LMFDB - Embedded newform 1012.1.r.a.417.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1012,1,Mod(197,1012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1012, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 14]))

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

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

chi := DirichletCharacter("1012.197");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1012 = 2^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1012.r (of order $22$, degree $10$, 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:	$0.505053792785$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Embedding invariants

Embedding label			417.2
Root			$0.981929 + 0.189251i$ of defining polynomial
Character	$\chi$	$=$	1012.417
Dual form			1012.1.r.a.813.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0395325 - 0.0865641i) q^{3} +(1.02951 - 1.18812i) q^{5} +(0.648930 + 0.748905i) q^{9} +O(q^{10})$	$q+(0.0395325 - 0.0865641i) q^{3} +(1.02951 - 1.18812i) q^{5} +(0.648930 + 0.748905i) q^{9} +(-0.142315 + 0.989821i) q^{11} +(-0.0621493 - 0.136088i) q^{15} +(-0.786053 - 0.618159i) q^{23} +(-0.209419 - 1.45654i) q^{25} +(0.181791 - 0.0533787i) q^{27} +(-0.271738 - 0.595023i) q^{31} +(0.0800569 + 0.0514495i) q^{33} +(-0.308779 - 0.356349i) q^{37} +1.55787 q^{45} -1.30972 q^{47} +(0.415415 - 0.909632i) q^{49} +(-1.61435 + 1.03748i) q^{53} +(1.02951 + 1.18812i) q^{55} +(1.21769 + 0.782560i) q^{59} +(0.252989 + 1.75958i) q^{67} +(-0.0845850 + 0.0436066i) q^{69} +(-0.264241 - 1.83784i) q^{71} +(-0.134363 - 0.0394525i) q^{75} +(-0.138460 + 0.963011i) q^{81} +(-0.738471 + 1.61703i) q^{89} -0.0622501 q^{93} +(0.428368 - 0.494363i) q^{97} +(-0.833635 + 0.535745i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} + 2 q^{5}+O(q^{10}) $ 20 * q + 2 * q^3 + 2 * q^5	$ 20 q + 2 q^{3} + 2 q^{5} - 2 q^{11} - 13 q^{15} + q^{23} - 2 q^{27} + 2 q^{31} - 9 q^{33} + 2 q^{37} - 4 q^{47} - 2 q^{49} - 4 q^{53} + 2 q^{55} + 2 q^{59} + 2 q^{67} - 12 q^{69} - 9 q^{71} + 22 q^{75} + 2 q^{81} + 2 q^{89} - 2 q^{93} - 9 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 + 2 * q^5 - 2 * q^11 - 13 * q^15 + q^23 - 2 * q^27 + 2 * q^31 - 9 * q^33 + 2 * q^37 - 4 * q^47 - 2 * q^49 - 4 * q^53 + 2 * q^55 + 2 * q^59 + 2 * q^67 - 12 * q^69 - 9 * q^71 + 22 * q^75 + 2 * q^81 + 2 * q^89 - 2 * q^93 - 9 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1012\mathbb{Z}\right)^\times$.

