LMFDB - Embedded newform 1359.1.bp.a.532.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1359,1,Mod(28,1359)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1359, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([0, 19]))

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

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

chi := DirichletCharacter("1359.28");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1359 = 3^{2} \cdot 151 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1359.bp (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:	$0.678229352168$
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:	$\Z[a_1, \ldots, a_{13}]$
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			532.1
Root			$0.187381 - 0.982287i$ of defining polynomial
Character	$\chi$	$=$	1359.532
Dual form			1359.1.bp.a.820.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.809017 + 0.587785i) q^{4} +(-0.813516 + 1.47978i) q^{7} +O(q^{10})$	$q+(0.809017 + 0.587785i) q^{4} +(-0.813516 + 1.47978i) q^{7} +(-1.15596 - 0.733597i) q^{13} +(0.309017 + 0.951057i) q^{16} +(1.60528 + 1.16630i) q^{19} +(-0.0627905 + 0.998027i) q^{25} +(-1.52794 + 0.718995i) q^{28} +(-0.328407 - 1.72157i) q^{31} +(-0.996398 + 1.57007i) q^{37} +(1.11716 - 0.614163i) q^{43} +(-0.992115 - 1.56332i) q^{49} +(-0.503997 - 1.27295i) q^{52} +(0.700215 - 1.76854i) q^{61} +(-0.309017 + 0.951057i) q^{64} +(-0.340480 + 0.362574i) q^{67} +(0.566335 - 1.03016i) q^{73} +(0.613161 + 1.88711i) q^{76} +(1.96070 + 0.374023i) q^{79} +(2.02596 - 1.11378i) q^{91} +(0.450527 + 0.423073i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{4}+O(q^{10}) $ 20 * q + 5 * q^4	$ 20 q + 5 q^{4} - 5 q^{16} - 5 q^{31} - 5 q^{37} + 5 q^{64} + 5 q^{91}+O(q^{100}) $ 20 * q + 5 * q^4 - 5 * q^16 - 5 * q^31 - 5 * q^37 + 5 * q^64 + 5 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1359\mathbb{Z}\right)^\times$.

$n$	$605$	$1063$
$\chi(n)$	$1$	$e\left(\frac{49}{50}\right)$

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		−0.951057	−	0.309017i	$-0.900000\pi$
$2$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$3$		0			0
$3$		0			0
$4$	0.809017	+	0.587785i	0.809017	+	0.587785i
$5$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$5$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$6$		0			0
$7$	−0.813516	+	1.47978i	−0.813516	+	1.47978i	0.0627905	+	0.998027i	$0.480000\pi$
$7$	−0.813516	+	1.47978i	−0.813516	+	1.47978i	−0.876307	+	0.481754i	$0.840000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$11$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$12$		0			0
$13$	−1.15596	−	0.733597i	−1.15596	−	0.733597i	−0.187381	−	0.982287i	$-0.560000\pi$
$13$	−1.15596	−	0.733597i	−1.15596	−	0.733597i	−0.968583	+	0.248690i	$0.920000\pi$
$14$		0			0
$15$		0			0
$16$	0.309017	+	0.951057i	0.309017	+	0.951057i
$17$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$17$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$18$		0			0
$19$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$19$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$23$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$24$		0			0
$25$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$26$		0			0
$27$		0			0
$28$	−1.52794	+	0.718995i	−1.52794	+	0.718995i
$29$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$29$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$30$		0			0
$31$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	−0.637424	−	0.770513i	$-0.720000\pi$
$31$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	0.309017	−	0.951057i	$-0.400000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.187381	+	0.982287i	$0.560000\pi$
$37$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.809017	+	0.587785i	$0.800000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$41$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$42$		0			0
$43$	1.11716	−	0.614163i	1.11716	−	0.614163i	0.187381	−	0.982287i	$-0.440000\pi$
$43$	1.11716	−	0.614163i	1.11716	−	0.614163i	0.929776	+	0.368125i	$0.120000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$47$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$48$		0			0
