LMFDB - Embedded newform 1375.1.bz.a.934.1

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			934.1
Root			$-0.968583 + 0.248690i$ of defining polynomial
Character	$\chi$	$=$	1375.934
Dual form			1375.1.bz.a.714.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.723208 + 1.82662i) q^{3} +(-0.535827 + 0.844328i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(-2.08452 + 1.95750i) q^{9} +O(q^{10})$	\(q+(0.723208 + 1.82662i) q^{3} +(-0.535827 + 0.844328i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(-2.08452 + 1.95750i) q^{9} +(-0.876307 + 0.481754i) q^{11} +(-1.92978 - 0.368125i) q^{12} +(1.51373 - 1.25227i) q^{15} +(-0.425779 - 0.904827i) q^{16} +(0.968583 + 0.248690i) q^{20} +(-0.0922765 + 0.730444i) q^{23} +(-0.809017 + 0.587785i) q^{25} +(-3.30554 - 1.55547i) q^{27} +(1.06320 + 1.67534i) q^{31} +(-1.51373 - 1.25227i) q^{33} +(-0.535827 - 2.80890i) q^{36} +(-1.06369 + 0.500534i) q^{37} +(0.0627905 - 0.998027i) q^{44} +(2.50585 + 1.37760i) q^{45} +(1.89836 - 0.119435i) q^{47} +(1.34484 - 1.43211i) q^{48} +(0.809017 - 0.587785i) q^{49} +(0.742395 - 0.614163i) q^{53} +(0.728969 + 0.684547i) q^{55} +(0.159566 - 0.836475i) q^{59} +(0.246226 + 1.94908i) q^{60} +(0.992115 + 0.125333i) q^{64} +(0.383238 + 1.49261i) q^{67} +(-1.40097 + 0.359709i) q^{69} +(-0.0235315 - 0.374023i) q^{71} +(-1.65875 - 1.05267i) q^{75} +(-0.728969 + 0.684547i) q^{80} +(0.271097 - 4.30897i) q^{81} +(0.0235315 + 0.123357i) q^{89} +(-0.567290 - 0.469303i) q^{92} +(-2.29128 + 3.15368i) q^{93} +(-0.450043 + 1.75280i) q^{97} +(0.883650 - 2.71960i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$.

$n$	$376$	$1002$
$\chi(n)$	$-1$	$e\left(\frac{17}{50}\right)$

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		−0.481754	−	0.876307i	$-0.660000\pi$
$2$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$3$	0.723208	+	1.82662i	0.723208	+	1.82662i	0.535827	+	0.844328i	$0.320000\pi$
$3$	0.723208	+	1.82662i	0.723208	+	1.82662i	0.187381	+	0.982287i	$0.440000\pi$
$4$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$5$	−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
$9$	−2.08452	+	1.95750i	−2.08452	+	1.95750i
$10$		0			0
$11$	−0.876307	+	0.481754i	−0.876307	+	0.481754i
$11$	−0.876307	+	0.481754i	−0.876307	+	0.481754i
$12$	−1.92978	−	0.368125i	−1.92978	−	0.368125i
$13$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$13$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$14$		0			0
$15$	1.51373	−	1.25227i	1.51373	−	1.25227i
$16$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$17$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$17$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$18$		0			0
$19$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$19$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$20$	0.968583	+	0.248690i	0.968583	+	0.248690i
$21$		0			0
$22$		0			0
$23$	−0.0922765	+	0.730444i	−0.0922765	+	0.730444i	0.876307	+	0.481754i	$0.160000\pi$
$23$	−0.0922765	+	0.730444i	−0.0922765	+	0.730444i	−0.968583	+	0.248690i	$0.920000\pi$
$24$		0			0
$25$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$26$		0			0
$27$	−3.30554	−	1.55547i	−3.30554	−	1.55547i
$28$		0			0
$29$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$29$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$30$		0			0
$31$	1.06320	+	1.67534i	1.06320	+	1.67534i	0.637424	+	0.770513i	$0.280000\pi$
$31$	1.06320	+	1.67534i	1.06320	+	1.67534i	0.425779	+	0.904827i	$0.360000\pi$
$32$		0			0
$33$	−1.51373	−	1.25227i	−1.51373	−	1.25227i
$34$		0			0
$35$		0			0
$36$	−0.535827	−	2.80890i	−0.535827	−	2.80890i
$37$	−1.06369	+	0.500534i	−1.06369	+	0.500534i	−0.876307	−	0.481754i	$-0.840000\pi$
$37$	−1.06369	+	0.500534i	−1.06369	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$41$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$42$		0			0
$43$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$43$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$44$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$45$	2.50585	+	1.37760i	2.50585	+	1.37760i
$46$		0			0
$47$	1.89836	−	0.119435i	1.89836	−	0.119435i	0.929776	−	0.368125i	$-0.120000\pi$
$47$	1.89836	−	0.119435i	1.89836	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$48$	1.34484	−	1.43211i	1.34484	−	1.43211i
$49$	0.809017	−	0.587785i	0.809017	−	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.742395	−	0.614163i	0.742395	−	0.614163i	−0.187381	−	0.982287i	$-0.560000\pi$
$53$	0.742395	−	0.614163i	0.742395	−	0.614163i	0.929776	+	0.368125i	$0.120000\pi$
$54$		0			0
$55$	0.728969	+	0.684547i	0.728969	+	0.684547i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.159566	−	0.836475i	0.159566	−	0.836475i	−0.809017	−	0.587785i	$-0.800000\pi$
$59$	0.159566	−	0.836475i	0.159566	−	0.836475i	0.968583	−	0.248690i	$-0.0800000\pi$
$60$	0.246226	+	1.94908i	0.246226	+	1.94908i
$61$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$61$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$62$		0			0
$63$		0			0
$64$	0.992115	+	0.125333i	0.992115	+	0.125333i