$n$	$277$	$507$	$925$
$\chi(n)$	$-1$	$1$	$e\left(\frac{8}{11}\right)$

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
$2$		0			0
$3$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	−0.888835	−	0.458227i	$-0.848485\pi$
$3$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	0.928368	+	0.371662i	$0.121212\pi$
$4$		0			0
$5$	1.02951	−	1.18812i	1.02951	−	1.18812i	0.0475819	−	0.998867i	$-0.484848\pi$
$5$	1.02951	−	1.18812i	1.02951	−	1.18812i	0.981929	−	0.189251i	$-0.0606061\pi$
$6$		0			0
$7$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$7$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$8$		0			0
$9$	0.648930	+	0.748905i	0.648930	+	0.748905i
$10$		0			0
$11$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$11$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$12$		0			0
$13$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$13$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$14$		0			0
$15$	−0.0621493	−	0.136088i	−0.0621493	−	0.136088i
$16$		0			0
$17$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$17$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$18$		0			0
$19$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$19$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$23$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$24$		0			0
$25$	−0.209419	−	1.45654i	−0.209419	−	1.45654i
$26$		0			0
$27$	0.181791	−	0.0533787i	0.181791	−	0.0533787i
$28$		0			0
$29$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$29$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$30$		0			0
$31$	−0.271738	−	0.595023i	−0.271738	−	0.595023i	0.723734	−	0.690079i	$-0.242424\pi$
$31$	−0.271738	−	0.595023i	−0.271738	−	0.595023i	−0.995472	+	0.0950560i	$0.969697\pi$
$32$		0			0
$33$	0.0800569	+	0.0514495i	0.0800569	+	0.0514495i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.308779	−	0.356349i	−0.308779	−	0.356349i	0.580057	−	0.814576i	$-0.303030\pi$
$37$	−0.308779	−	0.356349i	−0.308779	−	0.356349i	−0.888835	+	0.458227i	$0.848485\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$41$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$42$		0			0
$43$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$43$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$44$		0			0
$45$	1.55787			1.55787
$46$		0			0
$47$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$47$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$48$		0			0
$49$	0.415415	−	0.909632i	0.415415	−	0.909632i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$53$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$54$		0			0
$55$	1.02951	+	1.18812i	1.02951	+	1.18812i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.21769	+	0.782560i	1.21769	+	0.782560i	0.981929	−	0.189251i	$-0.0606061\pi$
$59$	1.21769	+	0.782560i	1.21769	+	0.782560i	0.235759	+	0.971812i	$0.424242\pi$
$60$		0			0
$61$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$61$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.252989	+	1.75958i	0.252989	+	1.75958i	0.580057	+	0.814576i	$0.303030\pi$
$67$	0.252989	+	1.75958i	0.252989	+	1.75958i	−0.327068	+	0.945001i	$0.606061\pi$
$68$		0			0
$69$	−0.0845850	+	0.0436066i	−0.0845850	+	0.0436066i
$70$		0			0
$71$	−0.264241	−	1.83784i	−0.264241	−	1.83784i	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−0.264241	−	1.83784i	−0.264241	−	1.83784i	0.235759	−	0.971812i	$-0.424242\pi$
$72$		0			0
$73$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$73$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$74$		0			0
$75$	−0.134363	−	0.0394525i	−0.134363	−	0.0394525i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$79$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$80$		0			0
$81$	−0.138460	+	0.963011i	−0.138460	+	0.963011i
$82$		0			0
$83$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$83$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	0.0475819	+	0.998867i	$0.484848\pi$
$89$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	−0.786053	+	0.618159i	$0.787879\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.0622501			−0.0622501
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.428368	−	0.494363i	0.428368	−	0.494363i	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	0.428368	−	0.494363i	0.428368	−	0.494363i	0.928368	+	0.371662i	$0.121212\pi$
$98$		0			0
$99$	−0.833635	+	0.535745i	−0.833635	+	0.535745i
$100$		0			0
$101$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$101$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$102$		0			0