$49$	−0.992115	−	1.56332i	−0.992115	−	1.56332i
$50$		0			0
$51$		0			0
$52$	−0.503997	−	1.27295i	−0.503997	−	1.27295i
$53$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$53$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$59$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$	0.700215	−	1.76854i	0.700215	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$61$	0.700215	−	1.76854i	0.700215	−	1.76854i	0.637424	−	0.770513i	$-0.280000\pi$
$62$		0			0
$63$		0			0
$64$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	−0.340480	+	0.362574i	−0.340480	+	0.362574i	−0.876307	−	0.481754i	$-0.840000\pi$
$67$	−0.340480	+	0.362574i	−0.340480	+	0.362574i	0.535827	+	0.844328i	$0.320000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$71$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$72$		0			0
$73$	0.566335	−	1.03016i	0.566335	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$73$	0.566335	−	1.03016i	0.566335	−	1.03016i	0.992115	−	0.125333i	$-0.0400000\pi$
$74$		0			0
$75$		0			0
$76$	0.613161	+	1.88711i	0.613161	+	1.88711i
$77$		0			0
$78$		0			0
$79$	1.96070	+	0.374023i	1.96070	+	0.374023i	0.992115	+	0.125333i	$0.0400000\pi$
$79$	1.96070	+	0.374023i	1.96070	+	0.374023i	0.968583	+	0.248690i	$0.0800000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$83$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$89$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$90$		0			0
$91$	2.02596	−	1.11378i	2.02596	−	1.11378i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.450527	+	0.423073i	0.450527	+	0.423073i	0.876307	−	0.481754i	$-0.160000\pi$
$97$	0.450527	+	0.423073i	0.450527	+	0.423073i	−0.425779	+	0.904827i	$0.640000\pi$
$98$		0			0
$99$		0			0
$100$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$101$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$101$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$102$		0			0
$103$	0.0800484	+	0.0967619i	0.0800484	+	0.0967619i	0.809017	−	0.587785i	$-0.200000\pi$
$103$	0.0800484	+	0.0967619i	0.0800484	+	0.0967619i	−0.728969	+	0.684547i	$0.760000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$107$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$108$		0			0
$109$	−1.77760	−	0.836475i	−1.77760	−	0.836475i	−0.968583	−	0.248690i	$-0.920000\pi$
$109$	−1.77760	−	0.836475i	−1.77760	−	0.836475i	−0.809017	−	0.587785i	$-0.800000\pi$
$110$		0			0
$111$		0			0
$112$	−1.65875	−	0.316423i	−1.65875	−	0.316423i
$113$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$113$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.728969	−	0.684547i	−0.728969	−	0.684547i
$122$		0			0
$123$		0			0
$124$	0.746226	−	1.58581i	0.746226	−	1.58581i
$125$		0			0
$126$		0			0
$127$	0.866986	−	1.36615i	0.866986	−	1.36615i	−0.0627905	−	0.998027i	$-0.520000\pi$
$127$	0.866986	−	1.36615i	0.866986	−	1.36615i	0.929776	−	0.368125i	$-0.120000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$131$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$132$		0			0
$133$	−3.03179	+	1.42665i	−3.03179	+	1.42665i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$137$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$138$		0			0
$139$	1.56720	+	0.402389i	1.56720	+	0.402389i	0.929776	−	0.368125i	$-0.120000\pi$
$139$	1.56720	+	0.402389i	1.56720	+	0.402389i	0.637424	+	0.770513i	$0.280000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.72897	+	0.684547i	−1.72897	+	0.684547i
$149$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$149$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$150$		0			0
$151$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$151$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.96070	−	0.123357i	1.96070	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$157$	1.96070	−	0.123357i	1.96070	−	0.123357i	0.992115	+	0.125333i	$0.0400000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.183098	−	0.713118i	0.183098	−	0.713118i	−0.809017	−	0.587785i	$-0.800000\pi$
$163$	0.183098	−	0.713118i	0.183098	−	0.713118i	0.992115	−	0.125333i	$-0.0400000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$167$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$168$		0			0
