LMFDB - Embedded newform 1191.1.z.a.83.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1191,1,Mod(65,1191)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1191, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 17]))

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

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

chi := DirichletCharacter("1191.65");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1191 = 3 \cdot 397 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1191.z (of order $66$, 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:	$0.594386430046$
Analytic rank:	$0$
Dimension:	$20$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{66}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{66} - \cdots)$

Embedding invariants

Embedding label			83.1
Root			$-0.995472 + 0.0950560i$ of defining polynomial
Character	$\chi$	$=$	1191.83
Dual form			1191.1.z.a.287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0475819 - 0.998867i) q^{3} +(0.654861 - 0.755750i) q^{4} +(0.164642 + 1.72421i) q^{7} +(-0.995472 - 0.0950560i) q^{9} +(-0.723734 - 0.690079i) q^{12} +(0.367833 - 1.90850i) q^{13} +(-0.142315 - 0.989821i) q^{16} +(1.07701 - 0.431171i) q^{19} +(1.73009 - 0.0824143i) q^{21} +(-0.981929 - 0.189251i) q^{25} +(-0.142315 + 0.989821i) q^{27} +(1.41089 + 1.00469i) q^{28} +(0.396666 - 0.254922i) q^{31} +(-0.723734 + 0.690079i) q^{36} +(-0.252989 - 0.130425i) q^{37} +(-1.88884 - 0.458227i) q^{39} +(0.827068 + 1.81103i) q^{43} +(-0.995472 + 0.0950560i) q^{48} +(-1.96386 + 0.378502i) q^{49} +(-1.20147 - 1.52779i) q^{52} +(-0.379436 - 1.09631i) q^{57} +(0.378074 + 1.96163i) q^{61} -1.73205i q^{63} +(-0.841254 - 0.540641i) q^{64} +(-1.16413 - 0.600149i) q^{67} +(-1.54370 + 1.21398i) q^{73} +(-0.235759 + 0.971812i) q^{75} +(0.379436 - 1.09631i) q^{76} +(0.0475819 + 0.0824143i) q^{79} +(0.981929 + 0.189251i) q^{81} +(1.07068 - 1.36148i) q^{84} +(3.35121 + 0.320002i) q^{91} +(-0.235759 - 0.408346i) q^{93} +(0.911911 + 1.28060i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{3} + 2 q^{4} + 3 q^{7} + q^{9} - q^{12} - 3 q^{13} - 2 q^{16} - q^{19} - 3 q^{21} - q^{25} - 2 q^{27} - 3 q^{28} + 2 q^{31} - q^{36} - 2 q^{37} - 19 q^{39} + 9 q^{43} + q^{48} - 2 q^{49}+ \cdots + q^{97}+O(q^{100}) $ 20 * q + q^3 + 2 * q^4 + 3 * q^7 + q^9 - q^12 - 3 * q^13 - 2 * q^16 - q^19 - 3 * q^21 - q^25 - 2 * q^27 - 3 * q^28 + 2 * q^31 - q^36 - 2 * q^37 - 19 * q^39 + 9 * q^43 + q^48 - 2 * q^49 + 3 * q^52 - q^57 + 3 * q^61 + 2 * q^64 - 2 * q^67 - q^73 - q^75 + q^76 + q^79 + q^81 + 3 * q^84 + 3 * q^91 - q^93 + q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1191\mathbb{Z}\right)^\times$.

$n$	$398$	$799$
$\chi(n)$	$-1$	$e\left(\frac{35}{66}\right)$

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		0.909632	−	0.415415i	$-0.136364\pi$
$2$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$3$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$3$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$4$	0.654861	−	0.755750i	0.654861	−	0.755750i
$5$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$5$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$6$		0			0
$7$	0.164642	+	1.72421i	0.164642	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$7$	0.164642	+	1.72421i	0.164642	+	1.72421i	−0.415415	+	0.909632i	$0.636364\pi$
$8$		0			0
$9$	−0.995472	−	0.0950560i	−0.995472	−	0.0950560i
$10$		0			0
$11$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$11$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$12$	−0.723734	−	0.690079i	−0.723734	−	0.690079i
$13$	0.367833	−	1.90850i	0.367833	−	1.90850i	−0.0475819	−	0.998867i	$-0.515152\pi$
$13$	0.367833	−	1.90850i	0.367833	−	1.90850i	0.415415	−	0.909632i	$-0.363636\pi$
$14$		0			0
$15$		0			0
$16$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$17$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$17$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$18$		0			0
$19$	1.07701	−	0.431171i	1.07701	−	0.431171i	0.235759	−	0.971812i	$-0.424242\pi$
$19$	1.07701	−	0.431171i	1.07701	−	0.431171i	0.841254	+	0.540641i	$0.181818\pi$
$20$		0			0
$21$	1.73009	−	0.0824143i	1.73009	−	0.0824143i
$22$		0			0
$23$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$23$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$24$		0			0
$25$	−0.981929	−	0.189251i	−0.981929	−	0.189251i
$26$		0			0
$27$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$28$	1.41089	+	1.00469i	1.41089	+	1.00469i
$29$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$29$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$30$		0			0
$31$	0.396666	−	0.254922i	0.396666	−	0.254922i	−0.327068	−	0.945001i	$-0.606061\pi$
$31$	0.396666	−	0.254922i	0.396666	−	0.254922i	0.723734	+	0.690079i	$0.242424\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.723734	+	0.690079i	−0.723734	+	0.690079i
$37$	−0.252989	−	0.130425i	−0.252989	−	0.130425i	0.327068	−	0.945001i	$-0.393939\pi$
$37$	−0.252989	−	0.130425i	−0.252989	−	0.130425i	−0.580057	+	0.814576i	$0.696970\pi$
$38$		0			0