$65$		0			0
$66$		0			0
$67$	0.383238	+	1.49261i	0.383238	+	1.49261i	0.809017	+	0.587785i	$0.200000\pi$
$67$	0.383238	+	1.49261i	0.383238	+	1.49261i	−0.425779	+	0.904827i	$0.640000\pi$
$68$		0			0
$69$	−1.40097	+	0.359709i	−1.40097	+	0.359709i
$70$		0			0
$71$	−0.0235315	−	0.374023i	−0.0235315	−	0.374023i	−0.992115	−	0.125333i	$-0.960000\pi$
$71$	−0.0235315	−	0.374023i	−0.0235315	−	0.374023i	0.968583	−	0.248690i	$-0.0800000\pi$
$72$		0			0
$73$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$73$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$74$		0			0
$75$	−1.65875	−	1.05267i	−1.65875	−	1.05267i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$79$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$80$	−0.728969	+	0.684547i	−0.728969	+	0.684547i
$81$	0.271097	−	4.30897i	0.271097	−	4.30897i
$82$		0			0
$83$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$83$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.0235315	+	0.123357i	0.0235315	+	0.123357i	0.992115	−	0.125333i	$-0.0400000\pi$
$89$	0.0235315	+	0.123357i	0.0235315	+	0.123357i	−0.968583	+	0.248690i	$0.920000\pi$
$90$		0			0
$91$		0			0
$92$	−0.567290	−	0.469303i	−0.567290	−	0.469303i
$93$	−2.29128	+	3.15368i	−2.29128	+	3.15368i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.450043	+	1.75280i	−0.450043	+	1.75280i	0.187381	+	0.982287i	$0.440000\pi$
$97$	−0.450043	+	1.75280i	−0.450043	+	1.75280i	−0.637424	+	0.770513i	$0.720000\pi$
$98$		0			0
$99$	0.883650	−	2.71960i	0.883650	−	2.71960i
$100$	−0.0627905	−	0.998027i	−0.0627905	−	0.998027i
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.992115	−	0.125333i	$-0.0400000\pi$
$103$	1.30113	+	0.825723i	1.30113	+	0.825723i	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	$0.100000\pi$
$108$	3.08452	−	1.95750i	3.08452	−	1.95750i
$109$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$109$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$110$		0			0
$111$	−1.68355	−	1.58096i	−1.68355	−	1.58096i
$112$		0			0
$113$	−1.96070	−	0.374023i	−1.96070	−	0.374023i	−0.992115	−	0.125333i	$-0.960000\pi$
$113$	−1.96070	−	0.374023i	−1.96070	−	0.374023i	−0.968583	−	0.248690i	$-0.920000\pi$
$114$		0			0
$115$	0.723208	−	0.137959i	0.723208	−	0.137959i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.535827	−	0.844328i	0.535827	−	0.844328i
$122$		0			0
$123$		0			0
$124$	−1.98423			−1.98423
$125$	0.809017	+	0.587785i	0.809017	+	0.587785i
$126$		0			0
$127$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$127$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$131$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$132$	1.86842	−	0.607087i	1.86842	−	0.607087i
$133$		0			0
$134$		0			0
$135$	−0.457871	+	3.62442i	−0.457871	+	3.62442i
$136$		0			0
$137$	0.488570	+	0.0931997i	0.488570	+	0.0931997i	0.425779	−	0.904827i	$-0.360000\pi$
$137$	0.488570	+	0.0931997i	0.488570	+	0.0931997i	0.0627905	+	0.998027i	$0.480000\pi$
$138$		0			0
$139$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$139$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$140$		0			0
$141$	1.59107	+	3.38120i	1.59107	+	3.38120i
$142$		0			0
$143$		0			0
$144$	2.65875	+	1.05267i	2.65875	+	1.05267i
$145$		0			0
$146$		0			0
$147$	1.65875	+	1.05267i	1.65875	+	1.05267i
$148$	0.147338	−	1.16630i	0.147338	−	1.16630i
$149$		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
$155$	1.26480	−	1.52888i	1.26480	−	1.52888i
$156$		0			0
$157$	−0.992567	+	1.36615i	−0.992567	+	1.36615i	−0.0627905	+	0.998027i	$0.520000\pi$
$157$	−0.992567	+	1.36615i	−0.992567	+	1.36615i	−0.929776	+	0.368125i	$0.880000\pi$
$158$		0			0
$159$	1.65875	+	0.911903i	1.65875	+	0.911903i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.0314168	−	0.248690i	−0.0314168	−	0.248690i	0.968583	−	0.248690i	$-0.0800000\pi$
$163$	−0.0314168	−	0.248690i	−0.0314168	−	0.248690i	−1.00000			$\pi$
$164$		0			0
$165$	−0.723208	+	1.82662i	−0.723208	+	1.82662i
$166$		0			0
$167$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$167$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$168$		0			0
$169$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$173$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$174$		0			0
$175$		0			0
$176$	0.809017	+	0.587785i	0.809017	+	0.587785i
$177$	1.64332	−	0.313480i	1.64332	−	0.313480i
$178$		0			0
$179$	−0.121636	−	1.93334i	−0.121636	−	1.93334i	−0.309017	−	0.951057i	$-0.600000\pi$
$179$	−0.121636	−	1.93334i	−0.121636	−	1.93334i	0.187381	−	0.982287i	$-0.440000\pi$
$180$	−2.50585	+	1.37760i	−2.50585	+	1.37760i
$181$	1.56720	−	0.402389i	1.56720	−	0.402389i	0.637424	−	0.770513i	$-0.280000\pi$
$181$	1.56720	−	0.402389i	1.56720	−	0.402389i	0.929776	+	0.368125i	$0.120000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.804733	+	0.856954i	0.804733	+	0.856954i
$186$		0			0
$187$		0			0
$188$	−0.916350	+	1.66683i	−0.916350	+	1.66683i
$189$		0			0
$190$		0			0
$191$	0.0235315	−	0.123357i	0.0235315	−	0.123357i	−0.968583	−	0.248690i	$-0.920000\pi$
