LMFDB - Embedded newform 1375.1.bz.a.384.1

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			384.1
Root			$-0.535827 - 0.844328i$ of defining polynomial
Character	$\chi$	$=$	1375.384
Dual form			1375.1.bz.a.1264.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.36639 + 0.0859661i) q^{3} +(0.637424 + 0.770513i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(0.867524 - 0.109594i) q^{9} +O(q^{10})$	\(q+(-1.36639 + 0.0859661i) q^{3} +(0.637424 + 0.770513i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(0.867524 - 0.109594i) q^{9} +(0.425779 - 0.904827i) q^{11} +(-0.937209 - 0.998027i) q^{12} +(0.503997 + 1.27295i) q^{15} +(-0.187381 + 0.982287i) q^{16} +(0.535827 - 0.844328i) q^{20} +(-0.961606 - 1.74915i) q^{23} +(-0.809017 + 0.587785i) q^{25} +(0.168887 - 0.0322169i) q^{27} +(1.11716 - 1.35041i) q^{31} +(-0.503997 + 1.27295i) q^{33} +(0.637424 + 0.598581i) q^{36} +(1.15475 + 0.220280i) q^{37} +(0.968583 - 0.248690i) q^{44} +(-0.372309 - 0.791198i) q^{45} +(0.473036 - 1.84235i) q^{47} +(0.171593 - 1.35830i) q^{48} +(0.809017 - 0.587785i) q^{49} +(0.666178 + 1.68257i) q^{53} +(-0.992115 - 0.125333i) q^{55} +(-0.273190 + 0.256543i) q^{59} +(-0.659566 + 1.19975i) q^{60} +(-0.876307 + 0.481754i) q^{64} +(0.621636 - 0.394502i) q^{67} +(1.46430 + 2.30737i) q^{69} +(1.41213 + 0.362574i) q^{71} +(1.05491 - 0.872693i) q^{75} +(0.992115 - 0.125333i) q^{80} +(-1.07494 + 0.275999i) q^{81} +(-1.41213 - 1.32608i) q^{89} +(0.734796 - 1.85588i) q^{92} +(-1.41039 + 1.94123i) q^{93} +(-1.65875 - 1.05267i) q^{97} +(0.270210 - 0.831622i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$.

$n$	$376$	$1002$
$\chi(n)$	$-1$	$e\left(\frac{7}{50}\right)$

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		−0.904827	−	0.425779i	$-0.860000\pi$
$2$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$3$	−1.36639	+	0.0859661i	−1.36639	+	0.0859661i	−0.728969	−	0.684547i	$-0.760000\pi$
$3$	−1.36639	+	0.0859661i	−1.36639	+	0.0859661i	−0.637424	+	0.770513i	$0.720000\pi$
$4$	0.637424	+	0.770513i	0.637424	+	0.770513i
$5$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$5$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$6$		0			0
$7$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$7$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$8$		0			0
$9$	0.867524	−	0.109594i	0.867524	−	0.109594i
$10$		0			0
$11$	0.425779	−	0.904827i	0.425779	−	0.904827i
$11$	0.425779	−	0.904827i	0.425779	−	0.904827i
$12$	−0.937209	−	0.998027i	−0.937209	−	0.998027i
$13$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$13$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$14$		0			0
$15$	0.503997	+	1.27295i	0.503997	+	1.27295i
$16$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$17$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$17$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$18$		0			0
$19$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$19$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$20$	0.535827	−	0.844328i	0.535827	−	0.844328i
$21$		0			0
$22$		0			0
$23$	−0.961606	−	1.74915i	−0.961606	−	1.74915i	−0.535827	−	0.844328i	$-0.680000\pi$
$23$	−0.961606	−	1.74915i	−0.961606	−	1.74915i	−0.425779	−	0.904827i	$-0.640000\pi$
$24$		0			0
$25$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$26$		0			0
$27$	0.168887	−	0.0322169i	0.168887	−	0.0322169i
$28$		0			0
$29$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$29$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$30$		0			0
$31$	1.11716	−	1.35041i	1.11716	−	1.35041i	0.187381	−	0.982287i	$-0.440000\pi$
$31$	1.11716	−	1.35041i	1.11716	−	1.35041i	0.929776	−	0.368125i	$-0.120000\pi$
$32$		0			0
$33$	−0.503997	+	1.27295i	−0.503997	+	1.27295i
$34$		0			0
$35$		0			0
$36$	0.637424	+	0.598581i	0.637424	+	0.598581i
$37$	1.15475	+	0.220280i	1.15475	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$37$	1.15475	+	0.220280i	1.15475	+	0.220280i	0.425779	+	0.904827i	$0.360000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$41$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$42$		0			0
$43$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$43$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$44$	0.968583	−	0.248690i	0.968583	−	0.248690i
$45$	−0.372309	−	0.791198i	−0.372309	−	0.791198i
$46$		0			0
$47$	0.473036	−	1.84235i	0.473036	−	1.84235i	−0.0627905	−	0.998027i	$-0.520000\pi$
$47$	0.473036	−	1.84235i	0.473036	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$48$	0.171593	−	1.35830i	0.171593	−	1.35830i
$49$	0.809017	−	0.587785i	0.809017	−	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.666178	+	1.68257i	0.666178	+	1.68257i	0.728969	+	0.684547i	$0.240000\pi$
$53$	0.666178	+	1.68257i	0.666178	+	1.68257i	−0.0627905	+	0.998027i	$0.520000\pi$
$54$		0			0
$55$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	−0.809017	−	0.587785i	$-0.800000\pi$
$59$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	0.535827	+	0.844328i	$0.320000\pi$
$60$	−0.659566	+	1.19975i	−0.659566	+	1.19975i
$61$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$61$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$62$		0			0
$63$		0			0