$103$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$103$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$107$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$108$		0			0
$109$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$109$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$110$		0			0
$111$	−0.0430538	+	0.0126417i	−0.0430538	+	0.0126417i
$112$		0			0
$113$	−0.279486	−	1.94387i	−0.279486	−	1.94387i	−0.327068	−	0.945001i	$-0.606061\pi$
$113$	−0.279486	−	1.94387i	−0.279486	−	1.94387i	0.0475819	−	0.998867i	$-0.484848\pi$
$114$		0			0
$115$	−1.54370	+	0.297523i	−1.54370	+	0.297523i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.623601	−	0.400764i	−0.623601	−	0.400764i
$126$		0			0
$127$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$127$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$131$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.123736	−	0.270943i	0.123736	−	0.270943i
$136$		0			0
$137$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$137$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−0.0517765	+	0.113375i	−0.0517765	+	0.113375i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.0623191	−	0.0719200i	−0.0623191	−	0.0719200i
$148$		0			0
$149$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$149$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$150$		0			0
$151$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$151$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.986715	−	0.289726i	−0.986715	−	0.289726i
$156$		0			0
$157$	0.959493	−	0.281733i	0.959493	−	0.281733i	0.235759	−	0.971812i	$-0.424242\pi$
$157$	0.959493	−	0.281733i	0.959493	−	0.281733i	0.723734	+	0.690079i	$0.242424\pi$
$158$		0			0
$159$	0.0259893	+	0.180759i	0.0259893	+	0.180759i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$163$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$164$		0			0
$165$	0.143547	−	0.0421493i	0.143547	−	0.0421493i
$166$		0			0
$167$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$167$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$168$		0			0
$169$	0.415415	+	0.909632i	0.415415	+	0.909632i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$173$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.115880	−	0.0744714i	0.115880	−	0.0744714i
$178$		0			0
$179$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	−0.995472	−	0.0950560i	$-0.969697\pi$
$179$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	0.235759	+	0.971812i	$0.424242\pi$
$180$		0			0
$181$	−0.827068	+	1.81103i	−0.827068	+	1.81103i	−0.327068	+	0.945001i	$0.606061\pi$
$181$	−0.827068	+	1.81103i	−0.827068	+	1.81103i	−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.741276			−0.741276
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	0.580057	+	0.814576i	$0.303030\pi$
$192$		0			0
$193$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$193$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$197$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$198$		0			0
$199$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$199$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$200$		0			0
$201$	0.162317	+	0.0476607i	0.162317	+	0.0476607i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.0471510	−	0.989821i	−0.0471510	−	0.989821i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$211$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$212$		0			0
$213$	−0.169537	−	0.0497805i	−0.169537	−	0.0497805i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.975950	−	0.627205i	0.975950	−	0.627205i	0.0475819	−	0.998867i	$-0.484848\pi$
$223$	0.975950	−	0.627205i	0.975950	−	0.627205i	0.928368	+	0.371662i	$0.121212\pi$
$224$		0			0
$225$	0.954912	−	1.10203i	0.954912	−	1.10203i
$226$		0			0
$227$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$227$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$228$		0			0
$229$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$229$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$233$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$234$		0			0
$235$	−1.34837	+	1.55610i	−1.34837	+	1.55610i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$239$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$240$		0			0
$241$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$241$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$242$		0			0
$243$	0.237277	+	0.152489i	0.237277	+	0.152489i
$244$		0			0
$245$	−0.653077	−	1.43004i	−0.653077	−	1.43004i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$251$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$252$		0			0