$191$	0.0235315	−	0.123357i	0.0235315	−	0.123357i	0.992115	+	0.125333i	$0.0400000\pi$
$192$	0.488570	+	1.90285i	0.488570	+	1.90285i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$197$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$197$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$198$		0			0
$199$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$199$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$200$		0			0
$201$	−2.44927	+	1.77950i	−2.44927	+	1.77950i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.23749	−	1.70326i	−1.23749	−	1.70326i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$211$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$212$	0.120759	+	0.955910i	0.120759	+	0.955910i
$213$	0.666178	−	0.313480i	0.666178	−	0.313480i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.968583	+	0.248690i	−0.968583	+	0.248690i
$221$		0			0
$222$		0			0
$223$	1.23879	+	0.582932i	1.23879	+	0.582932i	0.929776	−	0.368125i	$-0.120000\pi$
$223$	1.23879	+	0.582932i	1.23879	+	0.582932i	0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	0.535827	−	2.80890i	0.535827	−	2.80890i
$226$		0			0
$227$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$227$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$228$		0			0
$229$	−0.683098	−	0.825723i	−0.683098	−	0.825723i	0.309017	−	0.951057i	$-0.400000\pi$
$229$	−0.683098	−	0.825723i	−0.683098	−	0.825723i	−0.992115	+	0.125333i	$0.960000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$233$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$234$		0			0
$235$	−0.700215	−	1.76854i	−0.700215	−	1.76854i
$236$	0.620759	+	0.582932i	0.620759	+	0.582932i
$237$		0			0
$238$		0			0
$239$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$239$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$240$	−1.77760	−	0.836475i	−1.77760	−	0.836475i
$241$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$241$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$242$		0			0
$243$	4.59246	−	1.49218i	4.59246	−	1.49218i
$244$		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
$251$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$251$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$252$		0			0
$253$	−0.271031	−	0.684547i	−0.271031	−	0.684547i
$254$		0			0
$255$		0			0
$256$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$257$	0.700215	−	0.227513i	0.700215	−	0.227513i	0.0627905	−	0.998027i	$-0.480000\pi$
$257$	0.700215	−	0.227513i	0.700215	−	0.227513i	0.637424	+	0.770513i	$0.280000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$263$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$264$		0			0
$265$	−0.813516	−	0.516273i	−0.813516	−	0.516273i
$266$		0			0
$267$	−0.208307	+	0.132196i	−0.208307	+	0.132196i
$268$	−1.46560	−	0.476203i	−1.46560	−	0.476203i
$269$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$269$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$271$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.425779	−	0.904827i	0.425779	−	0.904827i
$276$	0.446967	−	1.37562i	0.446967	−	1.37562i
$277$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$277$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$278$		0			0
$279$	−5.49575	−	1.41107i	−5.49575	−	1.41107i
$280$		0			0
$281$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$281$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$282$		0			0
$283$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$283$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$284$	0.328407	+	0.180543i	0.328407	+	0.180543i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.425779	−	0.904827i	0.425779	−	0.904827i
$290$		0			0
$291$	−3.52717	+	0.445585i	−3.52717	+	0.445585i
$292$		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
$295$	−0.844844	+	0.106729i	−0.844844	+	0.106729i
$296$		0			0
$297$	3.64602	−	0.229388i	3.64602	−	0.229388i
$298$		0			0
$299$		0			0
$300$	1.77760	−	0.836475i	1.77760	−	0.836475i
$301$		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
$309$	−0.567290	+	2.97384i	−0.567290	+	2.97384i
$310$		0			0
$311$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.728969	−	0.684547i	$-0.240000\pi$
$311$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.535827	+	0.844328i	$0.320000\pi$
$312$		0			0
$313$	0.239615	−	0.435857i	0.239615	−	0.435857i	−0.728969	−	0.684547i	$-0.760000\pi$
$313$	0.239615	−	0.435857i	0.239615	−	0.435857i	0.968583	+	0.248690i	$0.0800000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.340480	+	1.32608i	0.340480	+	1.32608i	0.876307	+	0.481754i	$0.160000\pi$
$317$	0.340480	+	1.32608i	0.340480	+	1.32608i	−0.535827	+	0.844328i	$0.680000\pi$
$318$		0			0
$319$		0			0
$320$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	3.49292	+	2.53776i	3.49292	+	2.53776i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0800484	−	1.27233i	0.0800484	−	1.27233i	−0.728969	−	0.684547i	$-0.760000\pi$
$331$	0.0800484	−	1.27233i	0.0800484	−	1.27233i	0.809017	−	0.587785i	$-0.200000\pi$
$332$		0			0