$169$	0.372309	+	0.791198i	0.372309	+	0.791198i
$170$		0			0
$171$		0			0
$172$	1.26480	+	0.159781i	1.26480	+	0.159781i
$173$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$173$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$174$		0			0
$175$	−1.42578	−	0.904827i	−1.42578	−	0.904827i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$179$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$180$		0			0
$181$	−1.23879	+	1.31918i	−1.23879	+	1.31918i	−0.309017	+	0.951057i	$0.600000\pi$
$181$	−1.23879	+	1.31918i	−1.23879	+	1.31918i	−0.929776	+	0.368125i	$0.880000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$191$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$192$		0			0
$193$	0.159566	−	0.339095i	0.159566	−	0.339095i	−0.809017	−	0.587785i	$-0.800000\pi$
$193$	0.159566	−	0.339095i	0.159566	−	0.339095i	0.968583	+	0.248690i	$0.0800000\pi$
$194$		0			0
$195$		0			0
$196$	0.116260	−	1.84790i	0.116260	−	1.84790i
$197$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$197$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$198$		0			0
$199$	−0.742395	+	0.614163i	−0.742395	+	0.614163i	−0.929776	−	0.368125i	$-0.880000\pi$
$199$	−0.742395	+	0.614163i	−0.742395	+	0.614163i	0.187381	+	0.982287i	$0.440000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.340480	−	1.32608i	0.340480	−	1.32608i
$209$		0			0
$210$		0			0
$211$	−0.354691	−	0.645180i	−0.354691	−	0.645180i	0.637424	−	0.770513i	$-0.280000\pi$
$211$	−0.354691	−	0.645180i	−0.354691	−	0.645180i	−0.992115	+	0.125333i	$0.960000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.81471	+	0.914555i	2.81471	+	0.914555i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.362989	−	1.90285i	0.362989	−	1.90285i	−0.0627905	−	0.998027i	$-0.520000\pi$
$223$	0.362989	−	1.90285i	0.362989	−	1.90285i	0.425779	−	0.904827i	$-0.360000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$		0			0
$229$	−0.541587	−	0.297740i	−0.541587	−	0.297740i	0.187381	−	0.982287i	$-0.440000\pi$
$229$	−0.541587	−	0.297740i	−0.541587	−	0.297740i	−0.728969	+	0.684547i	$0.760000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$233$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$239$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$240$		0			0
$241$	0.348445	+	0.137959i	0.348445	+	0.137959i	0.535827	−	0.844328i	$-0.320000\pi$
$241$	0.348445	+	0.137959i	0.348445	+	0.137959i	−0.187381	+	0.982287i	$0.560000\pi$
$242$		0			0
$243$		0			0
$244$	1.60601	−	1.01920i	1.60601	−	1.01920i
$245$		0			0
$246$		0			0
$247$	−1.00005	−	2.52583i	−1.00005	−	2.52583i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$251$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$257$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$257$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$258$		0			0
$259$	−1.51278	−	2.75173i	−1.51278	−	2.75173i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$263$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.488570	+	0.0931997i	−0.488570	+	0.0931997i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	0.488570	−	0.0931997i	0.488570	−	0.0931997i	0.0627905	−	0.998027i	$-0.480000\pi$
$271$	0.488570	−	0.0931997i	0.488570	−	0.0931997i	0.425779	+	0.904827i	$0.360000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.53799	−	0.0967619i	−1.53799	−	0.0967619i	−0.728969	−	0.684547i	$-0.760000\pi$
$277$	−1.53799	−	0.0967619i	−1.53799	−	0.0967619i	−0.809017	+	0.587785i	$0.800000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$281$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$282$		0			0
$283$	−1.17325	+	1.61484i	−1.17325	+	1.61484i	−0.535827	+	0.844328i	$0.680000\pi$
$283$	−1.17325	+	1.61484i	−1.17325	+	1.61484i	−0.637424	+	0.770513i	$0.720000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$290$		0			0
$291$		0			0
$292$	1.06369	−	0.500534i	1.06369	−	0.500534i
$293$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$293$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$			2.15278i			2.15278i
$302$		0			0
$303$		0			0
$304$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$305$		0			0