$253$	0.723734	−	0.690079i	0.723734	−	0.690079i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$257$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		0			0
$265$	−0.429342	+	2.98614i	−0.429342	+	2.98614i
$266$		0			0
$267$	0.110783	+	0.127850i	0.110783	+	0.127850i
$268$		0			0
$269$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$269$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$270$		0			0
$271$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$271$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.47152			1.47152
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	0.269277	−	0.589634i	0.269277	−	0.589634i
$280$		0			0
$281$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$281$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$282$		0			0
$283$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$283$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.841254	+	0.540641i	0.841254	+	0.540641i
$290$		0			0
$291$	−0.0258596	−	0.0566247i	−0.0258596	−	0.0566247i
$292$		0			0
$293$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$293$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$294$		0			0
$295$	2.18340	−	0.641103i	2.18340	−	0.641103i
$296$		0			0
$297$	0.0269638	+	0.187537i	0.0269638	+	0.187537i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$307$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$308$		0			0
$309$	0.0665137	+	0.0427457i	0.0665137	+	0.0427457i
$310$		0			0
$311$	0.273100	−	1.89945i	0.273100	−	1.89945i	−0.142315	−	0.989821i	$-0.545455\pi$
$311$	0.273100	−	1.89945i	0.273100	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$312$		0			0
$313$	−1.21590	−	1.40323i	−1.21590	−	1.40323i	−0.888835	−	0.458227i	$-0.848485\pi$
$313$	−1.21590	−	1.40323i	−1.21590	−	1.40323i	−0.327068	−	0.945001i	$-0.606061\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.28605	+	1.48418i	−1.28605	+	1.48418i	−0.500000	+	0.866025i	$0.666667\pi$
$317$	−1.28605	+	1.48418i	−1.28605	+	1.48418i	−0.786053	+	0.618159i	$0.787879\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.21590	−	1.40323i	−1.21590	−	1.40323i	−0.888835	−	0.458227i	$-0.848485\pi$
$331$	−1.21590	−	1.40323i	−1.21590	−	1.40323i	−0.327068	−	0.945001i	$-0.606061\pi$
$332$		0			0
$333$	0.0664963	−	0.462492i	0.0664963	−	0.462492i
$334$		0			0
$335$	2.35104	+	1.51092i	2.35104	+	1.51092i
$336$		0			0
$337$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$337$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$338$		0			0
$339$	−0.179318	−	0.0526525i	−0.179318	−	0.0526525i
$340$		0			0
$341$	0.627639	−	0.184291i	0.627639	−	0.184291i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.0352713	+	0.145390i	−0.0352713	+	0.145390i
$346$		0			0
$347$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$347$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$348$		0			0
$349$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$349$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.481929	+	1.05528i	0.481929	+	1.05528i	0.981929	+	0.189251i	$0.0606061\pi$
$353$	0.481929	+	1.05528i	0.481929	+	1.05528i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	−2.45561	−	1.57812i	−2.45561	−	1.57812i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$359$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$360$		0			0
$361$	0.841254	−	0.540641i	0.841254	−	0.540641i
$362$		0			0
$363$	−0.0623191	+	0.0719200i	−0.0623191	+	0.0719200i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$367$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$373$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$374$		0			0
$375$	−0.0593443	+	0.0381383i	−0.0593443	+	0.0381383i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.205996	+	1.43273i	−0.205996	+	1.43273i	0.580057	+	0.814576i	$0.303030\pi$
$379$	−0.205996	+	1.43273i	−0.205996	+	1.43273i	−0.786053	+	0.618159i	$0.787879\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.827068	−	1.81103i	−0.827068	−	1.81103i	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	−0.827068	−	1.81103i	−0.827068	−	1.81103i	−0.327068	−	0.945001i	$-0.606061\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.165101	−	1.14831i	−0.165101	−	1.14831i	−0.888835	−	0.458227i	$-0.848485\pi$
$389$	−0.165101	−	1.14831i	−0.165101	−	1.14831i	0.723734	−	0.690079i	$-0.242424\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$397$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$401$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00162	+	1.15594i	1.00162	+	1.15594i
$406$		0			0