$333$	1.23749	−	3.12554i	1.23749	−	3.12554i
$334$		0			0
$335$	1.30113	−	0.825723i	1.30113	−	0.825723i
$336$		0			0
$337$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$337$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$338$		0			0
$339$	−0.734796	−	3.85194i	−0.734796	−	3.85194i
$340$		0			0
$341$	−1.73879	−	0.955910i	−1.73879	−	0.955910i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.775029	+	1.22125i	0.775029	+	1.22125i
$346$		0			0
$347$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$347$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$348$		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		0.535827	−	0.844328i	$-0.320000\pi$
$353$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$354$		0			0
$355$	−0.348445	+	0.137959i	−0.348445	+	0.137959i
$356$	−0.116762	−	0.0462295i	−0.116762	−	0.0462295i
$357$		0			0
$358$		0			0
$359$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$359$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$360$		0			0
$361$	0.728969	+	0.684547i	0.728969	+	0.684547i
$362$		0			0
$363$	1.92978	+	0.368125i	1.92978	+	0.368125i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.496398	−	0.0312307i	−0.496398	−	0.0312307i	−0.187381	−	0.982287i	$-0.560000\pi$
$367$	−0.496398	−	0.0312307i	−0.496398	−	0.0312307i	−0.309017	+	0.951057i	$0.600000\pi$
$368$	0.700215	−	0.227513i	0.700215	−	0.227513i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.43501	−	3.62442i	−1.43501	−	3.62442i
$373$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$373$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$374$		0			0
$375$	−0.488570	+	1.90285i	−0.488570	+	1.90285i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.331159	−	0.521823i	0.331159	−	0.521823i	−0.637424	−	0.770513i	$-0.720000\pi$
$379$	0.331159	−	0.521823i	0.331159	−	0.521823i	0.968583	+	0.248690i	$0.0800000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.250172	+	0.0157395i	0.250172	+	0.0157395i	0.187381	−	0.982287i	$-0.440000\pi$
$383$	0.250172	+	0.0157395i	0.250172	+	0.0157395i	0.0627905	+	0.998027i	$0.480000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.23879	−	1.31918i	−1.23879	−	1.31918i
$389$	1.06279	+	0.998027i	1.06279	+	0.998027i	1.00000			$0$
$389$	1.06279	+	0.998027i	1.06279	+	0.998027i	0.0627905	+	0.998027i	$0.480000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	1.82275	+	2.20332i	1.82275	+	2.20332i
$397$	−1.68532	−	1.06954i	−1.68532	−	1.06954i	−0.876307	−	0.481754i	$-0.840000\pi$
$397$	−1.68532	−	1.06954i	−1.68532	−	1.06954i	−0.809017	−	0.587785i	$-0.800000\pi$
$398$		0			0
$399$		0			0
$400$	0.876307	+	0.481754i	0.876307	+	0.481754i
$401$	−0.541587	+	1.66683i	−0.541587	+	1.66683i	0.187381	+	0.982287i	$0.440000\pi$
$401$	−0.541587	+	1.66683i	−0.541587	+	1.66683i	−0.728969	+	0.684547i	$0.760000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−4.18185	+	1.07372i	−4.18185	+	1.07372i
$406$		0			0
$407$	0.690983	−	0.951057i	0.690983	−	0.951057i
$408$		0			0
$409$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$409$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$410$		0			0
$411$	0.183098	+	0.959832i	0.183098	+	0.959832i
$412$	−1.39436	+	0.656137i	−1.39436	+	0.656137i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.0534698	−	0.849878i	0.0534698	−	0.849878i	−0.876307	−	0.481754i	$-0.840000\pi$
$419$	0.0534698	−	0.849878i	0.0534698	−	0.849878i	0.929776	−	0.368125i	$-0.120000\pi$
$420$		0			0
$421$	0.116762	−	0.0462295i	0.116762	−	0.0462295i	−0.309017	−	0.951057i	$-0.600000\pi$
$421$	0.116762	−	0.0462295i	0.116762	−	0.0462295i	0.425779	+	0.904827i	$0.360000\pi$
$422$		0			0
$423$	−3.72338	+	3.96500i	−3.72338	+	3.96500i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$431$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$432$			3.65323i			3.65323i
$433$	−0.292352	−	1.13864i	−0.292352	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$433$	−0.292352	−	1.13864i	−0.292352	−	1.13864i	0.637424	−	0.770513i	$-0.280000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$439$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$440$		0			0
$441$	−0.535827	+	2.80890i	−0.535827	+	2.80890i
$442$		0			0
$443$		−	1.80965i		−	1.80965i	−0.425779	−	0.904827i	$-0.640000\pi$
$443$		−	1.80965i		−	1.80965i	0.425779	−	0.904827i	$-0.360000\pi$
$444$	2.23694	−	0.574348i	2.23694	−	0.574348i
$445$	0.110048	−	0.0604991i	0.110048	−	0.0604991i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.50441	−	1.09302i	−1.50441	−	1.09302i	−0.968583	−	0.248690i	$-0.920000\pi$
$449$	−1.50441	−	1.09302i	−1.50441	−	1.09302i	−0.535827	−	0.844328i	$-0.680000\pi$
$450$		0			0
$451$		0			0
$452$	1.36639	−	1.45506i	1.36639	−	1.45506i
$453$		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
$460$	−0.271031	+	0.684547i	−0.271031	+	0.684547i
$461$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$461$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$462$		0			0
$463$	−1.80608	+	0.849878i	−1.80608	+	0.849878i	−0.876307	+	0.481754i	$0.840000\pi$