$64$	−0.876307	+	0.481754i	−0.876307	+	0.481754i
$65$		0			0
$66$		0			0
$67$	0.621636	−	0.394502i	0.621636	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$67$	0.621636	−	0.394502i	0.621636	−	0.394502i	0.809017	+	0.587785i	$0.200000\pi$
$68$		0			0
$69$	1.46430	+	2.30737i	1.46430	+	2.30737i
$70$		0			0
$71$	1.41213	+	0.362574i	1.41213	+	0.362574i	0.876307	−	0.481754i	$-0.160000\pi$
$71$	1.41213	+	0.362574i	1.41213	+	0.362574i	0.535827	+	0.844328i	$0.320000\pi$
$72$		0			0
$73$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$73$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$74$		0			0
$75$	1.05491	−	0.872693i	1.05491	−	0.872693i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$79$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$80$	0.992115	−	0.125333i	0.992115	−	0.125333i
$81$	−1.07494	+	0.275999i	−1.07494	+	0.275999i
$82$		0			0
$83$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$83$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.41213	−	1.32608i	−1.41213	−	1.32608i	−0.876307	−	0.481754i	$-0.840000\pi$
$89$	−1.41213	−	1.32608i	−1.41213	−	1.32608i	−0.535827	−	0.844328i	$-0.680000\pi$
$90$		0			0
$91$		0			0
$92$	0.734796	−	1.85588i	0.734796	−	1.85588i
$93$	−1.41039	+	1.94123i	−1.41039	+	1.94123i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.65875	−	1.05267i	−1.65875	−	1.05267i	−0.929776	−	0.368125i	$-0.880000\pi$
$97$	−1.65875	−	1.05267i	−1.65875	−	1.05267i	−0.728969	−	0.684547i	$-0.760000\pi$
$98$		0			0
$99$	0.270210	−	0.831622i	0.270210	−	0.831622i
$100$	−0.968583	−	0.248690i	−0.968583	−	0.248690i
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$	−0.567290	+	0.469303i	−0.567290	+	0.469303i	−0.876307	−	0.481754i	$-0.840000\pi$
$103$	−0.567290	+	0.469303i	−0.567290	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$	0.132476	+	0.109594i	0.132476	+	0.109594i
$109$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$109$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$110$		0			0
$111$	−1.59678	−	0.201720i	−1.59678	−	0.201720i
$112$		0			0
$113$	0.340480	+	0.362574i	0.340480	+	0.362574i	0.876307	−	0.481754i	$-0.160000\pi$
$113$	0.340480	+	0.362574i	0.340480	+	0.362574i	−0.535827	+	0.844328i	$0.680000\pi$
$114$		0			0
$115$	−1.36639	+	1.45506i	−1.36639	+	1.45506i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$122$		0			0
$123$		0			0
$124$	1.75261			1.75261
$125$	0.809017	+	0.587785i	0.809017	+	0.587785i
$126$		0			0
$127$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$127$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$131$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$132$	−1.30209	+	0.423073i	−1.30209	+	0.423073i
$133$		0			0
$134$		0			0
$135$	−0.0828290	−	0.150665i	−0.0828290	−	0.150665i
$136$		0			0
$137$	1.15596	+	1.23098i	1.15596	+	1.23098i	0.968583	+	0.248690i	$0.0800000\pi$
$137$	1.15596	+	1.23098i	1.15596	+	1.23098i	0.187381	+	0.982287i	$0.440000\pi$
$138$		0			0
$139$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$139$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$140$		0			0
$141$	−0.487973	+	2.55804i	−0.487973	+	2.55804i
$142$		0			0
$143$		0			0
$144$	−0.0549052	+	0.872693i	−0.0549052	+	0.872693i
$145$		0			0
$146$		0			0
$147$	−1.05491	+	0.872693i	−1.05491	+	0.872693i
$148$	0.566335	+	1.03016i	0.566335	+	1.03016i
$149$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$149$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$150$		0			0
$151$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$151$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.62954	−	0.645180i	−1.62954	−	0.645180i
$156$		0			0
$157$	−0.905793	+	1.24672i	−0.905793	+	1.24672i	0.0627905	+	0.998027i	$0.480000\pi$
$157$	−0.905793	+	1.24672i	−0.905793	+	1.24672i	−0.968583	+	0.248690i	$0.920000\pi$
$158$		0			0
$159$	−1.05491	−	2.24179i	−1.05491	−	2.24179i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.464173	+	0.844328i	−0.464173	+	0.844328i	0.535827	+	0.844328i	$0.320000\pi$
$163$	−0.464173	+	0.844328i	−0.464173	+	0.844328i	−1.00000			$\pi$
$164$		0			0
$165$	1.36639	+	0.0859661i	1.36639	+	0.0859661i
$166$		0			0
$167$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$167$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$168$		0			0
$169$	−0.968583	+	0.248690i	−0.968583	+	0.248690i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$173$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$174$		0			0
$175$		0			0
$176$	0.809017	+	0.587785i	0.809017	+	0.587785i
$177$	0.351231	−	0.374023i	0.351231	−	0.374023i
$178$		0			0
$179$	−1.03799	−	0.266509i	−1.03799	−	0.266509i	−0.309017	−	0.951057i	$-0.600000\pi$
$179$	−1.03799	−	0.266509i	−1.03799	−	0.266509i	−0.728969	+	0.684547i	$0.760000\pi$
$180$	0.372309	−	0.791198i	0.372309	−	0.791198i
$181$	0.866986	+	1.36615i	0.866986	+	1.36615i	0.929776	+	0.368125i	$0.120000\pi$
$181$	0.866986	+	1.36615i	0.866986	+	1.36615i	−0.0627905	+	0.998027i	$0.520000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.147338	−	1.16630i	−0.147338	−	1.16630i
$186$		0			0
$187$		0			0
$188$	1.72108	−	0.809880i	1.72108	−	0.809880i
$189$		0			0
$190$		0			0