$306$		0			0
$307$	0.0915446	−	0.0859661i	0.0915446	−	0.0859661i	−0.637424	−	0.770513i	$-0.720000\pi$
$307$	0.0915446	−	0.0859661i	0.0915446	−	0.0859661i	0.728969	+	0.684547i	$0.240000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$311$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$312$		0			0
$313$	−1.50441	−	0.595638i	−1.50441	−	0.595638i	−0.535827	−	0.844328i	$-0.680000\pi$
$313$	−1.50441	−	0.595638i	−1.50441	−	0.595638i	−0.968583	+	0.248690i	$0.920000\pi$
$314$		0			0
$315$		0			0
$316$	1.36639	+	1.45506i	1.36639	+	1.45506i
$317$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$317$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.804733	−	1.10762i	0.804733	−	1.10762i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.393950	+	0.476203i	−0.393950	+	0.476203i	−0.929776	−	0.368125i	$-0.880000\pi$
$331$	−0.393950	+	0.476203i	−0.393950	+	0.476203i	0.535827	+	0.844328i	$0.320000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.183098	−	0.713118i	−0.183098	−	0.713118i	−0.992115	−	0.125333i	$-0.960000\pi$
$337$	−0.183098	−	0.713118i	−0.183098	−	0.713118i	0.809017	−	0.587785i	$-0.200000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.43515	−	0.0902921i	1.43515	−	0.0902921i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$347$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$348$		0			0
$349$	0.0534698	+	0.849878i	0.0534698	+	0.849878i	0.929776	+	0.368125i	$0.120000\pi$
$349$	0.0534698	+	0.849878i	0.0534698	+	0.849878i	−0.876307	+	0.481754i	$0.840000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$353$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$359$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$360$		0			0
$361$	0.907634	+	2.79341i	0.907634	+	2.79341i
$362$		0			0
$363$		0			0
$364$	2.29370	+	0.289762i	2.29370	+	0.289762i
$365$		0			0
$366$		0			0
$367$	0.946441	−	0.180543i	0.946441	−	0.180543i	0.309017	−	0.951057i	$-0.400000\pi$
$367$	0.946441	−	0.180543i	0.946441	−	0.180543i	0.637424	+	0.770513i	$0.280000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.120759	+	0.219661i	−0.120759	+	0.219661i	−0.929776	−	0.368125i	$-0.880000\pi$
$373$	−0.120759	+	0.219661i	−0.120759	+	0.219661i	0.809017	+	0.587785i	$0.200000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.39436	+	1.15352i	−1.39436	+	1.15352i	−0.425779	+	0.904827i	$0.640000\pi$
$379$	−1.39436	+	1.15352i	−1.39436	+	1.15352i	−0.968583	+	0.248690i	$0.920000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$383$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.115808	+	0.607087i	0.115808	+	0.607087i
$389$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$389$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.844844	+	0.106729i	−0.844844	+	0.106729i	−0.535827	−	0.844328i	$-0.680000\pi$
$397$	−0.844844	+	0.106729i	−0.844844	+	0.106729i	−0.309017	+	0.951057i	$0.600000\pi$
$398$		0			0
$399$		0			0
$400$	−0.968583	+	0.248690i	−0.968583	+	0.248690i
$401$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$401$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$402$		0			0
$403$	−0.883312	+	2.23099i	−0.883312	+	2.23099i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.419952	+	0.266509i	−0.419952	+	0.266509i	−0.728969	−	0.684547i	$-0.760000\pi$
$409$	−0.419952	+	0.266509i	−0.419952	+	0.266509i	0.309017	+	0.951057i	$0.400000\pi$
$410$		0			0
$411$		0			0
$412$	0.00788530	+	0.125333i	0.00788530	+	0.125333i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$419$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$420$		0			0
$421$			0.963507i			0.963507i	0.876307	+	0.481754i	$0.160000\pi$
$421$			0.963507i			0.963507i	−0.876307	+	0.481754i	$0.840000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.04741	+	2.47490i	2.04741	+	2.47490i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$431$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$432$		0			0
$433$	−0.238398	−	1.88711i	−0.238398	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$433$	−0.238398	−	1.88711i	−0.238398	−	1.88711i	0.187381	−	0.982287i	$-0.440000\pi$
$434$		0			0
$435$		0			0