$463$	−1.80608	+	0.849878i	−1.80608	+	0.849878i	−0.929776	+	0.368125i	$0.880000\pi$
$464$		0			0
$465$	3.70738	+	1.20460i	3.70738	+	1.20460i
$466$		0			0
$467$	−1.46560	−	1.21245i	−1.46560	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$467$	−1.46560	−	1.21245i	−1.46560	−	1.21245i	−0.535827	−	0.844328i	$-0.680000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−3.21327	−	0.825026i	−3.21327	−	0.825026i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.345318	+	2.73347i	−0.345318	+	2.73347i
$478$		0			0
$479$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$479$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.425779	+	0.904827i	0.425779	+	0.904827i
$485$	1.80608	−	0.113629i	1.80608	−	0.113629i
$486$		0			0
$487$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$487$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$488$		0			0
$489$	0.431540	−	0.237241i	0.431540	−	0.237241i
$490$		0			0
$491$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$491$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−2.85955			−2.85955
$496$	1.06320	−	1.67534i	1.06320	−	1.67534i
$497$		0			0
$498$		0			0
$499$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$499$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$500$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$501$		0			0
$502$		0			0
$503$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$503$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.86842	+	0.607087i	−1.86842	+	0.607087i
$508$		0			0
$509$	1.41213	−	1.32608i	1.41213	−	1.32608i	0.535827	−	0.844328i	$-0.320000\pi$
$509$	1.41213	−	1.32608i	1.41213	−	1.32608i	0.876307	−	0.481754i	$-0.160000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.383238	−	1.49261i	0.383238	−	1.49261i
$516$		0			0
$517$	−1.60601	+	1.01920i	−1.60601	+	1.01920i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.812619	−	0.982287i	−0.812619	−	0.982287i	0.187381	−	0.982287i	$-0.440000\pi$
$521$	−0.812619	−	0.982287i	−0.812619	−	0.982287i	−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$523$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.488570	+	1.90285i	−0.488570	+	1.90285i
$529$	0.443550	+	0.113884i	0.443550	+	0.113884i
$530$		0			0
$531$	1.30478	+	2.05600i	1.30478	+	2.05600i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.44351	−	1.62039i	3.44351	−	1.62039i
$538$		0			0
$539$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$540$	−2.81486	−	2.32866i	−2.81486	−	2.32866i
$541$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$541$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$542$		0			0
$543$	1.86842	+	2.57166i	1.86842	+	2.57166i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$547$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$548$	−0.340480	+	0.362574i	−0.340480	+	0.362574i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.983336	+	2.08969i	−0.983336	+	2.08969i
$556$		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		0.481754	−	0.876307i	$-0.340000\pi$
$563$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$564$	−3.70738	−	0.468351i	−3.70738	−	0.468351i
$565$	0.250172	+	1.98031i	0.250172	+	1.98031i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$569$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$570$		0			0
$571$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$571$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$572$		0			0
$573$	0.242343	−	0.0462295i	0.242343	−	0.0462295i
$574$		0			0
$575$	−0.354691	−	0.645180i	−0.354691	−	0.645180i
$576$	−2.31343	+	1.68080i	−2.31343	+	1.68080i
$577$	0.937209	−	0.998027i	0.937209	−	0.998027i	−0.0627905	−	0.998027i	$-0.520000\pi$
$577$	0.937209	−	0.998027i	0.937209	−	0.998027i	1.00000			$0$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.354691	+	0.895846i	−0.354691	+	0.895846i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.226810	+	1.79538i	0.226810	+	1.79538i	0.535827	+	0.844328i	$0.320000\pi$
$587$	0.226810	+	1.79538i	0.226810	+	1.79538i	−0.309017	+	0.951057i	$0.600000\pi$
$588$	−1.77760	+	0.836475i	−1.77760	+	0.836475i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.905793	+	0.749337i	0.905793	+	0.749337i
$593$		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
$597$	0.946441	−	3.68614i	0.946441	−	3.68614i
$598$		0			0
$599$	−0.450527	+	1.38658i	−0.450527	+	1.38658i	0.425779	+	0.904827i	$0.360000\pi$
$599$	−0.450527	+	1.38658i	−0.450527	+	1.38658i	−0.876307	+	0.481754i	$0.840000\pi$
$600$		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
$603$	−3.72066	−	2.36120i	−3.72066	−	2.36120i
$604$		0			0
$605$	−0.968583	−	0.248690i	−0.968583	−	0.248690i
$606$		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		−0.982287	−	0.187381i	$-0.940000\pi$
$613$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.36639	−	0.0859661i	−1.36639	−	0.0859661i	−0.637424	−	0.770513i	$-0.720000\pi$
$617$	−1.36639	−	0.0859661i	−1.36639	−	0.0859661i	−0.728969	+	0.684547i	$0.760000\pi$
$618$		0			0
$619$	−1.26480	+	1.52888i	−1.26480	+	1.52888i	−0.535827	+	0.844328i	$0.680000\pi$