$191$	−1.41213	+	1.32608i	−1.41213	+	1.32608i	−0.535827	+	0.844328i	$0.680000\pi$
$191$	−1.41213	+	1.32608i	−1.41213	+	1.32608i	−0.876307	+	0.481754i	$0.840000\pi$
$192$	1.15596	−	0.733597i	1.15596	−	0.733597i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	0.968583	+	0.248690i	0.968583	+	0.248690i
$197$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$197$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$198$		0			0
$199$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$199$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$200$		0			0
$201$	−0.815485	+	0.592484i	−0.815485	+	0.592484i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.02591	−	1.41205i	−1.02591	−	1.41205i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$211$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$212$	−0.871808	+	1.58581i	−0.871808	+	1.58581i
$213$	−1.96070	−	0.374023i	−1.96070	−	0.374023i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$221$		0			0
$222$		0			0
$223$	0.246226	−	0.0469702i	0.246226	−	0.0469702i	−0.0627905	−	0.998027i	$-0.520000\pi$
$223$	0.246226	−	0.0469702i	0.246226	−	0.0469702i	0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	−0.637424	+	0.598581i	−0.637424	+	0.598581i
$226$		0			0
$227$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$227$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$228$		0			0
$229$	1.18532	−	0.469303i	1.18532	−	0.469303i	0.309017	−	0.951057i	$-0.400000\pi$
$229$	1.18532	−	0.469303i	1.18532	−	0.469303i	0.876307	+	0.481754i	$0.160000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$233$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$234$		0			0
$235$	−1.89836	+	0.119435i	−1.89836	+	0.119435i
$236$	−0.371808	−	0.0469702i	−0.371808	−	0.0469702i
$237$		0			0
$238$		0			0
$239$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$239$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$240$	−1.34484	+	0.256543i	−1.34484	+	0.256543i
$241$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$241$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$242$		0			0
$243$	1.28155	−	0.416401i	1.28155	−	0.416401i
$244$		0			0
$245$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$251$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$252$		0			0
$253$	−1.99211	+	0.125333i	−1.99211	+	0.125333i
$254$		0			0
$255$		0			0
$256$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$257$	1.89836	−	0.616814i	1.89836	−	0.616814i	0.929776	−	0.368125i	$-0.120000\pi$
$257$	1.89836	−	0.616814i	1.89836	−	0.616814i	0.968583	−	0.248690i	$-0.0800000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$263$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$264$		0			0
$265$	1.39436	−	1.15352i	1.39436	−	1.15352i
$266$		0			0
$267$	2.04353	+	1.69055i	2.04353	+	1.69055i
$268$	0.700215	+	0.227513i	0.700215	+	0.227513i
$269$	−0.0800484	+	1.27233i	−0.0800484	+	1.27233i	0.728969	+	0.684547i	$0.240000\pi$
$269$	−0.0800484	+	1.27233i	−0.0800484	+	1.27233i	−0.809017	+	0.587785i	$0.800000\pi$
$270$		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
$274$		0			0
$275$	0.187381	+	0.982287i	0.187381	+	0.982287i
$276$	−0.844477	+	2.59903i	−0.844477	+	2.59903i
$277$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$277$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$278$		0			0
$279$	0.821164	−	1.29395i	0.821164	−	1.29395i
$280$		0			0
$281$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$281$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$282$		0			0
$283$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$283$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$284$	0.620759	+	1.31918i	0.620759	+	1.31918i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.187381	+	0.982287i	0.187381	+	0.982287i
$290$		0			0
$291$	2.35699	+	1.29577i	2.35699	+	1.29577i
$292$		0			0
$293$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$293$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$294$		0			0
$295$	0.328407	+	0.180543i	0.328407	+	0.180543i
$296$		0			0
$297$	0.0427578	−	0.166531i	0.0427578	−	0.166531i
$298$		0			0
$299$		0			0
$300$	1.34484	+	0.256543i	1.34484	+	0.256543i
$301$		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
$309$	0.734796	−	0.690020i	0.734796	−	0.690020i
$310$		0			0
$311$	−1.62954	+	0.895846i	−1.62954	+	0.895846i	−0.637424	+	0.770513i	$0.720000\pi$
$311$	−1.62954	+	0.895846i	−1.62954	+	0.895846i	−0.992115	+	0.125333i	$0.960000\pi$
$312$		0			0
$313$	1.52794	−	0.718995i	1.52794	−	0.718995i	0.535827	−	0.844328i	$-0.320000\pi$
$313$	1.52794	−	0.718995i	1.52794	−	0.718995i	0.992115	+	0.125333i	$0.0400000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.211645	−	0.134314i	0.211645	−	0.134314i	−0.425779	−	0.904827i	$-0.640000\pi$
$317$	0.211645	−	0.134314i	0.211645	−	0.134314i	0.637424	+	0.770513i	$0.280000\pi$
$318$		0			0
$319$		0			0
$320$	0.728969	+	0.684547i	0.728969	+	0.684547i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.897856	−	0.652330i	−0.897856	−	0.652330i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.80113	−	0.462452i	1.80113	−	0.462452i	0.809017	−	0.587785i	$-0.200000\pi$
$331$	1.80113	−	0.462452i	1.80113	−	0.462452i	0.992115	+	0.125333i	$0.0400000\pi$