$436$	−0.946441	−	1.72157i	−0.946441	−	1.72157i
$437$		0			0
$438$		0			0
$439$	1.06279	−	0.998027i	1.06279	−	0.998027i	0.0627905	−	0.998027i	$-0.480000\pi$
$439$	1.06279	−	0.998027i	1.06279	−	0.998027i	1.00000			$0$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$443$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.15596	−	1.23098i	−1.15596	−	1.23098i
$449$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$449$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$457$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$461$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$462$		0			0
$463$	0.791759	+	1.68257i	0.791759	+	1.68257i	0.728969	+	0.684547i	$0.240000\pi$
$463$	0.791759	+	1.68257i	0.791759	+	1.68257i	0.0627905	+	0.998027i	$0.480000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$467$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$468$		0			0
$469$	−0.259544	−	0.798795i	−0.259544	−	0.798795i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.26480	+	1.52888i	−1.26480	+	1.52888i
$476$		0			0
$477$		0			0
$478$		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
$481$	2.30360	−	1.08399i	2.30360	−	1.08399i
$482$		0			0
$483$		0			0
$484$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$485$		0			0
$486$		0			0
$487$	0.273190	+	1.43211i	0.273190	+	1.43211i	0.809017	+	0.587785i	$0.200000\pi$
$487$	0.273190	+	1.43211i	0.273190	+	1.43211i	−0.535827	+	0.844328i	$0.680000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$491$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.53583	−	0.844328i	1.53583	−	0.844328i
$497$		0			0
$498$		0			0
$499$	−0.147338	−	0.202793i	−0.147338	−	0.202793i	0.728969	−	0.684547i	$-0.240000\pi$
$499$	−0.147338	−	0.202793i	−0.147338	−	0.202793i	−0.876307	+	0.481754i	$0.840000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$503$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.50441	−	0.595638i	1.50441	−	0.595638i
$509$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$509$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$510$		0			0
$511$	1.06369	+	1.67610i	1.06369	+	1.67610i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$521$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$522$		0			0
$523$	1.80608	+	0.113629i	1.80608	+	0.113629i	0.929776	−	0.368125i	$-0.120000\pi$
$523$	1.80608	+	0.113629i	1.80608	+	0.113629i	0.876307	+	0.481754i	$0.160000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.309017	+	0.951057i	0.309017	+	0.951057i
$530$		0			0
$531$		0			0
$532$	−3.29133	−	0.627855i	−3.29133	−	0.627855i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.23480	−	0.317042i	1.23480	−	0.317042i	0.425779	−	0.904827i	$-0.360000\pi$
$541$	1.23480	−	0.317042i	1.23480	−	0.317042i	0.809017	+	0.587785i	$0.200000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.824805	+	0.211774i	0.824805	+	0.211774i	0.637424	−	0.770513i	$-0.280000\pi$
$547$	0.824805	+	0.211774i	0.824805	+	0.211774i	0.187381	+	0.982287i	$0.440000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.14853	+	2.59713i	−2.14853	+	2.59713i
$554$		0			0
$555$		0			0
$556$	1.03137	+	1.24672i	1.03137	+	1.24672i
$557$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$557$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$558$		0			0
$559$	−1.74194	−	0.109594i	−1.74194	−	0.109594i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$563$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$569$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$570$		0			0
$571$	−1.75261			−1.75261			−0.876307	−	0.481754i	$-0.840000\pi$
$571$	−1.75261			−1.75261			−0.876307	+	0.481754i	$0.840000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	0.309017	+	0.951057i	$0.400000\pi$
$577$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	−0.929776	+	0.368125i	$0.880000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$587$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$588$		0			0
$589$	1.48068	−	3.14661i	1.48068	−	3.14661i
$590$		0			0
$591$		0			0
$592$	−1.80113	−	0.462452i	−1.80113	−	0.462452i