$39$	−1.88884	−	0.458227i	−1.88884	−	0.458227i
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	0.827068	+	1.81103i	0.827068	+	1.81103i	0.500000	+	0.866025i	$0.333333\pi$
$43$	0.827068	+	1.81103i	0.827068	+	1.81103i	0.327068	+	0.945001i	$0.393939\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$47$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$48$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$49$	−1.96386	+	0.378502i	−1.96386	+	0.378502i
$50$		0			0
$51$		0			0
$52$	−1.20147	−	1.52779i	−1.20147	−	1.52779i
$53$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$53$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.379436	−	1.09631i	−0.379436	−	1.09631i
$58$		0			0
$59$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$59$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$60$		0			0
$61$	0.378074	+	1.96163i	0.378074	+	1.96163i	0.235759	+	0.971812i	$0.424242\pi$
$61$	0.378074	+	1.96163i	0.378074	+	1.96163i	0.142315	+	0.989821i	$0.454545\pi$
$62$		0			0
$63$		−	1.73205i		−	1.73205i
$64$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$65$		0			0
$66$		0			0
$67$	−1.16413	−	0.600149i	−1.16413	−	0.600149i	−0.235759	−	0.971812i	$-0.575758\pi$
$67$	−1.16413	−	0.600149i	−1.16413	−	0.600149i	−0.928368	+	0.371662i	$0.878788\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$71$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$72$		0			0
$73$	−1.54370	+	1.21398i	−1.54370	+	1.21398i	−0.654861	+	0.755750i	$0.727273\pi$
$73$	−1.54370	+	1.21398i	−1.54370	+	1.21398i	−0.888835	+	0.458227i	$0.848485\pi$
$74$		0			0
$75$	−0.235759	+	0.971812i	−0.235759	+	0.971812i
$76$	0.379436	−	1.09631i	0.379436	−	1.09631i
$77$		0			0
$78$		0			0
$79$	0.0475819	+	0.0824143i	0.0475819	+	0.0824143i	0.888835	−	0.458227i	$-0.151515\pi$
$79$	0.0475819	+	0.0824143i	0.0475819	+	0.0824143i	−0.841254	+	0.540641i	$0.818182\pi$
$80$		0			0
$81$	0.981929	+	0.189251i	0.981929	+	0.189251i
$82$		0			0
$83$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$83$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$84$	1.07068	−	1.36148i	1.07068	−	1.36148i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$89$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$90$		0			0
$91$	3.35121	+	0.320002i	3.35121	+	0.320002i
$92$		0			0
$93$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.911911	+	1.28060i	0.911911	+	1.28060i	0.959493	+	0.281733i	$0.0909091\pi$
$97$	0.911911	+	1.28060i	0.911911	+	1.28060i	−0.0475819	+	0.998867i	$0.515152\pi$
$98$		0			0
$99$		0			0
$100$	−0.786053	+	0.618159i	−0.786053	+	0.618159i
$101$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$101$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$102$		0			0
$103$	0.495472	+	0.770969i	0.495472	+	0.770969i	0.995472	−	0.0950560i	$-0.0303030\pi$
$103$	0.495472	+	0.770969i	0.495472	+	0.770969i	−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$107$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$108$	0.654861	+	0.755750i	0.654861	+	0.755750i
$109$	−0.173440	−	0.336426i	−0.173440	−	0.336426i	0.786053	−	0.618159i	$-0.212121\pi$
$109$	−0.173440	−	0.336426i	−0.173440	−	0.336426i	−0.959493	+	0.281733i	$0.909091\pi$
$110$		0			0
$111$	−0.142315	+	0.246497i	−0.142315	+	0.246497i
$112$	1.68323	−	0.408346i	1.68323	−	0.408346i
$113$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$113$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.547582	+	1.86489i	−0.547582	+	1.86489i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$122$		0			0
$123$		0			0
$124$	0.0671040	−	0.466718i	0.0671040	−	0.466718i
$125$		0			0
$126$		0			0
$127$	−0.815816	+	0.157236i	−0.815816	+	0.157236i	−0.580057	−	0.814576i	$-0.696970\pi$
$127$	−0.815816	+	0.157236i	−0.815816	+	0.157236i	−0.235759	+	0.971812i	$0.575758\pi$
$128$		0			0
$129$	1.84833	−	0.739959i	1.84833	−	0.739959i
$130$		0			0
$131$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$131$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$132$		0			0
$133$	0.920749	+	1.78600i	0.920749	+	1.78600i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$137$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$138$		0			0
$139$	1.12424	−	1.17907i	1.12424	−	1.17907i	0.142315	−	0.989821i	$-0.454545\pi$
$139$	1.12424	−	1.17907i	1.12424	−	1.17907i	0.981929	−	0.189251i	$-0.0606061\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$145$		0			0
$146$		0			0
$147$	0.284630	+	1.97964i	0.284630	+	1.97964i
$148$	−0.264241	+	0.105786i	−0.264241	+	0.105786i
$149$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$149$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$150$		0			0
$151$	0.264241	−	1.83784i	0.264241	−	1.83784i	−0.235759	−	0.971812i	$-0.575758\pi$
$151$	0.264241	−	1.83784i	0.264241	−	1.83784i	0.500000	−	0.866025i	$-0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.58323	+	1.12741i	−1.58323	+	1.12741i
$157$			1.23632i			1.23632i	0.786053	+	0.618159i	$0.212121\pi$
$157$			1.23632i			1.23632i	−0.786053	+	0.618159i	$0.787879\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0930932	+	0.647478i	0.0930932	+	0.647478i	0.981929	+	0.189251i	$0.0606061\pi$
