LMFDB - Embedded newform 1127.1.x.a.79.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,1,Mod(30,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([22, 57]))

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

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

chi := DirichletCharacter("1127.30");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1127.x (of order $66$, degree $20$, not 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:	$0.562446269237$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

Embedding invariants

Embedding label			79.1
Root			$0.928368 - 0.371662i$ of defining polynomial
Character	$\chi$	$=$	1127.79
Dual form			1127.1.x.a.214.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.279486 - 0.0538665i) q^{2} +(-0.853157 + 0.341553i) q^{4} +(-0.459493 + 0.295298i) q^{8} +(0.995472 + 0.0950560i) q^{9} +O(q^{10})$	$q+(0.279486 - 0.0538665i) q^{2} +(-0.853157 + 0.341553i) q^{4} +(-0.459493 + 0.295298i) q^{8} +(0.995472 + 0.0950560i) q^{9} +(-0.204634 + 1.06174i) q^{11} +(0.552586 - 0.526890i) q^{16} +(0.283341 - 0.0270558i) q^{18} +0.307765i q^{22} +(0.580057 + 0.814576i) q^{23} +(-0.327068 + 0.945001i) q^{25} +(0.186393 + 1.29639i) q^{29} +(0.442886 - 0.621946i) q^{32} +(-0.881761 + 0.258908i) q^{36} +(-0.0535608 + 0.560914i) q^{37} +(0.983568 - 1.53046i) q^{43} +(-0.188056 - 0.975725i) q^{44} +(0.205996 + 0.196417i) q^{46} +(-0.0405070 + 0.281733i) q^{50} +(1.46889 - 0.356349i) q^{53} +(0.121926 + 0.352283i) q^{58} +(-0.226900 + 0.496841i) q^{64} +(-1.87076 - 0.647478i) q^{67} +(-1.10181 - 1.27155i) q^{71} +(-0.485482 + 0.250283i) q^{72} +(0.0152450 + 0.159653i) q^{74} +(-1.46889 - 0.356349i) q^{79} +(0.981929 + 0.189251i) q^{81} +(0.192453 - 0.480724i) q^{86} +(-0.219503 - 0.548291i) q^{88} +(-0.773100 - 0.496841i) q^{92} +(-0.304632 + 1.03748i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 3 q^{4} + 8 q^{8} - q^{9}+O(q^{10}) $ 20 * q - 2 * q^2 + 3 * q^4 + 8 * q^8 - q^9	$ 20 q - 2 q^{2} + 3 q^{4} + 8 q^{8} - q^{9} - 6 q^{16} + 2 q^{18} + q^{23} + q^{25} - 4 q^{29} + 5 q^{32} - 16 q^{36} + 11 q^{44} - 2 q^{46} - 18 q^{50} + 7 q^{58} - 14 q^{64} - 4 q^{71} + 4 q^{72} - 11 q^{74} + q^{81} - 11 q^{88} - 6 q^{92}+O(q^{100}) $ 20 * q - 2 * q^2 + 3 * q^4 + 8 * q^8 - q^9 - 6 * q^16 + 2 * q^18 + q^23 + q^25 - 4 * q^29 + 5 * q^32 - 16 * q^36 + 11 * q^44 - 2 * q^46 - 18 * q^50 + 7 * q^58 - 14 * q^64 - 4 * q^71 + 4 * q^72 - 11 * q^74 + q^81 - 11 * q^88 - 6 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times$.

$n$	$346$	$442$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{3}{22}\right)$