$619$	−1.26480	+	1.52888i	−1.26480	+	1.52888i	−0.728969	+	0.684547i	$0.760000\pi$
$620$	0.613161	+	1.88711i	0.613161	+	1.88711i
$621$	1.44121	−	2.27098i	1.44121	−	2.27098i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.309017	−	0.951057i	0.309017	−	0.951057i
$626$		0			0
$627$		0			0
$628$	−0.621636	−	1.57007i	−0.621636	−	1.57007i
$629$		0			0
$630$		0			0
$631$	0.812619	−	0.982287i	0.812619	−	0.982287i	−0.187381	−	0.982287i	$-0.560000\pi$
$631$	0.812619	−	0.982287i	0.812619	−	0.982287i	1.00000			$0$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.65875	+	0.911903i	−1.65875	+	0.911903i
$637$		0			0
$638$		0			0
$639$	0.781202	+	0.733597i	0.781202	+	0.733597i
$640$		0			0
$641$	−0.159566	−	0.339095i	−0.159566	−	0.339095i	0.809017	−	0.587785i	$-0.200000\pi$
$641$	−0.159566	−	0.339095i	−0.159566	−	0.339095i	−0.968583	+	0.248690i	$0.920000\pi$
$642$		0			0
$643$	−0.238398	−	0.0774602i	−0.238398	−	0.0774602i	0.187381	−	0.982287i	$-0.440000\pi$
$643$	−0.238398	−	0.0774602i	−0.238398	−	0.0774602i	−0.425779	+	0.904827i	$0.640000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.992115	−	0.125333i	$-0.0400000\pi$
$647$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.309017	+	0.951057i	$0.400000\pi$
$648$		0			0
$649$	0.263146	+	0.809880i	0.263146	+	0.809880i
$650$		0			0
$651$		0			0
$652$	0.226810	+	0.106729i	0.226810	+	0.106729i
$653$	0.0623382	−	0.242791i	0.0623382	−	0.242791i	−0.929776	−	0.368125i	$-0.880000\pi$
$653$	0.0623382	−	0.242791i	0.0623382	−	0.242791i	0.992115	+	0.125333i	$0.0400000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$659$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$660$	−1.15475	−	1.58937i	−1.15475	−	1.58937i
$661$	−0.362989	−	1.90285i	−0.362989	−	1.90285i	−0.425779	−	0.904827i	$-0.640000\pi$
$661$	−0.362989	−	1.90285i	−0.362989	−	1.90285i	0.0627905	−	0.998027i	$-0.480000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.168887	+	2.68438i	−0.168887	+	2.68438i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$673$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$674$		0			0
$675$	3.58852	−	0.684547i	3.58852	−	0.684547i
$676$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$677$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$677$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.419952	+	1.63560i	0.419952	+	1.63560i	0.728969	+	0.684547i	$0.240000\pi$
$683$	0.419952	+	1.63560i	0.419952	+	1.63560i	−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$	−0.0623382	−	0.493458i	−0.0623382	−	0.493458i
$686$		0			0
$687$	1.01426	−	1.84493i	1.01426	−	1.84493i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.348445	−	1.82662i	0.348445	−	1.82662i	−0.187381	−	0.982287i	$-0.560000\pi$
$691$	0.348445	−	1.82662i	0.348445	−	1.82662i	0.535827	−	0.844328i	$-0.320000\pi$
$692$		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
$704$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$705$	2.72404	−	2.55804i	2.72404	−	2.55804i
$706$		0			0
$707$		0			0
$708$	−0.615854	+	1.55547i	−0.615854	+	1.55547i
$709$	1.73879	−	0.219661i	1.73879	−	0.219661i	0.809017	−	0.587785i	$-0.200000\pi$
$709$	1.73879	−	0.219661i	1.73879	−	0.219661i	0.929776	+	0.368125i	$0.120000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.32185	+	0.622015i	−1.32185	+	0.622015i
$714$		0			0
$715$		0			0
$716$	1.69755	+	0.933237i	1.69755	+	0.933237i
$717$		0			0
$718$		0			0
$719$	−0.574221	−	0.904827i	−0.574221	−	0.904827i	0.425779	−	0.904827i	$-0.360000\pi$
$719$	−0.574221	−	0.904827i	−0.574221	−	0.904827i	−1.00000			$\pi$
$720$	0.179553	−	2.85391i	0.179553	−	2.85391i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$725$		0			0
$726$		0			0
$727$	−0.226810	+	1.79538i	−0.226810	+	1.79538i	0.309017	+	0.951057i	$0.400000\pi$
$727$	−0.226810	+	1.79538i	−0.226810	+	1.79538i	−0.535827	+	0.844328i	$0.680000\pi$
$728$		0			0
$729$	3.29488	+	3.98282i	3.29488	+	3.98282i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$733$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$734$		0			0
$735$	0.488570	−	1.90285i	0.488570	−	1.90285i
$736$		0			0
$737$	−1.05491	−	1.12336i	−1.05491	−	1.12336i
$738$		0			0
$739$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$739$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$740$	−1.15475	+	0.220280i	−1.15475	+	0.220280i
$741$		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
$751$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$751$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$752$	−0.916350	−	1.66683i	−0.916350	−	1.66683i
$753$	1.26750	+	3.20135i	1.26750	+	3.20135i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.72108	−	0.559214i	1.72108	−	0.559214i	0.728969	−	0.684547i	$-0.240000\pi$
$757$	1.72108	−	0.559214i	1.72108	−	0.559214i	0.992115	+	0.125333i	$0.0400000\pi$
$758$		0			0
$759$	1.05439	−	0.990140i	1.05439	−	0.990140i
$760$		0			0
$761$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$761$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$762$		0			0