$163$	0.0930932	+	0.647478i	0.0930932	+	0.647478i	−0.888835	+	0.458227i	$0.848485\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$167$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$168$		0			0
$169$	−2.57870	−	1.03236i	−2.57870	−	1.03236i
$170$		0			0
$171$	−1.11312	+	0.326842i	−1.11312	+	0.326842i
$172$	1.91030	+	0.560914i	1.91030	+	0.560914i
$173$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$173$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$174$		0			0
$175$	0.164642	−	1.72421i	0.164642	−	1.72421i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$179$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$180$		0			0
$181$	−1.80075	−	0.822373i	−1.80075	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$181$	−1.80075	−	0.822373i	−1.80075	−	0.822373i	−0.841254	−	0.540641i	$-0.818182\pi$
$182$		0			0
$183$	1.97740	−	0.284307i	1.97740	−	0.284307i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.73009	−	0.0824143i	−1.73009	−	0.0824143i
$190$		0			0
$191$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$191$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$192$	−0.580057	+	0.814576i	−0.580057	+	0.814576i
$193$	−0.561767	−	1.40323i	−0.561767	−	1.40323i	−0.888835	−	0.458227i	$-0.848485\pi$
$193$	−0.561767	−	1.40323i	−0.561767	−	1.40323i	0.327068	−	0.945001i	$-0.393939\pi$
$194$		0			0
$195$		0			0
$196$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$197$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$197$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$198$		0			0
$199$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$199$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$200$		0			0
$201$	−0.654861	+	1.13425i	−0.654861	+	1.13425i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.94142	−	0.0924813i	−1.94142	−	0.0924813i
$209$		0			0
$210$		0			0
$211$	0.880786	+	0.627205i	0.880786	+	0.627205i	0.928368	−	0.371662i	$-0.121212\pi$
$211$	0.880786	+	0.627205i	0.880786	+	0.627205i	−0.0475819	+	0.998867i	$0.515152\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.504846	+	0.641964i	0.504846	+	0.641964i
$218$		0			0
$219$	1.13915	+	1.59971i	1.13915	+	1.59971i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.05080	−	0.254922i	−1.05080	−	0.254922i	−0.327068	−	0.945001i	$-0.606061\pi$
$223$	−1.05080	−	0.254922i	−1.05080	−	0.254922i	−0.723734	+	0.690079i	$0.757576\pi$
$224$		0			0
$225$	0.959493	+	0.281733i	0.959493	+	0.281733i
$226$		0			0
$227$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$227$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$228$	−1.07701	−	0.431171i	−1.07701	−	0.431171i
$229$	−0.396666	+	1.63508i	−0.396666	+	1.63508i	0.327068	+	0.945001i	$0.393939\pi$
$229$	−0.396666	+	1.63508i	−0.396666	+	1.63508i	−0.723734	+	0.690079i	$0.757576\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$233$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.0845850	−	0.0436066i	0.0845850	−	0.0436066i
$238$		0			0
$239$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$239$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$240$		0			0
$241$	1.61257	−	1.14831i	1.61257	−	1.14831i	0.723734	−	0.690079i	$-0.242424\pi$
$241$	1.61257	−	1.14831i	1.61257	−	1.14831i	0.888835	−	0.458227i	$-0.151515\pi$
$242$		0			0
$243$	0.235759	−	0.971812i	0.235759	−	0.971812i
$244$	1.73009	+	0.998867i	1.73009	+	0.998867i
$245$		0			0
$246$		0			0
$247$	−0.426728	−	2.21408i	−0.426728	−	2.21408i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$251$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$252$	−1.30900	−	1.13425i	−1.30900	−	1.13425i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$257$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$257$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$258$		0			0
$259$	0.183227	−	0.457679i	0.183227	−	0.457679i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$263$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.21590	+	0.486774i	−1.21590	+	0.486774i
$269$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$269$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$270$		0			0
$271$	−0.771316	−	1.68895i	−0.771316	−	1.68895i	−0.723734	−	0.690079i	$-0.757576\pi$
$271$	−0.771316	−	1.68895i	−0.771316	−	1.68895i	−0.0475819	−	0.998867i	$-0.515152\pi$
$272$		0			0
$273$	0.479096	−	3.33219i	0.479096	−	3.33219i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$277$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$278$		0			0
$279$	−0.419102	+	0.216062i	−0.419102	+	0.216062i
$280$		0			0
$281$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$281$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$282$		0			0
$283$	1.30425	+	1.36786i	1.30425	+	1.36786i	0.888835	+	0.458227i	$0.151515\pi$
$283$	1.30425	+	1.36786i	1.30425	+	1.36786i	0.415415	+	0.909632i	$0.363636\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.654861	+	0.755750i	0.654861	+	0.755750i
$290$		0			0
$291$	1.32254	−	0.849945i	1.32254	−	0.849945i