$407$	0.396666	−	0.254922i	0.396666	−	0.254922i
$408$		0			0
$409$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$409$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$410$		0			0
$411$	−0.0787070	+	0.172344i	−0.0787070	+	0.172344i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	−0.959493	−	0.281733i	$-0.909091\pi$
$419$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$420$		0			0
$421$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$421$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$422$		0			0
$423$	−0.849918	−	0.980857i	−0.849918	−	0.980857i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$431$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$432$		0			0
$433$	−1.11312	+	0.326842i	−1.11312	+	0.326842i	−0.786053	−	0.618159i	$-0.787879\pi$
$433$	−1.11312	+	0.326842i	−1.11312	+	0.326842i	−0.327068	+	0.945001i	$0.606061\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$439$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$440$		0			0
$441$	0.950804	−	0.279181i	0.950804	−	0.279181i
$442$		0			0
$443$	1.50842	+	0.442913i	1.50842	+	0.442913i	0.928368	−	0.371662i	$-0.121212\pi$
$443$	1.50842	+	0.442913i	1.50842	+	0.442913i	0.580057	+	0.814576i	$0.303030\pi$
$444$		0			0
$445$	1.16096	+	2.54214i	1.16096	+	2.54214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.0930932	−	0.647478i	0.0930932	−	0.647478i	−0.888835	−	0.458227i	$-0.848485\pi$
$449$	0.0930932	−	0.647478i	0.0930932	−	0.647478i	0.981929	−	0.189251i	$-0.0606061\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$457$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	0.815816	−	1.78639i	0.815816	−	1.78639i	0.235759	−	0.971812i	$-0.424242\pi$
$463$	0.815816	−	1.78639i	0.815816	−	1.78639i	0.580057	−	0.814576i	$-0.303030\pi$
$464$		0			0
$465$	−0.0640871	+	0.0739605i	−0.0640871	+	0.0739605i
$466$		0			0
$467$	1.65210	−	1.06174i	1.65210	−	1.06174i	0.723734	−	0.690079i	$-0.242424\pi$
$467$	1.65210	−	1.06174i	1.65210	−	1.06174i	0.928368	−	0.371662i	$-0.121212\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.0135432	−	0.0941952i	0.0135432	−	0.0941952i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.82458	−	0.535745i	−1.82458	−	0.535745i
$478$		0			0
$479$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$479$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.146352	−	1.01790i	−0.146352	−	1.01790i
$486$		0			0
$487$	1.70566	−	0.500828i	1.70566	−	0.500828i	0.723734	−	0.690079i	$-0.242424\pi$
$487$	1.70566	−	0.500828i	1.70566	−	0.500828i	0.981929	+	0.189251i	$0.0606061\pi$
$488$		0			0
$489$	−0.0758623	−	0.0222752i	−0.0758623	−	0.0222752i
$490$		0			0
$491$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$491$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.221708	+	1.54201i	−0.221708	+	1.54201i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$499$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$503$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.0951638			0.0951638
$508$		0			0
$509$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	0.0475819	+	0.998867i	$0.484848\pi$
$509$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	−0.786053	+	0.618159i	$0.787879\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.855348	+	0.987125i	0.855348	+	0.987125i
$516$		0			0
$517$	0.186393	−	1.29639i	0.186393	−	1.29639i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.601300	+	1.31666i	0.601300	+	1.31666i	0.928368	+	0.371662i	$0.121212\pi$
$521$	0.601300	+	1.31666i	0.601300	+	1.31666i	−0.327068	+	0.945001i	$0.606061\pi$
$522$		0			0
$523$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$523$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.235759	+	0.971812i	0.235759	+	0.971812i
$530$		0			0
$531$	0.204131	+	1.41976i	0.204131	+	1.41976i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.0458622	+	0.100424i	0.0458622	+	0.100424i
$538$		0			0
$539$	0.841254	+	0.540641i	0.841254	+	0.540641i
$540$		0			0
$541$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$541$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$542$		0			0
$543$	0.124074	+	0.143189i	0.124074	+	0.143189i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$547$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.0293045	+	0.0641679i	−0.0293045	+	0.0641679i
$556$		0			0
$557$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$557$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$563$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$564$		0			0
$565$	−2.59728	−	1.66917i	−2.59728	−	1.66917i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$569$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$570$		0			0