$332$		0			0
$333$	1.02591	+	0.0645450i	1.02591	+	0.0645450i
$334$		0			0
$335$	−0.567290	−	0.469303i	−0.567290	−	0.469303i
$336$		0			0
$337$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$337$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$338$		0			0
$339$	−0.496398	−	0.466149i	−0.496398	−	0.466149i
$340$		0			0
$341$	−0.746226	−	1.58581i	−0.746226	−	1.58581i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.74194	−	2.10565i	1.74194	−	2.10565i
$346$		0			0
$347$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$347$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$348$		0			0
$349$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$349$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$353$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$354$		0			0
$355$	−0.0915446	−	1.45506i	−0.0915446	−	1.45506i
$356$	0.121636	−	1.93334i	0.121636	−	1.93334i
$357$		0			0
$358$		0			0
$359$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$359$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$360$		0			0
$361$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$362$		0			0
$363$	0.937209	+	0.998027i	0.937209	+	0.998027i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.419952	+	1.63560i	0.419952	+	1.63560i	0.728969	+	0.684547i	$0.240000\pi$
$367$	0.419952	+	1.63560i	0.419952	+	1.63560i	−0.309017	+	0.951057i	$0.600000\pi$
$368$	1.89836	−	0.616814i	1.89836	−	0.616814i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.39476	+	0.150665i	−2.39476	+	0.150665i
$373$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$373$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$374$		0			0
$375$	−1.15596	−	0.733597i	−1.15596	−	0.733597i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	0.535827	−	0.844328i	$-0.320000\pi$
$379$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	−0.929776	+	0.368125i	$0.880000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.239615	+	0.933237i	0.239615	+	0.933237i	0.968583	+	0.248690i	$0.0800000\pi$
$383$	0.239615	+	0.933237i	0.239615	+	0.933237i	−0.728969	+	0.684547i	$0.760000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.246226	−	1.94908i	−0.246226	−	1.94908i
$389$	1.96858	+	0.248690i	1.96858	+	0.248690i	1.00000			$0$
$389$	1.96858	+	0.248690i	1.96858	+	0.248690i	0.968583	+	0.248690i	$0.0800000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.813014	−	0.321895i	0.813014	−	0.321895i
$397$	−0.383238	+	0.317042i	−0.383238	+	0.317042i	−0.809017	−	0.587785i	$-0.800000\pi$
$397$	−0.383238	+	0.317042i	−0.383238	+	0.317042i	0.425779	+	0.904827i	$0.360000\pi$
$398$		0			0
$399$		0			0
$400$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$401$	0.263146	−	0.809880i	0.263146	−	0.809880i	−0.728969	−	0.684547i	$-0.760000\pi$
$401$	0.263146	−	0.809880i	0.263146	−	0.809880i	0.992115	−	0.125333i	$-0.0400000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.594666	+	0.937044i	0.594666	+	0.937044i
$406$		0			0
$407$	0.690983	−	0.951057i	0.690983	−	0.951057i
$408$		0			0
$409$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$409$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$410$		0			0
$411$	−1.68532	−	1.58262i	−1.68532	−	1.58262i
$412$	−0.723208	−	0.137959i	−0.723208	−	0.137959i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.362989	−	0.0931997i	0.362989	−	0.0931997i	−0.0627905	−	0.998027i	$-0.520000\pi$
$419$	0.362989	−	0.0931997i	0.362989	−	0.0931997i	0.425779	+	0.904827i	$0.360000\pi$
$420$		0			0
$421$	−0.121636	−	1.93334i	−0.121636	−	1.93334i	−0.309017	−	0.951057i	$-0.600000\pi$
$421$	−0.121636	−	1.93334i	−0.121636	−	1.93334i	0.187381	−	0.982287i	$-0.440000\pi$
$422$		0			0
$423$	0.208460	−	1.65013i	0.208460	−	1.65013i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$431$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$432$			0.171932i			0.171932i
$433$	0.992567	−	0.629902i	0.992567	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$433$	0.992567	−	0.629902i	0.992567	−	0.629902i	0.929776	+	0.368125i	$0.120000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$439$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$440$		0			0
$441$	0.637424	−	0.598581i	0.637424	−	0.598581i
$442$		0			0
$443$			1.96457i			1.96457i	0.187381	+	0.982287i	$0.440000\pi$
$443$			1.96457i			1.96457i	−0.187381	+	0.982287i	$0.560000\pi$
$444$	−0.862395	−	1.35892i	−0.862395	−	1.35892i
$445$	−0.824805	+	1.75280i	−0.824805	+	1.75280i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.101597	+	0.0738147i	0.101597	+	0.0738147i	0.637424	−	0.770513i	$-0.280000\pi$
$449$	0.101597	+	0.0738147i	0.101597	+	0.0738147i	−0.535827	+	0.844328i	$0.680000\pi$
$450$		0			0
$451$		0			0
$452$	−0.0623382	+	0.493458i	−0.0623382	+	0.493458i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$457$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$458$		0			0
$459$		0			0
$460$	−1.99211	−	0.125333i	−1.99211	−	0.125333i
$461$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$461$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$462$		0			0
$463$	0.488570	+	0.0931997i	0.488570	+	0.0931997i	0.425779	−	0.904827i	$-0.360000\pi$