$292$	−0.0934441	+	1.96163i	−0.0934441	+	1.96163i
$293$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$293$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.580057	+	0.814576i	0.580057	+	0.814576i
$301$	−2.98642	+	1.72421i	−2.98642	+	1.72421i
$302$		0			0
$303$		0			0
$304$	−0.580057	−	1.00469i	−0.580057	−	1.00469i
$305$		0			0
$306$		0			0
$307$	0.419102	+	1.72756i	0.419102	+	1.72756i	0.654861	+	0.755750i	$0.272727\pi$
$307$	0.419102	+	1.72756i	0.419102	+	1.72756i	−0.235759	+	0.971812i	$0.575758\pi$
$308$		0			0
$309$	0.793672	−	0.458227i	0.793672	−	0.458227i
$310$		0			0
$311$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$311$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$312$		0			0
$313$	−0.853157	+	1.08488i	−0.853157	+	1.08488i	0.142315	+	0.989821i	$0.454545\pi$
$313$	−0.853157	+	1.08488i	−0.853157	+	1.08488i	−0.995472	+	0.0950560i	$0.969697\pi$
$314$		0			0
$315$		0			0
$316$	0.0934441	+	0.0180099i	0.0934441	+	0.0180099i
$317$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$317$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.786053	−	0.618159i	0.786053	−	0.618159i
$325$	−0.722372	+	1.80440i	−0.722372	+	1.80440i
$326$		0			0
$327$	−0.344298	+	0.157236i	−0.344298	+	0.157236i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.702443	−	1.75462i	0.702443	−	1.75462i	0.0475819	−	0.998867i	$-0.484848\pi$
$331$	0.702443	−	1.75462i	0.702443	−	1.75462i	0.654861	−	0.755750i	$-0.272727\pi$
$332$		0			0
$333$	0.239446	+	0.153882i	0.239446	+	0.153882i
$334$		0			0
$335$		0			0
$336$	−0.327793	−	1.70075i	−0.327793	−	1.70075i
$337$	−0.143677	+	0.124497i	−0.143677	+	0.124497i	−0.723734	−	0.690079i	$-0.757576\pi$
$337$	−0.143677	+	0.124497i	−0.143677	+	0.124497i	0.580057	+	0.814576i	$0.303030\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.487975	−	1.66189i	−0.487975	−	1.66189i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$347$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$348$		0			0
$349$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i	−0.327068	+	0.945001i	$0.606061\pi$
$350$		0			0
$351$	1.83673	+	0.635697i	1.83673	+	0.635697i
$352$		0			0
$353$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$353$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$359$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$360$		0			0
$361$	0.250314	−	0.238674i	0.250314	−	0.238674i
$362$		0			0
$363$	1.00000			1.00000
$364$	2.43642	−	2.32312i	2.43642	−	2.32312i
$365$		0			0
$366$		0			0
$367$	0.627639	−	0.184291i	0.627639	−	0.184291i	0.0475819	−	0.998867i	$-0.484848\pi$
$367$	0.627639	−	0.184291i	0.627639	−	0.184291i	0.580057	+	0.814576i	$0.303030\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.462997	−	0.0892353i	−0.462997	−	0.0892353i
$373$	−1.78605	−	0.618159i	−1.78605	−	0.618159i	−0.786053	−	0.618159i	$-0.787879\pi$
$373$	−1.78605	−	0.618159i	−1.78605	−	0.618159i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.30425	−	0.451405i	1.30425	−	0.451405i	0.415415	−	0.909632i	$-0.363636\pi$
$379$	1.30425	−	0.451405i	1.30425	−	0.451405i	0.888835	+	0.458227i	$0.151515\pi$
$380$		0			0
$381$	0.118239	+	0.822373i	0.118239	+	0.822373i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.651174	−	1.88144i	−0.651174	−	1.88144i
$388$	1.56499	+	0.149438i	1.56499	+	0.149438i
$389$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$389$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.00000			1.00000
$397$	1.00000			1.00000
$398$		0			0
$399$	1.82779	−	0.834725i	1.82779	−	0.834725i
$400$	−0.0475819	+	0.998867i	−0.0475819	+	0.998867i
$401$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$401$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$402$		0			0
$403$	−0.340611	−	0.850806i	−0.340611	−	0.850806i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.13779	−	1.08488i	−1.13779	−	1.08488i	−0.995472	−	0.0950560i	$-0.969697\pi$
$409$	−1.13779	−	1.08488i	−1.13779	−	1.08488i	−0.142315	−	0.989821i	$-0.545455\pi$
$410$		0			0
$411$		0			0
$412$	0.907125	+	0.130425i	0.907125	+	0.130425i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.12424	−	1.17907i	−1.12424	−	1.17907i
$418$		0			0
$419$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$419$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$420$		0			0
$421$	−0.357685	−	0.123796i	−0.357685	−	0.123796i	0.142315	−	0.989821i	$-0.454545\pi$
$421$	−0.357685	−	0.123796i	−0.357685	−	0.123796i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.32002	+	0.974845i	−3.32002	+	0.974845i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	1.00000			1.00000
$433$	1.04758	−	0.998867i	1.04758	−	0.998867i	0.0475819	−	0.998867i	$-0.484848\pi$
$433$	1.04758	−	0.998867i	1.04758	−	0.998867i	1.00000			$0$
$434$		0			0
$435$		0			0
$436$	−0.367833	−	0.0892353i	−0.367833	−	0.0892353i
$437$		0			0
$438$		0			0
$439$	−1.70566	+	0.879330i	−1.70566	+	0.879330i	−0.723734	+	0.690079i	$0.757576\pi$
$439$	−1.70566	+	0.879330i	−1.70566	+	0.879330i	−0.981929	+	0.189251i	$0.939394\pi$
$440$		0			0
$441$	1.99094	−	0.190112i	1.99094	−	0.190112i