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 $

$2$	0.279486	−	0.0538665i	0.279486	−	0.0538665i	−0.0475819	−	0.998867i	$-0.515152\pi$
$2$	0.279486	−	0.0538665i	0.279486	−	0.0538665i	0.327068	+	0.945001i	$0.393939\pi$
$3$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$3$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$4$	−0.853157	+	0.341553i	−0.853157	+	0.341553i
$5$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$5$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−0.459493	+	0.295298i	−0.459493	+	0.295298i
$9$	0.995472	+	0.0950560i	0.995472	+	0.0950560i
$10$		0			0
$11$	−0.204634	+	1.06174i	−0.204634	+	1.06174i	0.723734	+	0.690079i	$0.242424\pi$
$11$	−0.204634	+	1.06174i	−0.204634	+	1.06174i	−0.928368	+	0.371662i	$0.878788\pi$
$12$		0			0
$13$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$13$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$14$		0			0
$15$		0			0
$16$	0.552586	−	0.526890i	0.552586	−	0.526890i
$17$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$17$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$18$	0.283341	−	0.0270558i	0.283341	−	0.0270558i
$19$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$19$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$20$		0			0
$21$		0			0
$22$			0.307765i			0.307765i
$23$	0.580057	+	0.814576i	0.580057	+	0.814576i
$23$	0.580057	+	0.814576i	0.580057	+	0.814576i
$24$		0			0
$25$	−0.327068	+	0.945001i	−0.327068	+	0.945001i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$29$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$30$		0			0
$31$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$31$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$32$	0.442886	−	0.621946i	0.442886	−	0.621946i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.881761	+	0.258908i	−0.881761	+	0.258908i
$37$	−0.0535608	+	0.560914i	−0.0535608	+	0.560914i	0.928368	+	0.371662i	$0.121212\pi$
$37$	−0.0535608	+	0.560914i	−0.0535608	+	0.560914i	−0.981929	+	0.189251i	$0.939394\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$41$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$42$		0			0
$43$	0.983568	−	1.53046i	0.983568	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$43$	0.983568	−	1.53046i	0.983568	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$44$	−0.188056	−	0.975725i	−0.188056	−	0.975725i
$45$		0			0
$46$	0.205996	+	0.196417i	0.205996	+	0.196417i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$51$		0			0
$52$		0			0
$53$	1.46889	−	0.356349i	1.46889	−	0.356349i	0.580057	−	0.814576i	$-0.303030\pi$
$53$	1.46889	−	0.356349i	1.46889	−	0.356349i	0.888835	+	0.458227i	$0.151515\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.121926	+	0.352283i	0.121926	+	0.352283i
$59$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$59$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$60$		0			0
$61$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$61$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$62$		0			0
$63$		0			0
$64$	−0.226900	+	0.496841i	−0.226900	+	0.496841i
$65$		0			0
$66$		0			0
$67$	−1.87076	−	0.647478i	−1.87076	−	0.647478i	−0.981929	−	0.189251i	$-0.939394\pi$
$67$	−1.87076	−	0.647478i	−1.87076	−	0.647478i	−0.888835	−	0.458227i	$-0.848485\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$71$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$72$	−0.485482	+	0.250283i	−0.485482	+	0.250283i
$73$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$73$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$74$	0.0152450	+	0.159653i	0.0152450	+	0.159653i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.46889	−	0.356349i	−1.46889	−	0.356349i	−0.580057	−	0.814576i	$-0.696970\pi$
$79$	−1.46889	−	0.356349i	−1.46889	−	0.356349i	−0.888835	+	0.458227i	$0.848485\pi$
$80$		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	$0.363636\pi$
$84$		0			0
$85$		0			0
$86$	0.192453	−	0.480724i	0.192453	−	0.480724i
$87$		0			0
$88$	−0.219503	−	0.548291i	−0.219503	−	0.548291i
$89$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$89$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$90$		0			0
$91$		0			0
$92$	−0.773100	−	0.496841i	−0.773100	−	0.496841i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$97$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$98$		0			0
$99$	−0.304632	+	1.03748i	−0.304632	+	1.03748i
$100$	−0.0437271	−	0.917945i	−0.0437271	−	0.917945i
$101$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$101$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$102$		0			0
$103$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$103$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$104$		0			0
$105$		0			0
$106$	0.391340	−	0.178719i	0.391340	−	0.178719i
$107$	−0.562827	+	0.0268107i	−0.562827	+	0.0268107i	−0.327068	−	0.945001i	$-0.606061\pi$
$107$	−0.562827	+	0.0268107i	−0.562827	+	0.0268107i	−0.235759	+	0.971812i	$0.575758\pi$
$108$		0			0
$109$	−1.22373	−	1.55610i	−1.22373	−	1.55610i	−0.723734	−	0.690079i	$-0.757576\pi$
$109$	−1.22373	−	1.55610i	−1.22373	−	1.55610i	−0.500000	−	0.866025i	$-0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.37491	−	1.19136i	1.37491	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$113$	1.37491	−	1.19136i	1.37491	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$114$		0			0
$115$		0			0
$116$	−0.601808	−	1.04236i	−0.601808	−	1.04236i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.157052	−	0.0628741i	−0.157052	−	0.0628741i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$127$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$128$	−0.216659	+	0.893081i	−0.216659	+	0.893081i
$129$		0			0
$130$		0			0
$131$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$131$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$132$		0			0
$133$		0			0
$134$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$135$		0			0
$136$		0			0
$137$	1.71442	−	0.989821i	1.71442	−	0.989821i	0.786053	−	0.618159i	$-0.212121\pi$
$137$	1.71442	−	0.989821i	1.71442	−	0.989821i	0.928368	−	0.371662i	$-0.121212\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−0.376434	−	0.296031i	−0.376434	−	0.296031i
$143$		0			0
$144$	0.600168	−	0.471977i	0.600168	−	0.471977i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.145886	−	0.496841i	−0.145886	−	0.496841i
$149$	−1.71921	+	0.595023i	−1.71921	+	0.595023i	−0.995472	−	0.0950560i	$-0.969697\pi$