$463$	0.488570	+	0.0931997i	0.488570	+	0.0931997i	0.0627905	+	0.998027i	$0.480000\pi$
$464$		0			0
$465$	2.28205	+	0.741484i	2.28205	+	0.741484i
$466$		0			0
$467$	0.700215	−	1.76854i	0.700215	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$467$	0.700215	−	1.76854i	0.700215	−	1.76854i	0.637424	−	0.770513i	$-0.280000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.13049	−	1.78137i	1.13049	−	1.78137i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.762325	+	1.38666i	0.762325	+	1.38666i
$478$		0			0
$479$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$479$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.187381	−	0.982287i	0.187381	−	0.982287i
$485$	−0.488570	+	1.90285i	−0.488570	+	1.90285i
$486$		0			0
$487$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$487$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$488$		0			0
$489$	0.561659	−	1.19359i	0.561659	−	1.19359i
$490$		0			0
$491$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$491$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.874419			−0.874419
$496$	1.11716	+	1.35041i	1.11716	+	1.35041i
$497$		0			0
$498$		0			0
$499$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$499$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$500$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$501$		0			0
$502$		0			0
$503$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$503$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.30209	−	0.423073i	1.30209	−	0.423073i
$508$		0			0
$509$	−1.06320	+	0.134314i	−1.06320	+	0.134314i	−0.637424	−	0.770513i	$-0.720000\pi$
$509$	−1.06320	+	0.134314i	−1.06320	+	0.134314i	−0.425779	+	0.904827i	$0.640000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.621636	+	0.394502i	0.621636	+	0.394502i
$516$		0			0
$517$	−1.46560	−	1.21245i	−1.46560	−	1.21245i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.72897	+	0.684547i	−1.72897	+	0.684547i	−0.728969	+	0.684547i	$0.760000\pi$
$521$	−1.72897	+	0.684547i	−1.72897	+	0.684547i	−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$523$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.15596	−	0.733597i	−1.15596	−	0.733597i
$529$	−1.59903	+	2.51967i	−1.59903	+	2.51967i
$530$		0			0
$531$	−0.208884	+	0.252497i	−0.208884	+	0.252497i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.44121	+	0.274925i	1.44121	+	0.274925i
$538$		0			0
$539$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$540$	0.0632925	−	0.159859i	0.0632925	−	0.159859i
$541$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$541$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$542$		0			0
$543$	−1.30209	−	1.79217i	−1.30209	−	1.79217i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$547$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$548$	−0.211645	+	1.67534i	−0.211645	+	1.67534i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.301584	+	1.58096i	0.301584	+	1.58096i
$556$		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		0.904827	−	0.425779i	$-0.140000\pi$
$563$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$564$	−2.28205	+	1.25457i	−2.28205	+	1.25457i
$565$	0.239615	−	0.435857i	0.239615	−	0.435857i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$569$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$570$		0			0
$571$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$571$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$572$		0			0
$573$	1.81553	−	1.93334i	1.81553	−	1.93334i
$574$		0			0
$575$	1.80608	+	0.849878i	1.80608	+	0.849878i
$576$	−0.707420	+	0.513971i	−0.707420	+	0.513971i
$577$	0.0314168	−	0.248690i	0.0314168	−	0.248690i	−0.968583	−	0.248690i	$-0.920000\pi$
$577$	0.0314168	−	0.248690i	0.0314168	−	0.248690i	1.00000			$0$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.80608	+	0.113629i	1.80608	+	0.113629i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.946441	+	1.72157i	−0.946441	+	1.72157i	−0.309017	+	0.951057i	$0.600000\pi$
$587$	−0.946441	+	1.72157i	−0.946441	+	1.72157i	−0.637424	+	0.770513i	$0.720000\pi$
$588$	−1.34484	−	0.256543i	−1.34484	−	0.256543i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.432756	+	1.09302i	−0.432756	+	1.09302i
$593$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$593$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.23879	+	0.786162i	1.23879	+	0.786162i
$598$		0			0
$599$	0.613161	−	1.88711i	0.613161	−	1.88711i	0.187381	−	0.982287i	$-0.440000\pi$
$599$	0.613161	−	1.88711i	0.613161	−	1.88711i	0.425779	−	0.904827i	$-0.360000\pi$
$600$		0			0
$601$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$601$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$	0.496049	−	0.410367i	0.496049	−	0.410367i
$604$		0			0
$605$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$606$		0			0
$607$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$607$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$613$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.0623382	+	0.242791i	0.0623382	+	0.242791i	0.992115	−	0.125333i	$-0.0400000\pi$
$617$	0.0623382	+	0.242791i	0.0623382	+	0.242791i	−0.929776	+	0.368125i	$0.880000\pi$
$618$		0			0