$571$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$571$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$572$		0			0
$573$	−0.00128883	−	0.00896398i	−0.00128883	−	0.00896398i
$574$		0			0
$575$	−0.735759	+	1.27437i	−0.735759	+	1.27437i
$576$		0			0
$577$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$577$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.797176	−	1.74557i	−0.797176	−	1.74557i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$587$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$593$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.160114			0.160114
$598$		0			0
$599$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$599$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$600$		0			0
$601$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$601$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$602$		0			0
$603$	−1.15358	+	1.33131i	−1.15358	+	1.33131i
$604$		0			0
$605$	−1.32254	+	0.849945i	−1.32254	+	0.849945i
$606$		0			0
$607$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$607$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$613$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$617$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$618$		0			0
$619$	−0.264241	−	1.83784i	−0.264241	−	1.83784i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−0.264241	−	1.83784i	−0.264241	−	1.83784i	0.235759	−	0.971812i	$-0.424242\pi$
$620$		0			0
$621$	−0.175894	−	0.0704173i	−0.175894	−	0.0704173i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.293752	−	0.0862533i	0.293752	−	0.0862533i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.550294	−	0.353653i	−0.550294	−	0.353653i	0.235759	−	0.971812i	$-0.424242\pi$
$631$	−0.550294	−	0.353653i	−0.550294	−	0.353653i	−0.786053	+	0.618159i	$0.787879\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1.20489	−	1.39052i	1.20489	−	1.39052i
$640$		0			0
$641$	0.195876	−	0.428908i	0.195876	−	0.428908i	−0.786053	−	0.618159i	$-0.787879\pi$
$641$	0.195876	−	0.428908i	0.195876	−	0.428908i	0.981929	+	0.189251i	$0.0606061\pi$
$642$		0			0
$643$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$643$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.827068	+	1.81103i	−0.827068	+	1.81103i	−0.327068	+	0.945001i	$0.606061\pi$
$647$	−0.827068	+	1.81103i	−0.827068	+	1.81103i	−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$	−0.947890	+	1.09392i	−0.947890	+	1.09392i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.759713	−	0.876756i	−0.759713	−	0.876756i	0.235759	−	0.971812i	$-0.424242\pi$
$653$	−0.759713	−	0.876756i	−0.759713	−	0.876756i	−0.995472	+	0.0950560i	$0.969697\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$659$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$660$		0			0
$661$	−1.11312	−	0.326842i	−1.11312	−	0.326842i	−0.327068	−	0.945001i	$-0.606061\pi$
$661$	−1.11312	−	0.326842i	−1.11312	−	0.326842i	−0.786053	+	0.618159i	$0.787879\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.0157117	−	0.109277i	−0.0157117	−	0.109277i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$673$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$674$		0			0
$675$	−0.115819	−	0.253608i	−0.115819	−	0.253608i
$676$		0			0
$677$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$677$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$683$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$684$		0			0
$685$	−2.04970	+	2.36548i	−2.04970	+	2.36548i
$686$		0			0
$687$	0.0776362	−	0.169999i	0.0776362	−	0.169999i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$691$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$692$		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		0.142315	−	0.989821i	$-0.454545\pi$
$701$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.0813982	+	0.178237i	0.0813982	+	0.178237i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i	−0.327068	−	0.945001i	$-0.606061\pi$
$709$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i	0.235759	+	0.971812i	$0.424242\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.154218	+	0.635697i	−0.154218	+	0.635697i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.70566	+	0.500828i	1.70566	+	0.500828i	0.981929	−	0.189251i	$-0.0606061\pi$
$719$	1.70566	+	0.500828i	1.70566	+	0.500828i	0.723734	+	0.690079i	$0.242424\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.947890	−	1.09392i	−0.947890	−	1.09392i	−0.995472	−	0.0950560i	$-0.969697\pi$