$593$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$593$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$599$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$600$		0			0
$601$	1.18532	−	0.469303i	1.18532	−	0.469303i	0.309017	−	0.951057i	$-0.400000\pi$
$601$	1.18532	−	0.469303i	1.18532	−	0.469303i	0.876307	+	0.481754i	$0.160000\pi$
$602$		0			0
$603$		0			0
$604$	0.309017	−	0.951057i	0.309017	−	0.951057i
$605$		0			0
$606$		0			0
$607$	−0.723208	−	1.82662i	−0.723208	−	1.82662i	−0.535827	−	0.844328i	$-0.680000\pi$
$607$	−0.723208	−	1.82662i	−0.723208	−	1.82662i	−0.187381	−	0.982287i	$-0.560000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.06279	+	0.998027i	−1.06279	+	0.998027i	−0.0627905	+	0.998027i	$0.520000\pi$
$613$	−1.06279	+	0.998027i	−1.06279	+	0.998027i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$617$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$618$		0			0
$619$	−1.39436	−	0.656137i	−1.39436	−	0.656137i	−0.425779	−	0.904827i	$-0.640000\pi$
$619$	−1.39436	−	0.656137i	−1.39436	−	0.656137i	−0.968583	+	0.248690i	$0.920000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$626$		0			0
$627$		0			0
$628$	1.65875	+	1.05267i	1.65875	+	1.05267i
$629$		0			0
$630$		0			0
$631$	1.39436	+	0.656137i	1.39436	+	0.656137i	0.968583	−	0.248690i	$-0.0800000\pi$
$631$	1.39436	+	0.656137i	1.39436	+	0.656137i	0.425779	+	0.904827i	$0.360000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			2.53496i			2.53496i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$641$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$642$		0			0
$643$	−0.0702235	+	0.368125i	−0.0702235	+	0.368125i	0.929776	+	0.368125i	$0.120000\pi$
$643$	−0.0702235	+	0.368125i	−0.0702235	+	0.368125i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$647$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.567290	−	0.469303i	0.567290	−	0.469303i
$653$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$653$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$659$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$660$		0			0
$661$	0.292352	−	1.13864i	0.292352	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$661$	0.292352	−	1.13864i	0.292352	−	1.13864i	0.929776	−	0.368125i	$-0.120000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.598617	−	0.153699i	−0.598617	−	0.153699i	−0.0627905	−	0.998027i	$-0.520000\pi$
$673$	−0.598617	−	0.153699i	−0.598617	−	0.153699i	−0.535827	+	0.844328i	$0.680000\pi$
$674$		0			0
$675$		0			0
$676$	−0.163850	+	0.858931i	−0.163850	+	0.858931i
$677$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$677$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$678$		0			0
$679$	−0.992567	+	0.322505i	−0.992567	+	0.322505i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$683$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.929324	+	0.872693i	0.929324	+	0.872693i
$689$		0			0
$690$		0			0
$691$	−1.46560	−	0.476203i	−1.46560	−	0.476203i	−0.535827	−	0.844328i	$-0.680000\pi$
$691$	−1.46560	−	0.476203i	−1.46560	−	0.476203i	−0.929776	+	0.368125i	$0.880000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.621636	−	1.57007i	−0.621636	−	1.57007i
$701$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$701$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$702$		0			0
$703$	−3.43067	+	1.35830i	−3.43067	+	1.35830i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$709$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$719$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$720$		0			0
$721$	−0.208307	+	0.0397367i	−0.208307	+	0.0397367i
$722$		0			0
$723$		0			0
$724$	−1.77760	+	0.339095i	−1.77760	+	0.339095i
$725$		0			0
$726$		0			0
$727$	0.263146	+	0.559214i	0.263146	+	0.559214i	0.992115	−	0.125333i	$-0.0400000\pi$
$727$	0.263146	+	0.559214i	0.263146	+	0.559214i	−0.728969	+	0.684547i	$0.760000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.567290	+	0.469303i	0.567290	+	0.469303i	0.876307	−	0.481754i	$-0.160000\pi$
$733$	0.567290	+	0.469303i	0.567290	+	0.469303i	−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.916350	+	0.297740i	−0.916350	+	0.297740i	−0.728969	−	0.684547i	$-0.760000\pi$