$619$	1.62954	+	0.645180i	1.62954	+	0.645180i	0.992115	−	0.125333i	$-0.0400000\pi$
$619$	1.62954	+	0.645180i	1.62954	+	0.645180i	0.637424	+	0.770513i	$0.280000\pi$
$620$	−0.541587	−	1.66683i	−0.541587	−	1.66683i
$621$	−0.218755	−	0.264429i	−0.218755	−	0.264429i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.309017	−	0.951057i	0.309017	−	0.951057i
$626$		0			0
$627$		0			0
$628$	−1.53799	+	0.0967619i	−1.53799	+	0.0967619i
$629$		0			0
$630$		0			0
$631$	1.72897	+	0.684547i	1.72897	+	0.684547i	1.00000			$0$
$631$	1.72897	+	0.684547i	1.72897	+	0.684547i	0.728969	+	0.684547i	$0.240000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.05491	−	2.24179i	1.05491	−	2.24179i
$637$		0			0
$638$		0			0
$639$	1.26480	+	0.159781i	1.26480	+	0.159781i
$640$		0			0
$641$	0.273190	−	1.43211i	0.273190	−	1.43211i	−0.535827	−	0.844328i	$-0.680000\pi$
$641$	0.273190	−	1.43211i	0.273190	−	1.43211i	0.809017	−	0.587785i	$-0.200000\pi$
$642$		0			0
$643$	−0.916350	−	0.297740i	−0.916350	−	0.297740i	−0.187381	−	0.982287i	$-0.560000\pi$
$643$	−0.916350	−	0.297740i	−0.916350	−	0.297740i	−0.728969	+	0.684547i	$0.760000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.567290	+	0.469303i	−0.567290	+	0.469303i	−0.876307	−	0.481754i	$-0.840000\pi$
$647$	−0.567290	+	0.469303i	−0.567290	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$648$		0			0
$649$	0.115808	+	0.356420i	0.115808	+	0.356420i
$650$		0			0
$651$		0			0
$652$	−0.946441	+	0.180543i	−0.946441	+	0.180543i
$653$	−0.813516	−	0.516273i	−0.813516	−	0.516273i	0.0627905	−	0.998027i	$-0.480000\pi$
$653$	−0.813516	−	0.516273i	−0.813516	−	0.516273i	−0.876307	+	0.481754i	$0.840000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$659$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$660$	0.804733	+	1.10762i	0.804733	+	1.10762i
$661$	0.781202	+	0.733597i	0.781202	+	0.733597i	0.968583	−	0.248690i	$-0.0800000\pi$
$661$	0.781202	+	0.733597i	0.781202	+	0.733597i	−0.187381	+	0.982287i	$0.560000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.332404	+	0.0853469i	−0.332404	+	0.0853469i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$673$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$674$		0			0
$675$	−0.117696	+	0.125333i	−0.117696	+	0.125333i
$676$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$677$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$677$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.30113	+	0.825723i	−1.30113	+	0.825723i	−0.992115	−	0.125333i	$-0.960000\pi$
$683$	−1.30113	+	0.825723i	−1.30113	+	0.825723i	−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$	0.813516	−	1.47978i	0.813516	−	1.47978i
$686$		0			0
$687$	−1.57927	+	0.743150i	−1.57927	+	0.743150i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.0915446	−	0.0859661i	0.0915446	−	0.0859661i	−0.637424	−	0.770513i	$-0.720000\pi$
$691$	0.0915446	−	0.0859661i	0.0915446	−	0.0859661i	0.728969	+	0.684547i	$0.240000\pi$
$692$		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		0.809017	−	0.587785i	$-0.200000\pi$
$701$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$702$		0			0
$703$		0			0
$704$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$705$	2.58364	−	0.326389i	2.58364	−	0.326389i
$706$		0			0
$707$		0			0
$708$	0.512073	+	0.0322169i	0.512073	+	0.0322169i
$709$	0.746226	+	0.410241i	0.746226	+	0.410241i	0.809017	−	0.587785i	$-0.200000\pi$
$709$	0.746226	+	0.410241i	0.746226	+	0.410241i	−0.0627905	+	0.998027i	$0.520000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.43635	−	0.655518i	−3.43635	−	0.655518i
$714$		0			0
$715$		0			0
$716$	−0.456288	−	0.969661i	−0.456288	−	0.969661i
$717$		0			0
$718$		0			0
$719$	−0.812619	+	0.982287i	−0.812619	+	0.982287i	0.187381	+	0.982287i	$0.440000\pi$
$719$	−0.812619	+	0.982287i	−0.812619	+	0.982287i	−1.00000			$\pi$
$720$	0.846947	−	0.217459i	0.846947	−	0.217459i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$725$		0			0
$726$		0			0
$727$	0.946441	+	1.72157i	0.946441	+	1.72157i	0.637424	+	0.770513i	$0.280000\pi$
$727$	0.946441	+	1.72157i	0.946441	+	1.72157i	0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	−0.683430	+	0.270589i	−0.683430	+	0.270589i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$733$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$734$		0			0
$735$	1.15596	+	0.733597i	1.15596	+	0.733597i
$736$		0			0
$737$	−0.0922765	−	0.730444i	−0.0922765	−	0.730444i
$738$		0			0
$739$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$739$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$740$	0.804733	−	0.856954i	0.804733	−	0.856954i
$741$		0			0
$742$		0			0
$743$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$743$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$751$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$752$	1.72108	+	0.809880i	1.72108	+	0.809880i
$753$	1.16356	−	0.0732052i	1.16356	−	0.0732052i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.86842	+	0.607087i	−1.86842	+	0.607087i	−0.876307	+	0.481754i	$0.840000\pi$
$757$	−1.86842	+	0.607087i	−1.86842	+	0.607087i	−0.992115	+	0.125333i	$0.960000\pi$
$758$		0			0
$759$	2.71124	−	0.342509i	2.71124	−	0.342509i
$760$		0			0