$442$		0			0
$443$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$443$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$444$	0.0930932	+	0.268975i	0.0930932	+	0.268975i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.793672	−	1.53951i	0.793672	−	1.53951i
$449$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$449$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.82318	−	0.351390i	−1.82318	−	0.351390i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.561767	−	0.486774i	0.561767	−	0.486774i	−0.327068	−	0.945001i	$-0.606061\pi$
$457$	0.561767	−	0.486774i	0.561767	−	0.486774i	0.888835	+	0.458227i	$0.151515\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$461$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$462$		0			0
$463$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$463$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$467$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$468$	1.05080	+	1.63508i	1.05080	+	1.63508i
$469$	0.843118	−	2.10601i	0.843118	−	2.10601i
$470$		0			0
$471$	1.23492	+	0.0588264i	1.23492	+	0.0588264i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.13915	+	0.219553i	−1.13915	+	0.219553i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$479$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$480$		0			0
$481$	−0.341973	+	0.434855i	−0.341973	+	0.434855i
$482$		0			0
$483$		0			0
$484$	0.786053	+	0.618159i	0.786053	+	0.618159i
$485$		0			0
$486$		0			0
$487$	0.195876	+	0.807410i	0.195876	+	0.807410i	0.981929	+	0.189251i	$0.0606061\pi$
$487$	0.195876	+	0.807410i	0.195876	+	0.807410i	−0.786053	+	0.618159i	$0.787879\pi$
$488$		0			0
$489$	0.651174	−	0.0621796i	0.651174	−	0.0621796i
$490$		0			0
$491$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$491$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.308779	−	0.356349i	−0.308779	−	0.356349i
$497$		0			0
$498$		0			0
$499$	−0.00452808	+	0.0950560i	−0.00452808	+	0.0950560i	0.995472	+	0.0950560i	$0.0303030\pi$
$499$	−0.00452808	+	0.0950560i	−0.00452808	+	0.0950560i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$503$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.15389	+	2.52666i	−1.15389	+	2.52666i
$508$	−0.415415	+	0.719520i	−0.415415	+	0.719520i
$509$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$509$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$510$		0			0
$511$	−2.34730	−	2.46178i	−2.34730	−	2.46178i
$512$		0			0
$513$	0.273507	+	1.12741i	0.273507	+	1.12741i
$514$		0			0
$515$		0			0
$516$	0.651174	−	1.88144i	0.651174	−	1.88144i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$521$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$522$		0			0
$523$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$523$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$524$		0			0
$525$	−1.71442	−	0.246497i	−1.71442	−	0.246497i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$530$		0			0
$531$		0			0
$532$	1.95273	+	0.473728i	1.95273	+	0.473728i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.0934441	−	1.96163i	−0.0934441	−	1.96163i	−0.235759	−	0.971812i	$-0.575758\pi$
$541$	−0.0934441	−	1.96163i	−0.0934441	−	1.96163i	0.142315	−	0.989821i	$-0.454545\pi$
$542$		0			0
$543$	−0.907125	+	1.75958i	−0.907125	+	1.75958i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.308319	−	1.59971i	−0.308319	−	1.59971i	−0.723734	−	0.690079i	$-0.757576\pi$
$547$	−0.308319	−	1.59971i	−0.308319	−	1.59971i	0.415415	−	0.909632i	$-0.363636\pi$
$548$		0			0
$549$	−0.189897	−	1.98869i	−0.189897	−	1.98869i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.134265	+	0.0956100i	−0.134265	+	0.0956100i
$554$		0			0
$555$		0			0
$556$	−0.154861	−	1.62177i	−0.154861	−	1.62177i
$557$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$557$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$558$		0			0
$559$	3.76057	−	0.912303i	3.76057	−	0.912303i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$563$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.164642	+	1.72421i	−0.164642	+	1.72421i
$568$		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
$571$	−1.34125	−	0.325385i	−1.34125	−	0.325385i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.34125	−	0.325385i	−1.34125	−	0.325385i	−0.841254	+	0.540641i	$0.818182\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.786053	+	0.618159i	0.786053	+	0.618159i
$577$	−0.668404	−	0.849945i	−0.668404	−	0.849945i	0.327068	−	0.945001i	$-0.393939\pi$
$577$	−0.668404	−	0.849945i	−0.668404	−	0.849945i	−0.995472	+	0.0950560i	$0.969697\pi$
$578$		0			0
$579$	−1.42837	+	0.494363i	−1.42837	+	0.494363i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$587$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$588$	1.68251	+	1.08128i	1.68251	+	1.08128i
$589$	0.317299	−	0.445585i	0.317299	−	0.445585i
$590$		0			0
$591$		0			0
$592$	−0.0930932	+	0.268975i	−0.0930932	+	0.268975i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.258195	+	0.500828i	−0.258195	+	0.500828i
$598$		0			0