$763$		0			0
$764$	0.0915446	+	0.0859661i	0.0915446	+	0.0859661i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.86842	−	0.607087i	−1.86842	−	0.607087i
$769$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$769$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$770$		0			0
$771$	0.921980	+	1.11448i	0.921980	+	1.11448i
$772$		0			0
$773$	−0.120759	+	0.955910i	−0.120759	+	0.955910i	0.809017	+	0.587785i	$0.200000\pi$
$773$	−0.120759	+	0.955910i	−0.120759	+	0.955910i	−0.929776	+	0.368125i	$0.880000\pi$
$774$		0			0
$775$	−1.84489	−	0.730444i	−1.84489	−	0.730444i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.200808	+	0.316423i	0.200808	+	0.316423i
$782$		0			0
$783$		0			0
$784$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$785$	1.60601	+	0.521823i	1.60601	+	0.521823i
$786$		0			0
$787$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$787$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.354691	−	1.85935i	0.354691	−	1.85935i
$796$	1.80113	−	0.713118i	1.80113	−	0.713118i
$797$	−1.99211	+	0.125333i	−1.99211	+	0.125333i	−0.992115	+	0.125333i	$0.960000\pi$
$797$	−1.99211	+	0.125333i	−1.99211	+	0.125333i	−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.290523	−	0.211077i	−0.290523	−	0.211077i
$802$		0			0
$803$		0			0
$804$	−0.190096	−	3.02149i	−0.190096	−	3.02149i
$805$		0			0
$806$		0			0
$807$		−	2.10534i		−	2.10534i
$808$		0			0
$809$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$809$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$810$		0			0
$811$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$811$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.226810	+	0.106729i	−0.226810	+	0.106729i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$821$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$822$		0			0
$823$	0.946441	−	0.180543i	0.946441	−	0.180543i	0.309017	−	0.951057i	$-0.400000\pi$
$823$	0.946441	−	0.180543i	0.946441	−	0.180543i	0.637424	+	0.770513i	$0.280000\pi$
$824$		0			0
$825$	1.96070	+	0.123357i	1.96070	+	0.123357i
$826$		0			0
$827$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$827$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$828$	2.10119	−	0.132196i	2.10119	−	0.132196i
$829$	1.35556	−	0.536702i	1.35556	−	0.536702i	0.425779	−	0.904827i	$-0.360000\pi$
$829$	1.35556	−	0.536702i	1.35556	−	0.536702i	0.929776	+	0.368125i	$0.120000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.908521	−	7.19169i	−0.908521	−	7.19169i
$838$		0			0
$839$	0.238883	+	1.25227i	0.238883	+	1.25227i	0.876307	+	0.481754i	$0.160000\pi$
$839$	0.238883	+	1.25227i	0.238883	+	1.25227i	−0.637424	+	0.770513i	$0.720000\pi$
$840$		0			0
$841$	0.876307	+	0.481754i	0.876307	+	0.481754i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.968583	−	0.248690i	0.968583	−	0.248690i
$846$		0			0
$847$		0			0
$848$	−0.871808	−	0.410241i	−0.871808	−	0.410241i
$849$		0			0
$850$		0			0
$851$	−0.267458	−	0.823152i	−0.267458	−	0.823152i
$852$	−0.0922765	+	0.730444i	−0.0922765	+	0.730444i
$853$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$853$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$854$		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
$859$	0.746226	+	1.58581i	0.746226	+	1.58581i	0.809017	+	0.587785i	$0.200000\pi$
$859$	0.746226	+	1.58581i	0.746226	+	1.58581i	−0.0627905	+	0.998027i	$0.520000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.246226	+	0.0469702i	0.246226	+	0.0469702i	0.309017	−	0.951057i	$-0.400000\pi$
$863$	0.246226	+	0.0469702i	0.246226	+	0.0469702i	−0.0627905	+	0.998027i	$0.520000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.96070	+	0.123357i	1.96070	+	0.123357i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−2.49298	−	4.53471i	−2.49298	−	4.53471i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$877$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$878$		0			0
$879$		0			0
$880$	0.309017	−	0.951057i	0.309017	−	0.951057i
$881$	−0.929324	+	1.12336i	−0.929324	+	1.12336i	0.0627905	+	0.998027i	$0.480000\pi$
$881$	−0.929324	+	1.12336i	−0.929324	+	1.12336i	−0.992115	+	0.125333i	$0.960000\pi$
$882$		0			0
$883$	−0.250172	−	0.0157395i	−0.250172	−	0.0157395i	−0.0627905	−	0.998027i	$-0.520000\pi$
$883$	−0.250172	−	0.0157395i	−0.250172	−	0.0157395i	−0.187381	+	0.982287i	$0.560000\pi$
$884$		0			0
$885$	−0.805950	−	1.46602i	−0.805950	−	1.46602i
$886$		0			0
$887$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$887$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.83830	+	3.90658i	1.83830	+	3.90658i
$892$	−1.15596	+	0.733597i	−1.15596	+	0.733597i
$893$		0			0
$894$		0			0
$895$	−1.80113	+	0.713118i	−1.80113	+	0.713118i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2.08452	+	1.95750i	2.08452	+	1.95750i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.866986	−	1.36615i	−0.866986	−	1.36615i
$906$		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
$911$	−0.238883	−	1.25227i	−0.238883	−	1.25227i	−0.876307	−	0.481754i	$-0.840000\pi$