$761$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$761$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$762$		0			0
$763$		0			0
$764$	−1.92189	−	0.242791i	−1.92189	−	0.242791i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.30209	+	0.423073i	1.30209	+	0.423073i
$769$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$769$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$770$		0			0
$771$	−2.54088	+	1.00601i	−2.54088	+	1.00601i
$772$		0			0
$773$	0.871808	+	1.58581i	0.871808	+	1.58581i	0.809017	+	0.587785i	$0.200000\pi$
$773$	0.871808	+	1.58581i	0.871808	+	1.58581i	0.0627905	+	0.998027i	$0.480000\pi$
$774$		0			0
$775$	−0.110048	+	1.74915i	−0.110048	+	1.74915i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.929324	−	1.12336i	0.929324	−	1.12336i
$782$		0			0
$783$		0			0
$784$	0.425779	+	0.904827i	0.425779	+	0.904827i
$785$	1.46560	+	0.476203i	1.46560	+	0.476203i
$786$		0			0
$787$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$787$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−1.80608	+	1.69602i	−1.80608	+	1.69602i
$796$	−0.0672897	−	1.06954i	−0.0672897	−	1.06954i
$797$	−0.123693	+	0.481754i	−0.123693	+	0.481754i	0.876307	+	0.481754i	$0.160000\pi$
$797$	−0.123693	+	0.481754i	−0.123693	+	0.481754i	−1.00000			$1.00000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.37039	−	0.995646i	−1.37039	−	0.995646i
$802$		0			0
$803$		0			0
$804$	−0.976326	−	0.250678i	−0.976326	−	0.250678i
$805$		0			0
$806$		0			0
$807$		−	1.74539i		−	1.74539i
$808$		0			0
$809$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$809$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$810$		0			0
$811$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$811$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.946441	+	0.180543i	0.946441	+	0.180543i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$821$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$822$		0			0
$823$	1.23879	−	1.31918i	1.23879	−	1.31918i	0.309017	−	0.951057i	$-0.400000\pi$
$823$	1.23879	−	1.31918i	1.23879	−	1.31918i	0.929776	−	0.368125i	$-0.120000\pi$
$824$		0			0
$825$	−0.340480	−	1.32608i	−0.340480	−	1.32608i
$826$		0			0
$827$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$827$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$828$	0.434060	−	1.69055i	0.434060	−	1.69055i
$829$	0.124591	+	1.98031i	0.124591	+	1.98031i	0.187381	+	0.982287i	$0.440000\pi$
$829$	0.124591	+	1.98031i	0.124591	+	1.98031i	−0.0627905	+	0.998027i	$0.520000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.145167	−	0.264058i	0.145167	−	0.264058i
$838$		0			0
$839$	−1.35556	−	1.27295i	−1.35556	−	1.27295i	−0.929776	−	0.368125i	$-0.880000\pi$
$839$	−1.35556	−	1.27295i	−1.35556	−	1.27295i	−0.425779	−	0.904827i	$-0.640000\pi$
$840$		0			0
$841$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.535827	+	0.844328i	0.535827	+	0.844328i
$846$		0			0
$847$		0			0
$848$	−1.77760	+	0.339095i	−1.77760	+	0.339095i
$849$		0			0
$850$		0			0
$851$	−0.725109	−	2.23166i	−0.725109	−	2.23166i
$852$	−0.961606	−	1.74915i	−0.961606	−	1.74915i
$853$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$853$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$857$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$858$		0			0
$859$	−0.159566	+	0.836475i	−0.159566	+	0.836475i	0.809017	+	0.587785i	$0.200000\pi$
$859$	−0.159566	+	0.836475i	−0.159566	+	0.836475i	−0.968583	+	0.248690i	$0.920000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.659566	−	0.702367i	−0.659566	−	0.702367i	0.309017	−	0.951057i	$-0.400000\pi$
$863$	−0.659566	−	0.702367i	−0.659566	−	0.702367i	−0.968583	+	0.248690i	$0.920000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.340480	−	1.32608i	−0.340480	−	1.32608i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.55437	−	0.731430i	−1.55437	−	0.731430i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$877$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$878$		0			0
$879$		0			0
$880$	0.309017	−	0.951057i	0.309017	−	0.951057i
$881$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.968583	+	0.248690i	$0.0800000\pi$
$881$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.876307	+	0.481754i	$0.160000\pi$
$882$		0			0
$883$	−0.239615	−	0.933237i	−0.239615	−	0.933237i	−0.968583	−	0.248690i	$-0.920000\pi$
$883$	−0.239615	−	0.933237i	−0.239615	−	0.933237i	0.728969	−	0.684547i	$-0.240000\pi$
$884$		0			0
$885$	−0.464254	−	0.218461i	−0.464254	−	0.218461i
$886$		0			0
$887$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$887$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.207958	+	1.09015i	−0.207958	+	1.09015i
$892$	0.193142	+	0.159781i	0.193142	+	0.159781i
$893$		0			0
$894$		0			0
$895$	0.0672897	+	1.06954i	0.0672897	+	1.06954i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.867524	−	0.109594i	−0.867524	−	0.109594i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.03137	−	1.24672i	1.03137	−	1.24672i
$906$		0			0
$907$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$907$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.35556	+	1.27295i	1.35556	+	1.27295i	0.929776	+	0.368125i	$0.120000\pi$