$599$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$599$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$600$		0			0
$601$	0.209419	+	0.523103i	0.209419	+	0.523103i	0.995472	−	0.0950560i	$-0.0303030\pi$
$601$	0.209419	+	0.523103i	0.209419	+	0.523103i	−0.786053	+	0.618159i	$0.787879\pi$
$602$		0			0
$603$	1.10181	+	0.708089i	1.10181	+	0.708089i
$604$	−1.21590	−	1.40323i	−1.21590	−	1.40323i
$605$		0			0
$606$		0			0
$607$	0.459493	+	1.14776i	0.459493	+	1.14776i	0.959493	+	0.281733i	$0.0909091\pi$
$607$	0.459493	+	1.14776i	0.459493	+	1.14776i	−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$613$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	0.143677	−	1.50465i	0.143677	−	1.50465i	−0.580057	−	0.814576i	$-0.696970\pi$
$619$	0.143677	−	1.50465i	0.143677	−	1.50465i	0.723734	−	0.690079i	$-0.242424\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.184753	+	1.93482i	−0.184753	+	1.93482i
$625$	0.928368	+	0.371662i	0.928368	+	0.371662i
$626$		0			0
$627$		0			0
$628$	0.934347	+	0.809616i	0.934347	+	0.809616i
$629$		0			0
$630$		0			0
$631$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$631$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$632$		0			0
$633$	0.668404	−	0.849945i	0.668404	−	0.849945i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			3.88725i			3.88725i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−0.223734	+	1.55610i	−0.223734	+	1.55610i	0.500000	+	0.866025i	$0.333333\pi$
$643$	−0.223734	+	1.55610i	−0.223734	+	1.55610i	−0.723734	+	0.690079i	$0.757576\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$647$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	0.665258	−	0.473728i	0.665258	−	0.473728i
$652$	0.550294	+	0.353653i	0.550294	+	0.353653i
$653$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$653$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.65210	−	1.06174i	1.65210	−	1.06174i
$658$		0			0
$659$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$659$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$660$		0			0
$661$	−0.915415	−	1.77566i	−0.915415	−	1.77566i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−0.915415	−	1.77566i	−0.915415	−	1.77566i	−0.415415	−	0.909632i	$-0.636364\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.304632	+	1.03748i	−0.304632	+	1.03748i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0800569	−	1.68060i	−0.0800569	−	1.68060i	−0.580057	−	0.814576i	$-0.696970\pi$
$673$	−0.0800569	−	1.68060i	−0.0800569	−	1.68060i	0.500000	−	0.866025i	$-0.333333\pi$
$674$		0			0
$675$	0.327068	−	0.945001i	0.327068	−	0.945001i
$676$	−2.46889	+	1.27280i	−2.46889	+	1.27280i
$677$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$677$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$678$		0			0
$679$	−2.05788	+	1.78316i	−2.05788	+	1.78316i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.481929	+	1.05528i	−0.481929	+	1.05528i
$685$		0			0
$686$		0			0
$687$	1.61435	+	0.474017i	1.61435	+	0.474017i
$688$	1.67489	−	1.07639i	1.67489	−	1.07639i
$689$		0			0
$690$		0			0
$691$	−0.401872	−	0.625325i	−0.401872	−	0.625325i	0.580057	−	0.814576i	$-0.303030\pi$
$691$	−0.401872	−	0.625325i	−0.401872	−	0.625325i	−0.981929	+	0.189251i	$0.939394\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.19525	−	1.25354i	−1.19525	−	1.25354i
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$	−0.328708	−	0.0313878i	−0.328708	−	0.0313878i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.723734	−	1.25354i	0.723734	−	1.25354i	−0.235759	−	0.971812i	$-0.575758\pi$
$709$	0.723734	−	1.25354i	0.723734	−	1.25354i	0.959493	−	0.281733i	$-0.0909091\pi$
$710$		0			0
$711$	−0.0395325	−	0.0865641i	−0.0395325	−	0.0865641i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$719$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$720$		0			0
$721$	−1.24774	+	0.981230i	−1.24774	+	0.981230i
$722$		0			0
$723$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$724$	−1.80075	+	0.822373i	−1.80075	+	0.822373i
$725$		0			0
$726$		0			0
$727$	−1.74555	−	0.899892i	−1.74555	−	0.899892i	−0.959493	−	0.281733i	$-0.909091\pi$
$727$	−1.74555	−	0.899892i	−1.74555	−	0.899892i	−0.786053	−	0.618159i	$-0.787879\pi$
$728$		0			0
$729$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$730$		0			0
$731$		0			0
$732$	1.08006	−	1.68060i	1.08006	−	1.68060i
$733$	−0.261197	−	1.35522i	−0.261197	−	1.35522i	−0.841254	−	0.540641i	$-0.818182\pi$
$733$	−0.261197	−	1.35522i	−0.261197	−	1.35522i	0.580057	−	0.814576i	$-0.303030\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.50979	−	1.30824i	1.50979	−	1.30824i	0.723734	−	0.690079i	$-0.242424\pi$
$739$	1.50979	−	1.30824i	1.50979	−	1.30824i	0.786053	−	0.618159i	$-0.212121\pi$
$740$		0			0
$741$	−2.23187	+	0.320895i	−2.23187	+	0.320895i
$742$		0			0
$743$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$743$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	0.928368	−	0.371662i	$-0.121212\pi$
$751$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	−0.888835	+	0.458227i	$0.848485\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−1.19525	+	1.25354i	−1.19525	+	1.25354i