$727$	−0.947890	−	1.09392i	−0.947890	−	1.09392i	0.0475819	−	0.998867i	$-0.484848\pi$
$728$		0			0
$729$	−0.795887	+	0.511485i	−0.795887	+	0.511485i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$733$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$734$		0			0
$735$	−0.149608			−0.149608
$736$		0			0
$737$	−1.77767			−1.77767
$738$		0			0
$739$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$739$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$743$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.415415	−	0.909632i	−0.415415	−	0.909632i	−0.995472	−	0.0950560i	$-0.969697\pi$
$751$	−0.415415	−	0.909632i	−0.415415	−	0.909632i	0.580057	−	0.814576i	$-0.303030\pi$
$752$		0			0
$753$	−0.132167	−	0.0388077i	−0.132167	−	0.0388077i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$757$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$758$		0			0
$759$	−0.0311250	−	0.0899299i	−0.0311250	−	0.0899299i
$760$		0			0
$761$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$761$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$769$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$770$		0			0
$771$	0.0259893	−	0.180759i	0.0259893	−	0.180759i
$772$		0			0
$773$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$773$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$774$		0			0
$775$	−0.809768	+	0.520406i	−0.809768	+	0.520406i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	1.85674			1.85674
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.653077	−	1.43004i	0.653077	−	1.43004i
$786$		0			0
$787$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$787$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.241520	+	0.155215i	0.241520	+	0.155215i
$796$		0			0
$797$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	0.928368	−	0.371662i	$-0.121212\pi$
$797$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	−0.888835	+	0.458227i	$0.848485\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.69022	+	0.496292i	−1.69022	+	0.496292i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.00385480	+	0.0268107i	0.00385480	+	0.0268107i
$808$		0			0
$809$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$809$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$810$		0			0
$811$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$811$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.09881	−	0.706160i	−1.09881	−	0.706160i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$821$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$822$		0			0
$823$	−0.947890	+	1.09392i	−0.947890	+	1.09392i	0.0475819	+	0.998867i	$0.484848\pi$
$823$	−0.947890	+	1.09392i	−0.947890	+	1.09392i	−0.995472	+	0.0950560i	$0.969697\pi$
$824$		0			0
$825$	0.0581728	−	0.127381i	0.0581728	−	0.127381i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$829$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.0811611	−	0.0936649i	−0.0811611	−	0.0936649i
$838$		0			0
$839$	−0.279486	+	1.94387i	−0.279486	+	1.94387i	0.0475819	+	0.998867i	$0.484848\pi$
$839$	−0.279486	+	1.94387i	−0.279486	+	1.94387i	−0.327068	+	0.945001i	$0.606061\pi$
$840$		0			0
$841$	0.841254	+	0.540641i	0.841254	+	0.540641i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.50842	+	0.442913i	1.50842	+	0.442913i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.0224357	+	0.470984i	0.0224357	+	0.470984i
$852$		0			0
$853$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$853$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$857$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$858$		0			0
$859$	−0.738471	−	1.61703i	−0.738471	−	1.61703i	−0.786053	−	0.618159i	$-0.787879\pi$
$859$	−0.738471	−	1.61703i	−0.738471	−	1.61703i	0.0475819	−	0.998867i	$-0.484848\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$863$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.648212			0.648212
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$877$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.975950	−	0.627205i	0.975950	−	0.627205i	0.0475819	−	0.998867i	$-0.484848\pi$
$881$	0.975950	−	0.627205i	0.975950	−	0.627205i	0.928368	+	0.371662i	$0.121212\pi$
$882$		0			0
$883$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$883$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$884$		0			0
$885$	0.0308186	−	0.214348i	0.0308186	−	0.214348i
$886$		0			0
$887$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$887$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.933504	−	0.274101i	−0.933504	−	0.274101i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.259557	+	1.80526i	0.259557	+	1.80526i