$739$	−0.916350	+	0.297740i	−0.916350	+	0.297740i	−0.187381	+	0.982287i	$0.560000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$743$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.15475	−	1.58937i	1.15475	−	1.58937i	0.425779	−	0.904827i	$-0.360000\pi$
$751$	1.15475	−	1.58937i	1.15475	−	1.58937i	0.728969	−	0.684547i	$-0.240000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$757$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$761$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$762$		0			0
$763$	2.68391	−	1.94997i	2.68391	−	1.94997i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.171593	+	0.182728i	0.171593	+	0.182728i	0.809017	−	0.587785i	$-0.200000\pi$
$769$	0.171593	+	0.182728i	0.171593	+	0.182728i	−0.637424	+	0.770513i	$0.720000\pi$
$770$		0			0
$771$		0			0
$772$	0.328407	−	0.180543i	0.328407	−	0.180543i
$773$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$773$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$774$		0			0
$775$	1.73879	−	0.219661i	1.73879	−	0.219661i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.18023	−	1.42665i	1.18023	−	1.42665i
$785$		0			0
$786$		0			0
$787$	0.851559			0.851559			0.425779	−	0.904827i	$-0.360000\pi$
$787$	0.851559			0.851559			0.425779	+	0.904827i	$0.360000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.10682	+	1.53069i	−2.10682	+	1.53069i
$794$		0			0
$795$		0			0
$796$	−0.961606	+	0.0604991i	−0.961606	+	0.0604991i
$797$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$797$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$809$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$810$		0			0
$811$	−0.120759	−	0.955910i	−0.120759	−	0.955910i	−0.929776	−	0.368125i	$-0.880000\pi$
$811$	−0.120759	−	0.955910i	−0.120759	−	0.955910i	0.809017	−	0.587785i	$-0.200000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.50964	+	0.317042i	2.50964	+	0.317042i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$	−1.06320	−	0.134314i	−1.06320	−	0.134314i	−0.425779	−	0.904827i	$-0.640000\pi$
$823$	−1.06320	−	0.134314i	−1.06320	−	0.134314i	−0.637424	+	0.770513i	$0.720000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$827$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$828$		0			0
$829$	−1.18532	−	1.43281i	−1.18532	−	1.43281i	−0.876307	−	0.481754i	$-0.840000\pi$
$829$	−1.18532	−	1.43281i	−1.18532	−	1.43281i	−0.309017	−	0.951057i	$-0.600000\pi$
$830$		0			0
$831$		0			0
$832$	1.05491	−	0.872693i	1.05491	−	0.872693i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$839$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$840$		0			0
$841$	0.187381	+	0.982287i	0.187381	+	0.982287i
$842$		0			0
$843$		0			0
$844$	0.0922765	−	0.730444i	0.0922765	−	0.730444i
$845$		0			0
$846$		0			0
$847$	1.60601	−	0.521823i	1.60601	−	0.521823i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.03799	+	0.266509i	−1.03799	+	0.266509i	−0.728969	−	0.684547i	$-0.760000\pi$
$853$	−1.03799	+	0.266509i	−1.03799	+	0.266509i	−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$857$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$858$		0			0
$859$	0.250172	−	0.0157395i	0.250172	−	0.0157395i	0.0627905	−	0.998027i	$-0.480000\pi$
$859$	0.250172	−	0.0157395i	0.250172	−	0.0157395i	0.187381	+	0.982287i	$0.440000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$863$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	1.73959	+	2.39433i	1.73959	+	2.39433i
$869$		0			0
$870$		0			0
$871$	0.659566	−	0.169348i	0.659566	−	0.169348i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.39436	−	1.15352i	−1.39436	−	1.15352i	−0.968583	−	0.248690i	$-0.920000\pi$
$877$	−1.39436	−	1.15352i	−1.39436	−	1.15352i	−0.425779	−	0.904827i	$-0.640000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$881$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$882$		0			0
$883$	−0.303189	−	0.220280i	−0.303189	−	0.220280i	0.425779	−	0.904827i	$-0.360000\pi$
$883$	−0.303189	−	0.220280i	−0.303189	−	0.220280i	−0.728969	+	0.684547i	$0.760000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$887$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$888$		0			0
$889$	1.31630	+	2.39433i	1.31630	+	2.39433i
$890$		0			0