$757$	−1.39734	−	0.720381i	−1.39734	−	0.720381i	−0.415415	−	0.909632i	$-0.636364\pi$
$757$	−1.39734	−	0.720381i	−1.39734	−	0.720381i	−0.981929	+	0.189251i	$0.939394\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$761$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$762$		0			0
$763$	0.551514	−	0.354436i	0.551514	−	0.354436i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.235759	+	0.971812i	0.235759	+	0.971812i
$769$	0.642315	+	0.123796i	0.642315	+	0.123796i	0.500000	−	0.866025i	$-0.333333\pi$
$769$	0.642315	+	0.123796i	0.642315	+	0.123796i	0.142315	+	0.989821i	$0.454545\pi$
$770$		0			0
$771$		0			0
$772$	−1.42837	−	0.494363i	−1.42837	−	0.494363i
$773$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$773$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$774$		0			0
$775$	−0.437742	+	0.175245i	−0.437742	+	0.175245i
$776$		0			0
$777$	−0.448442	−	0.204797i	−0.448442	−	0.204797i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.654136	+	1.89000i	0.654136	+	1.89000i
$785$		0			0
$786$		0			0
$787$	0.172932	+	1.81103i	0.172932	+	1.81103i	0.500000	+	0.866025i	$0.333333\pi$
$787$	0.172932	+	1.81103i	0.172932	+	1.81103i	−0.327068	+	0.945001i	$0.606061\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	3.88284			3.88284
$794$		0			0
$795$		0			0
$796$	−0.512546	+	0.234072i	−0.512546	+	0.234072i
$797$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$797$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.428368	+	1.23769i	0.428368	+	1.23769i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$809$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$810$		0			0
$811$	1.12459	+	0.513585i	1.12459	+	0.513585i	0.888835	−	0.458227i	$-0.151515\pi$
$811$	1.12459	+	0.513585i	1.12459	+	0.513585i	0.235759	+	0.971812i	$0.424242\pi$
$812$		0			0
$813$	−1.72373	+	0.690079i	−1.72373	+	0.690079i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.67162	+	1.59389i	1.67162	+	1.59389i
$818$		0			0
$819$	−3.30562	−	0.637106i	−3.30562	−	0.637106i
$820$		0			0
$821$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$821$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$822$		0			0
$823$	1.03115	−	1.44805i	1.03115	−	1.44805i	0.142315	−	0.989821i	$-0.454545\pi$
$823$	1.03115	−	1.44805i	1.03115	−	1.44805i	0.888835	−	0.458227i	$-0.151515\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$827$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$828$		0			0
$829$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$829$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$830$		0			0
$831$	−1.49547	−	0.770969i	−1.49547	−	0.770969i
$832$	−1.34125	+	1.40667i	−1.34125	+	1.40667i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.195876	+	0.428908i	0.195876	+	0.428908i
$838$		0			0
$839$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$839$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$840$		0			0
$841$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$842$		0			0
$843$		0			0
$844$	1.05080	−	0.254922i	1.05080	−	0.254922i
$845$		0			0
$846$		0			0
$847$	−1.71442	+	0.246497i	−1.71442	+	0.246497i
$848$		0			0
$849$	1.42837	−	1.23769i	1.42837	−	1.23769i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.23492	+	0.0588264i	−1.23492	+	0.0588264i	−0.654861	−	0.755750i	$-0.727273\pi$
$853$	−1.23492	+	0.0588264i	−1.23492	+	0.0588264i	−0.580057	+	0.814576i	$0.696970\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	1.78153	+	0.523103i	1.78153	+	0.523103i	0.995472	−	0.0950560i	$-0.0303030\pi$
$859$	1.78153	+	0.523103i	1.78153	+	0.523103i	0.786053	+	0.618159i	$0.212121\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$863$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.786053	−	0.618159i	0.786053	−	0.618159i
$868$	0.815768	+	0.0388598i	0.815768	+	0.0388598i
$869$		0			0
$870$		0			0
$871$	−1.57359	+	2.00098i	−1.57359	+	2.00098i
$872$		0			0
$873$	−0.786053	−	1.36148i	−0.786053	−	1.36148i
$874$		0			0
$875$		0			0
$876$	1.95496	+	0.186677i	1.95496	+	0.186677i
$877$	0.415415	+	0.909632i	0.415415	+	0.909632i	0.995472	+	0.0950560i	$0.0303030\pi$
$877$	0.415415	+	0.909632i	0.415415	+	0.909632i	−0.580057	+	0.814576i	$0.696970\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$881$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$882$		0			0
$883$	0.286053	+	1.48418i	0.286053	+	1.48418i	0.786053	+	0.618159i	$0.212121\pi$
$883$	0.286053	+	1.48418i	0.286053	+	1.48418i	−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−0.405424	−	1.38075i	−0.405424	−	1.38075i
$890$		0			0
$891$		0			0
$892$	−0.880786	+	0.627205i	−0.880786	+	0.627205i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.841254	−	0.540641i	0.841254	−	0.540641i
$901$		0			0
$902$		0			0
$903$	1.58016	+	3.06507i	1.58016	+	3.06507i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.07028	−	0.153882i	−1.07028	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$907$	−1.07028	−	0.153882i	−1.07028	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$911$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$912$	−1.03115	+	0.531595i	−1.03115	+	0.531595i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.975950	+	1.37053i	0.975950	+	1.37053i