$911$	−0.238883	−	1.25227i	−0.238883	−	1.25227i	0.637424	−	0.770513i	$-0.280000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.06320	−	0.134314i	1.06320	−	0.134314i
$917$		0			0
$918$		0			0
$919$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$919$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.566335	−	1.03016i	0.566335	−	1.03016i
$926$		0			0
$927$	−4.32859	+	0.825723i	−4.32859	+	0.825723i
$928$		0			0
$929$	0.121636	+	1.93334i	0.121636	+	1.93334i	0.309017	+	0.951057i	$0.400000\pi$
$929$	0.121636	+	1.93334i	0.121636	+	1.93334i	−0.187381	+	0.982287i	$0.560000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.622852	+	2.42585i	0.622852	+	2.42585i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$937$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$938$		0			0
$939$	0.969435	+	0.122468i	0.969435	+	0.122468i
$940$	1.86842	+	0.356420i	1.86842	+	0.356420i
$941$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$941$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$942$		0			0
$943$		0			0
$944$	−0.824805	+	0.211774i	−0.824805	+	0.211774i
$945$		0			0
$946$		0			0
$947$	−1.05491	+	0.872693i	−1.05491	+	0.872693i	−0.992115	−	0.125333i	$-0.960000\pi$
$947$	−1.05491	+	0.872693i	−1.05491	+	0.872693i	−0.0627905	+	0.998027i	$0.520000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2.17600	+	1.58096i	−2.17600	+	1.58096i
$952$		0			0
$953$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$953$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$954$		0			0
$955$	−0.124591	+	0.0157395i	−0.124591	+	0.0157395i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.65875	−	1.05267i	1.65875	−	1.05267i
$961$	−1.25058	+	2.65763i	−1.25058	+	2.65763i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$967$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.69755	−	0.435857i	−1.69755	−	0.435857i	−0.728969	−	0.684547i	$-0.760000\pi$
$971$	−1.69755	−	0.435857i	−1.69755	−	0.435857i	−0.968583	+	0.248690i	$0.920000\pi$
$972$	−1.20087	+	4.67710i	−1.20087	+	4.67710i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.147338	−	1.16630i	0.147338	−	1.16630i	−0.728969	−	0.684547i	$-0.760000\pi$
$977$	0.147338	−	1.16630i	0.147338	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$978$		0			0
$979$	−0.0800484	−	0.0967619i	−0.0800484	−	0.0967619i
$980$	0.929776	−	0.368125i	0.929776	−	0.368125i
$981$		0			0
$982$		0			0
$983$	0.419952	−	0.266509i	0.419952	−	0.266509i	−0.309017	−	0.951057i	$-0.600000\pi$
$983$	0.419952	−	0.266509i	0.419952	−	0.266509i	0.728969	+	0.684547i	$0.240000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.620759	−	0.582932i	0.620759	−	0.582932i	−0.309017	−	0.951057i	$-0.600000\pi$
$991$	0.620759	−	0.582932i	0.620759	−	0.582932i	0.929776	+	0.368125i	$0.120000\pi$
$992$		0			0
$993$	2.38195	−	0.773944i	2.38195	−	0.773944i
$994$		0			0
$995$	−0.598617	+	1.84235i	−0.598617	+	1.84235i
$996$		0			0
$997$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$997$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$998$		0			0
$999$	4.29463			4.29463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1375.1.bz.a.934.1	yes	20
11.10	odd	2	CM	1375.1.bz.a.934.1	yes	20
125.89	even	50	inner	1375.1.bz.a.714.1	✓	20
1375.714	odd	50	inner	1375.1.bz.a.714.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1375.1.bz.a.714.1	✓	20	125.89	even	50	inner
1375.1.bz.a.714.1	✓	20	1375.714	odd	50	inner
1375.1.bz.a.934.1	yes	20	1.1	even	1	trivial
1375.1.bz.a.934.1	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)

\(f(q)\)	\(=\)	\(q+(0.723208 + 1.82662i) q^{3} +(-0.535827 + 0.844328i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(-2.08452 + 1.95750i) q^{9} +O(q^{10})\)	\(q+(0.723208 + 1.82662i) q^{3} +(-0.535827 + 0.844328i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(-2.08452 + 1.95750i) q^{9} +(-0.876307 + 0.481754i) q^{11} +(-1.92978 - 0.368125i) q^{12} +(1.51373 - 1.25227i) q^{15} +(-0.425779 - 0.904827i) q^{16} +(0.968583 + 0.248690i) q^{20} +(-0.0922765 + 0.730444i) q^{23} +(-0.809017 + 0.587785i) q^{25} +(-3.30554 - 1.55547i) q^{27} +(1.06320 + 1.67534i) q^{31} +(-1.51373 - 1.25227i) q^{33} +(-0.535827 - 2.80890i) q^{36} +(-1.06369 + 0.500534i) q^{37} +(0.0627905 - 0.998027i) q^{44} +(2.50585 + 1.37760i) q^{45} +(1.89836 - 0.119435i) q^{47} +(1.34484 - 1.43211i) q^{48} +(0.809017 - 0.587785i) q^{49} +(0.742395 - 0.614163i) q^{53} +(0.728969 + 0.684547i) q^{55} +(0.159566 - 0.836475i) q^{59} +(0.246226 + 1.94908i) q^{60} +(0.992115 + 0.125333i) q^{64} +(0.383238 + 1.49261i) q^{67} +(-1.40097 + 0.359709i) q^{69} +(-0.0235315 - 0.374023i) q^{71} +(-1.65875 - 1.05267i) q^{75} +(-0.728969 + 0.684547i) q^{80} +(0.271097 - 4.30897i) q^{81} +(0.0235315 + 0.123357i) q^{89} +(-0.567290 - 0.469303i) q^{92} +(-2.29128 + 3.15368i) q^{93} +(-0.450043 + 1.75280i) q^{97} +(0.883650 - 2.71960i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{5}+O(q^{10}) \) 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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