$891$		0			0
$892$	1.41213	−	1.32608i	1.41213	−	1.32608i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.98423			1.98423			0.992115	−	0.125333i	$-0.0400000\pi$
$907$	1.98423			1.98423			0.992115	+	0.125333i	$0.0400000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$911$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.263146	−	0.559214i	−0.263146	−	0.559214i
$917$		0			0
$918$		0			0
$919$	1.52794	+	0.969661i	1.52794	+	0.969661i	0.992115	+	0.125333i	$0.0400000\pi$
$919$	1.52794	+	0.969661i	1.52794	+	0.969661i	0.535827	+	0.844328i	$0.320000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.50441	−	1.09302i	−1.50441	−	1.09302i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$	0.230687	−	3.66667i	0.230687	−	3.66667i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.371808	−	1.94908i	−0.371808	−	1.94908i	−0.309017	−	0.951057i	$-0.600000\pi$
$937$	−0.371808	−	1.94908i	−0.371808	−	1.94908i	−0.0627905	−	0.998027i	$-0.520000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$941$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$947$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$948$		0			0
$949$	−1.41039	+	0.775367i	−1.41039	+	0.775367i
$950$		0			0
$951$		0			0
$952$		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			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.92617	+	0.762627i	−1.92617	+	0.762627i
$962$		0			0
$963$		0			0
$964$	0.200808	+	0.316423i	0.200808	+	0.316423i
$965$		0			0
$966$		0			0
$967$	−0.734796	+	1.85588i	−0.734796	+	1.85588i	−0.309017	+	0.951057i	$0.600000\pi$
$967$	−0.734796	+	1.85588i	−0.734796	+	1.85588i	−0.425779	+	0.904827i	$0.640000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$971$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$972$		0			0
$973$	−1.87039	+	1.99176i	−1.87039	+	1.99176i
$974$		0			0
$975$		0			0
$976$	1.89836	+	0.119435i	1.89836	+	0.119435i
$977$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$977$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$983$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.675590	−	2.63125i	0.675590	−	2.63125i
$989$		0			0
$990$		0			0
$991$	−0.0388067	+	0.119435i	−0.0388067	+	0.119435i	−0.968583	−	0.248690i	$-0.920000\pi$
$991$	−0.0388067	+	0.119435i	−0.0388067	+	0.119435i	0.929776	+	0.368125i	$0.120000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.53583	+	0.844328i	−1.53583	+	0.844328i	−0.535827	+	0.844328i	$0.680000\pi$
$997$	−1.53583	+	0.844328i	−1.53583	+	0.844328i	−1.00000			$1.00000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1359.1.bp.a.532.1	✓	20
3.2	odd	2	CM	1359.1.bp.a.532.1	✓	20
151.65	odd	50	inner	1359.1.bp.a.820.1	yes	20
453.65	even	50	inner	1359.1.bp.a.820.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1359.1.bp.a.532.1	✓	20	1.1	even	1	trivial
1359.1.bp.a.532.1	✓	20	3.2	odd	2	CM
1359.1.bp.a.820.1	yes	20	151.65	odd	50	inner
1359.1.bp.a.820.1	yes	20	453.65	even	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:	\( N \)	\(=\)	\( 1359 = 3^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1359.bp (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 + 0.587785i) q^{4} +(-0.813516 + 1.47978i) q^{7} +O(q^{10})\)	\(q+(0.809017 + 0.587785i) q^{4} +(-0.813516 + 1.47978i) q^{7} +(-1.15596 - 0.733597i) q^{13} +(0.309017 + 0.951057i) q^{16} +(1.60528 + 1.16630i) q^{19} +(-0.0627905 + 0.998027i) q^{25} +(-1.52794 + 0.718995i) q^{28} +(-0.328407 - 1.72157i) q^{31} +(-0.996398 + 1.57007i) q^{37} +(1.11716 - 0.614163i) q^{43} +(-0.992115 - 1.56332i) q^{49} +(-0.503997 - 1.27295i) q^{52} +(0.700215 - 1.76854i) q^{61} +(-0.309017 + 0.951057i) q^{64} +(-0.340480 + 0.362574i) q^{67} +(0.566335 - 1.03016i) q^{73} +(0.613161 + 1.88711i) q^{76} +(1.96070 + 0.374023i) q^{79} +(2.02596 - 1.11378i) q^{91} +(0.450527 + 0.423073i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{4}+O(q^{10}) \) 20 * q + 5 * q^4	\( 20 q + 5 q^{4} - 5 q^{16} - 5 q^{31} - 5 q^{37} + 5 q^{64} + 5 q^{91}+O(q^{100}) \) 20 * q + 5 * q^4 - 5 * q^16 - 5 * q^31 - 5 * q^37 + 5 * q^64 + 5 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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