$917$		0			0
$918$		0			0
$919$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$919$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$920$		0			0
$921$	1.74555	−	0.336426i	1.74555	−	0.336426i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.223734	+	0.175946i	0.223734	+	0.175946i
$926$		0			0
$927$	−0.419943	−	0.814576i	−0.419943	−	0.814576i
$928$		0			0
$929$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$929$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$930$		0			0
$931$	−1.95190	+	1.25441i	−1.95190	+	1.25441i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.23123	−	0.876756i	1.23123	−	0.876756i	0.235759	−	0.971812i	$-0.424242\pi$
$937$	1.23123	−	0.876756i	1.23123	−	0.876756i	0.995472	+	0.0950560i	$0.0303030\pi$
$938$		0			0
$939$	1.04305	+	0.903811i	1.04305	+	0.903811i
$940$		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
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$	0.0224357	−	0.0924813i	0.0224357	−	0.0924813i
$949$	1.74905	+	3.39268i	1.74905	+	3.39268i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$953$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.323056	+	0.707394i	−0.323056	+	0.707394i
$962$		0			0
$963$		0			0
$964$	0.188177	−	1.97068i	0.188177	−	1.97068i
$965$		0			0
$966$		0			0
$967$	−0.514186	+	0.404360i	−0.514186	+	0.404360i	−0.841254	−	0.540641i	$-0.818182\pi$
$967$	−0.514186	+	0.404360i	−0.514186	+	0.404360i	0.327068	+	0.945001i	$0.393939\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−0.580057	−	0.814576i	−0.580057	−	0.814576i
$973$	2.21806	+	1.74430i	2.21806	+	1.74430i
$974$		0			0
$975$	1.76798	+	0.807410i	1.76798	+	0.807410i
$976$	1.88786	−	0.653395i	1.88786	−	0.653395i
$977$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$977$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.140675	+	0.351390i	0.140675	+	0.351390i
$982$		0			0
$983$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$983$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.95273	−	1.12741i	−1.95273	−	1.12741i
$989$		0			0
$990$		0			0
$991$	0.0871144	−	0.168978i	0.0871144	−	0.168978i	−0.841254	−	0.540641i	$-0.818182\pi$
$991$	0.0871144	−	0.168978i	0.0871144	−	0.168978i	0.928368	+	0.371662i	$0.121212\pi$
$992$		0			0
$993$	−1.71921	−	0.785135i	−1.71921	−	0.785135i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.71442	+	0.989821i	1.71442	+	0.989821i	0.928368	+	0.371662i	$0.121212\pi$
$997$	1.71442	+	0.989821i	1.71442	+	0.989821i	0.786053	+	0.618159i	$0.212121\pi$
$998$		0			0
$999$	0.165101	−	0.231852i	0.165101	−	0.231852i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1191.1.z.a.83.1	✓	20
3.2	odd	2	CM	1191.1.z.a.83.1	✓	20
397.287	even	66	inner	1191.1.z.a.287.1	yes	20
1191.287	odd	66	inner	1191.1.z.a.287.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1191.1.z.a.83.1	✓	20	1.1	even	1	trivial
1191.1.z.a.83.1	✓	20	3.2	odd	2	CM
1191.1.z.a.287.1	yes	20	397.287	even	66	inner
1191.1.z.a.287.1	yes	20	1191.287	odd	66	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:	\( N \)	\(=\)	\( 1191 = 3 \cdot 397 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1191.z (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0475819 - 0.998867i) q^{3} +(0.654861 - 0.755750i) q^{4} +(0.164642 + 1.72421i) q^{7} +(-0.995472 - 0.0950560i) q^{9} +(-0.723734 - 0.690079i) q^{12} +(0.367833 - 1.90850i) q^{13} +(-0.142315 - 0.989821i) q^{16} +(1.07701 - 0.431171i) q^{19} +(1.73009 - 0.0824143i) q^{21} +(-0.981929 - 0.189251i) q^{25} +(-0.142315 + 0.989821i) q^{27} +(1.41089 + 1.00469i) q^{28} +(0.396666 - 0.254922i) q^{31} +(-0.723734 + 0.690079i) q^{36} +(-0.252989 - 0.130425i) q^{37} +(-1.88884 - 0.458227i) q^{39} +(0.827068 + 1.81103i) q^{43} +(-0.995472 + 0.0950560i) q^{48} +(-1.96386 + 0.378502i) q^{49} +(-1.20147 - 1.52779i) q^{52} +(-0.379436 - 1.09631i) q^{57} +(0.378074 + 1.96163i) q^{61} -1.73205i q^{63} +(-0.841254 - 0.540641i) q^{64} +(-1.16413 - 0.600149i) q^{67} +(-1.54370 + 1.21398i) q^{73} +(-0.235759 + 0.971812i) q^{75} +(0.379436 - 1.09631i) q^{76} +(0.0475819 + 0.0824143i) q^{79} +(0.981929 + 0.189251i) q^{81} +(1.07068 - 1.36148i) q^{84} +(3.35121 + 0.320002i) q^{91} +(-0.235759 - 0.408346i) q^{93} +(0.911911 + 1.28060i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{3} + 2 q^{4} + 3 q^{7} + q^{9} - q^{12} - 3 q^{13} - 2 q^{16} - q^{19} - 3 q^{21} - q^{25} - 2 q^{27} - 3 q^{28} + 2 q^{31} - q^{36} - 2 q^{37} - 19 q^{39} + 9 q^{43} + q^{48} - 2 q^{49}+ \cdots + q^{97}+O(q^{100}) \) 20 * q + q^3 + 2 * q^4 + 3 * q^7 + q^9 - q^12 - 3 * q^13 - 2 * q^16 - q^19 - 3 * q^21 - q^25 - 2 * q^27 - 3 * q^28 + 2 * q^31 - q^36 - 2 * q^37 - 19 * q^39 + 9 * q^43 + q^48 - 2 * q^49 + 3 * q^52 - q^57 + 3 * q^61 + 2 * q^64 - 2 * q^67 - q^73 - q^75 + q^76 + q^79 + q^81 + 3 * q^84 + 3 * q^91 - q^93 + q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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