +(-0.340480 + 1.32608i) q^{23} +(0.309017 - 0.951057i) q^{25} +(0.827073 - 0.684214i) q^{27} +(0.824805 - 1.75280i) q^{31} +(-0.723208 + 0.137959i) q^{33} +(0.425779 - 0.168578i) q^{36} +(-1.46560 - 1.21245i) q^{37} +(-0.992115 - 0.125333i) q^{44} +(-0.245375 - 0.386649i) q^{45} +(0.147338 + 1.16630i) q^{47} +(-0.734796 + 0.0462295i) q^{48} +(-0.309017 + 0.951057i) q^{49} +(-1.65875 - 0.316423i) q^{53} +(0.0627905 + 0.998027i) q^{55} +(1.18532 + 0.469303i) q^{59} +(0.183098 + 0.713118i) q^{60} +(-0.968583 - 0.248690i) q^{64} +(-0.946441 - 1.72157i) q^{67} +(-0.883312 + 0.485605i) q^{69} +(1.84489 - 0.233064i) q^{71} +(0.666178 - 0.313480i) q^{75} +(-0.0627905 + 0.998027i) q^{80} +(0.329735 + 0.0416553i) q^{81} +(-1.84489 + 0.730444i) q^{89} +(-1.34484 + 0.256543i) q^{92} +(1.35643 - 0.440731i) q^{93} +(0.742395 - 1.35041i) q^{97} +(0.370479 + 0.269169i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{5}+O(q^{10}) \) 20 * q + 5 * q^5	\( 20 q + 5 q^{5} - 20 q^{12} - 5 q^{25} - 5 q^{27} + 5 q^{48} + 5 q^{49} - 5 q^{59} - 5 q^{60} + 5 q^{67} + 5 q^{69} + 5 q^{81} - 5 q^{92} - 5 q^{99}+O(q^{100}) \) 20 * q + 5 * q^5 - 20 * q^12 - 5 * q^25 - 5 * q^27 + 5 * q^48 + 5 * q^49 - 5 * q^59 - 5 * q^60 + 5 * q^67 + 5 * q^69 + 5 * q^81 - 5 * q^92 - 5 * q^99

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