$911$	1.35556	+	1.27295i	1.35556	+	1.27295i	0.425779	+	0.904827i	$0.360000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.11716	+	0.614163i	1.11716	+	0.614163i
$917$		0			0
$918$		0			0
$919$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$919$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.06369	+	0.500534i	−1.06369	+	0.500534i
$926$		0			0
$927$	−0.440705	+	0.469303i	−0.440705	+	0.469303i
$928$		0			0
$929$	1.03799	+	0.266509i	1.03799	+	0.266509i	0.728969	−	0.684547i	$-0.240000\pi$
$929$	1.03799	+	0.266509i	1.03799	+	0.266509i	0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	2.14958	−	1.36416i	2.14958	−	1.36416i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$937$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$938$		0			0
$939$	−2.02596	+	1.11378i	−2.02596	+	1.11378i
$940$	−1.30209	−	1.38658i	−1.30209	−	1.38658i
$941$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$941$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$942$		0			0
$943$		0			0
$944$	−0.200808	−	0.316423i	−0.200808	−	0.316423i
$945$		0			0
$946$		0			0
$947$	−0.0922765	−	0.233064i	−0.0922765	−	0.233064i	0.876307	−	0.481754i	$-0.160000\pi$
$947$	−0.0922765	−	0.233064i	−0.0922765	−	0.233064i	−0.968583	+	0.248690i	$0.920000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−0.277643	+	0.201720i	−0.277643	+	0.201720i
$952$		0			0
$953$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$953$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$954$		0			0
$955$	1.69755	+	0.933237i	1.69755	+	0.933237i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1.05491	−	0.872693i	−1.05491	−	0.872693i
$961$	−0.388189	−	2.03496i	−0.388189	−	2.03496i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$967$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.456288	−	0.718995i	0.456288	−	0.718995i	−0.535827	−	0.844328i	$-0.680000\pi$
$971$	0.456288	−	0.718995i	0.456288	−	0.718995i	0.992115	+	0.125333i	$0.0400000\pi$
$972$	1.13773	+	0.722028i	1.13773	+	0.722028i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.566335	+	1.03016i	0.566335	+	1.03016i	0.992115	+	0.125333i	$0.0400000\pi$
$977$	0.566335	+	1.03016i	0.566335	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$978$		0			0
$979$	−1.80113	+	0.713118i	−1.80113	+	0.713118i
$980$	−0.0627905	−	0.998027i	−0.0627905	−	0.998027i
$981$		0			0
$982$		0			0
$983$	−1.30113	−	1.07639i	−1.30113	−	1.07639i	−0.992115	−	0.125333i	$-0.960000\pi$
$983$	−1.30113	−	1.07639i	−1.30113	−	1.07639i	−0.309017	−	0.951057i	$-0.600000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.371808	+	0.0469702i	−0.371808	+	0.0469702i	−0.309017	−	0.951057i	$-0.600000\pi$
$991$	−0.371808	+	0.0469702i	−0.371808	+	0.0469702i	−0.0627905	+	0.998027i	$0.520000\pi$
$992$		0			0
$993$	−2.42130	+	0.786727i	−2.42130	+	0.786727i
$994$		0			0
$995$	−0.331159	+	1.01920i	−0.331159	+	1.01920i
$996$		0			0
$997$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$997$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$998$		0			0
$999$	0.202119			0.202119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1375.1.bz.a.384.1	✓	20
11.10	odd	2	CM	1375.1.bz.a.384.1	✓	20
125.14	even	50	inner	1375.1.bz.a.1264.1	yes	20
1375.1264	odd	50	inner	1375.1.bz.a.1264.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1375.1.bz.a.384.1	✓	20	1.1	even	1	trivial
1375.1.bz.a.384.1	✓	20	11.10	odd	2	CM
1375.1.bz.a.1264.1	yes	20	125.14	even	50	inner
1375.1.bz.a.1264.1	yes	20	1375.1264	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:	\( N \)	\(=\)	\( 1375 = 5^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1375.bz (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.36639 + 0.0859661i) q^{3} +(0.637424 + 0.770513i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(0.867524 - 0.109594i) q^{9} +O(q^{10})\)	\(q+(-1.36639 + 0.0859661i) q^{3} +(0.637424 + 0.770513i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(0.867524 - 0.109594i) q^{9} +(0.425779 - 0.904827i) q^{11} +(-0.937209 - 0.998027i) q^{12} +(0.503997 + 1.27295i) q^{15} +(-0.187381 + 0.982287i) q^{16} +(0.535827 - 0.844328i) q^{20} +(-0.961606 - 1.74915i) q^{23} +(-0.809017 + 0.587785i) q^{25} +(0.168887 - 0.0322169i) q^{27} +(1.11716 - 1.35041i) q^{31} +(-0.503997 + 1.27295i) q^{33} +(0.637424 + 0.598581i) q^{36} +(1.15475 + 0.220280i) q^{37} +(0.968583 - 0.248690i) q^{44} +(-0.372309 - 0.791198i) q^{45} +(0.473036 - 1.84235i) q^{47} +(0.171593 - 1.35830i) q^{48} +(0.809017 - 0.587785i) q^{49} +(0.666178 + 1.68257i) q^{53} +(-0.992115 - 0.125333i) q^{55} +(-0.273190 + 0.256543i) q^{59} +(-0.659566 + 1.19975i) q^{60} +(-0.876307 + 0.481754i) q^{64} +(0.621636 - 0.394502i) q^{67} +(1.46430 + 2.30737i) q^{69} +(1.41213 + 0.362574i) q^{71} +(1.05491 - 0.872693i) q^{75} +(0.992115 - 0.125333i) q^{80} +(-1.07494 + 0.275999i) q^{81} +(-1.41213 - 1.32608i) q^{89} +(0.734796 - 1.85588i) q^{92} +(-1.41039 + 1.94123i) q^{93} +(-1.65875 - 1.05267i) q^{97} +(0.270210 - 0.831622i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{5}+O(q^{10}) \) 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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