$896$		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
$905$	1.30024	+	2.84713i	1.30024	+	2.84713i
$906$		0			0
$907$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$907$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$911$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.454373	+	0.524375i	−0.454373	+	0.524375i
$926$		0			0
$927$	−0.692609	+	0.445113i	−0.692609	+	0.445113i
$928$		0			0
$929$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−0.153628	−	0.0987308i	−0.153628	−	0.0987308i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$937$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$938$		0			0
$939$	−0.169537	+	0.0497805i	−0.169537	+	0.0497805i
$940$		0			0
$941$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$941$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.91030	−	0.560914i	1.91030	−	0.560914i	0.928368	−	0.371662i	$-0.121212\pi$
$947$	1.91030	−	0.560914i	1.91030	−	0.560914i	0.981929	−	0.189251i	$-0.0606061\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.0776362	+	0.169999i	0.0776362	+	0.169999i
$952$		0			0
$953$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$953$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$	0.0212914	−	0.148085i	0.0212914	−	0.148085i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.374650	−	0.432369i	0.374650	−	0.432369i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.30379	−	1.50465i	1.30379	−	1.50465i	0.580057	−	0.814576i	$-0.303030\pi$
$971$	1.30379	−	1.50465i	1.30379	−	1.50465i	0.723734	−	0.690079i	$-0.242424\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.165101	+	1.14831i	−0.165101	+	1.14831i	0.723734	+	0.690079i	$0.242424\pi$
$977$	−0.165101	+	1.14831i	−0.165101	+	1.14831i	−0.888835	+	0.458227i	$0.848485\pi$
$978$		0			0
$979$	−1.49547	−	0.961081i	−1.49547	−	0.961081i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.888835	+	0.458227i	$0.848485\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$991$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$992$		0			0
$993$	−0.169537	+	0.0497805i	−0.169537	+	0.0497805i
$994$		0			0
$995$	2.53794	+	0.745205i	2.53794	+	0.745205i
$996$		0			0
$997$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$997$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$998$		0			0
$999$	−0.0751547	−	0.0482990i	−0.0751547	−	0.0482990i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1012.1.r.a.417.2	✓	20
11.10	odd	2	CM	1012.1.r.a.417.2	✓	20
23.8	even	11	inner	1012.1.r.a.813.2	yes	20
253.54	odd	22	inner	1012.1.r.a.813.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1012.1.r.a.417.2	✓	20	1.1	even	1	trivial
1012.1.r.a.417.2	✓	20	11.10	odd	2	CM
1012.1.r.a.813.2	yes	20	23.8	even	11	inner
1012.1.r.a.813.2	yes	20	253.54	odd	22	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:	\( N \)	\(=\)	\( 1012 = 2^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1012.r (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0395325 - 0.0865641i) q^{3} +(1.02951 - 1.18812i) q^{5} +(0.648930 + 0.748905i) q^{9} +O(q^{10})\)	\(q+(0.0395325 - 0.0865641i) q^{3} +(1.02951 - 1.18812i) q^{5} +(0.648930 + 0.748905i) q^{9} +(-0.142315 + 0.989821i) q^{11} +(-0.0621493 - 0.136088i) q^{15} +(-0.786053 - 0.618159i) q^{23} +(-0.209419 - 1.45654i) q^{25} +(0.181791 - 0.0533787i) q^{27} +(-0.271738 - 0.595023i) q^{31} +(0.0800569 + 0.0514495i) q^{33} +(-0.308779 - 0.356349i) q^{37} +1.55787 q^{45} -1.30972 q^{47} +(0.415415 - 0.909632i) q^{49} +(-1.61435 + 1.03748i) q^{53} +(1.02951 + 1.18812i) q^{55} +(1.21769 + 0.782560i) q^{59} +(0.252989 + 1.75958i) q^{67} +(-0.0845850 + 0.0436066i) q^{69} +(-0.264241 - 1.83784i) q^{71} +(-0.134363 - 0.0394525i) q^{75} +(-0.138460 + 0.963011i) q^{81} +(-0.738471 + 1.61703i) q^{89} -0.0622501 q^{93} +(0.428368 - 0.494363i) q^{97} +(-0.833635 + 0.535745i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 20 * q + 2 * q^3 + 2 * q^5	\( 20 q + 2 q^{3} + 2 q^{5} - 2 q^{11} - 13 q^{15} + q^{23} - 2 q^{27} + 2 q^{31} - 9 q^{33} + 2 q^{37} - 4 q^{47} - 2 q^{49} - 4 q^{53} + 2 q^{55} + 2 q^{59} + 2 q^{67} - 12 q^{69} - 9 q^{71} + 22 q^{75} + 2 q^{81} + 2 q^{89} - 2 q^{93} - 9 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 + 2 * q^5 - 2 * q^11 - 13 * q^15 + q^23 - 2 * q^27 + 2 * q^31 - 9 * q^33 + 2 * q^37 - 4 * q^47 - 2 * q^49 - 4 * q^53 + 2 * q^55 + 2 * q^59 + 2 * q^67 - 12 * q^69 - 9 * q^71 + 22 * q^75 + 2 * q^81 + 2 * q^89 - 2 * q^93 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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