$149$	−1.71921	+	0.595023i	−1.71921	+	0.595023i	−0.723734	+	0.690079i	$0.757576\pi$
$150$		0			0
$151$	0.947890	+	0.903811i	0.947890	+	0.903811i	0.995472	−	0.0950560i	$-0.0303030\pi$
$151$	0.947890	+	0.903811i	0.947890	+	0.903811i	−0.0475819	+	0.998867i	$0.515152\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$157$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$158$	−0.429730	−	0.0204706i	−0.429730	−	0.0204706i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.284630			0.284630
$163$	−1.28605	+	0.247866i	−1.28605	+	0.247866i	−0.786053	−	0.618159i	$-0.787879\pi$
$163$	−1.28605	+	0.247866i	−1.28605	+	0.247866i	−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$167$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$168$		0			0
$169$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$170$		0			0
$171$		0			0
$172$	−0.316405	+	1.64167i	−0.316405	+	1.64167i
$173$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$173$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$174$		0			0
$175$		0			0
$176$	0.446343	+	0.694523i	0.446343	+	0.694523i
$177$		0			0
$178$		0			0
$179$	0.827068	−	0.0789754i	0.827068	−	0.0789754i	0.327068	−	0.945001i	$-0.393939\pi$
$179$	0.827068	−	0.0789754i	0.827068	−	0.0789754i	0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$181$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$182$		0			0
$183$		0			0
$184$	−0.507075	−	0.203002i	−0.507075	−	0.203002i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.388835	+	0.407799i	0.388835	+	0.407799i	0.888835	−	0.458227i	$-0.151515\pi$
$191$	0.388835	+	0.407799i	0.388835	+	0.407799i	−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	0.975950	−	1.37053i	0.975950	−	1.37053i	0.0475819	−	0.998867i	$-0.484848\pi$
$193$	0.975950	−	1.37053i	0.975950	−	1.37053i	0.928368	−	0.371662i	$-0.121212\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$197$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$198$	−0.0292549	+	0.306371i	−0.0292549	+	0.306371i
$199$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$199$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$200$	−0.128772	−	0.530804i	−0.128772	−	0.530804i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.500000	+	0.866025i	0.500000	+	0.866025i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.239446	−	1.66538i	0.239446	−	1.66538i	−0.415415	−	0.909632i	$-0.636364\pi$
$211$	0.239446	−	1.66538i	0.239446	−	1.66538i	0.654861	−	0.755750i	$-0.272727\pi$
$212$	−1.13148	+	0.805726i	−1.13148	+	0.805726i
$213$		0			0
$214$	−0.155858	+	0.0378108i	−0.155858	+	0.0378108i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.425839	−	0.368991i	−0.425839	−	0.368991i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$223$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$224$		0			0
$225$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$226$	0.320093	−	0.407031i	0.320093	−	0.407031i
$227$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$227$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	−0.468468	−	0.540641i	−0.468468	−	0.540641i
$233$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.723734	−	0.690079i	$-0.242424\pi$
$233$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.981929	−	0.189251i	$-0.0606061\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.654861	+	0.755750i	$0.272727\pi$
$239$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$240$		0			0
$241$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$241$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$242$	−0.0472806	−	0.00911260i	−0.0472806	−	0.00911260i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$251$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$252$		0			0
$253$	−0.983568	+	0.449181i	−0.983568	+	0.449181i
$254$	0.118239	−	0.204797i	0.118239	−	0.204797i
$255$		0			0
$256$	0.0135432	−	0.284307i	0.0135432	−	0.284307i
$257$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$257$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.0623191	+	1.30824i	0.0623191	+	1.30824i
$262$		0			0
$263$	−1.36611	+	1.43273i	−1.36611	+	1.43273i	−0.580057	+	0.814576i	$0.696970\pi$
$263$	−1.36611	+	1.43273i	−1.36611	+	1.43273i	−0.786053	+	0.618159i	$0.787879\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.81720	−	0.0865641i	1.81720	−	0.0865641i
$269$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$269$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$270$		0			0
$271$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$271$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$272$		0			0
$273$		0			0
$274$	0.425839	−	0.368991i	0.425839	−	0.368991i
$275$	−0.936417	−	0.540641i	−0.936417	−	0.540641i
$276$		0			0
$277$	−0.959493	−	1.66189i	−0.959493	−	1.66189i	−0.723734	−	0.690079i	$-0.757576\pi$
$277$	−0.959493	−	1.66189i	−0.959493	−	1.66189i	−0.235759	−	0.971812i	$-0.575758\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$281$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$282$		0			0
$283$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$283$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$284$	1.37432	+	0.708510i	1.37432	+	0.708510i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.500000	−	0.577031i	0.500000	−	0.577031i
$289$	0.235759	−	0.971812i	0.235759	−	0.971812i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$293$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$	−0.141026	−	0.273552i	−0.141026	−	0.273552i
$297$		0			0
$298$	−0.448442	+	0.258908i	−0.448442	+	0.258908i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.313607	+	0.201543i	0.313607	+	0.201543i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$307$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$311$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$312$		0			0
$313$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$313$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$314$		0			0
$315$		0			0
$316$	1.37491	−	0.197682i	1.37491	−	0.197682i
$317$	1.11312	+	1.56316i	1.11312	+	1.56316i	0.786053	+	0.618159i	$0.212121\pi$
$317$	1.11312	+	1.56316i	1.11312	+	1.56316i	0.327068	+	0.945001i	$0.393939\pi$
$318$		0			0
$319$	−1.41457	−	0.0673845i	−1.41457	−	0.0673845i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.902379	+	0.173919i	−0.902379	+	0.173919i
$325$		0			0
$326$	−0.346082	+	0.138550i	−0.346082	+	0.138550i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.827068	+	0.0789754i	0.827068	+	0.0789754i	0.500000	−	0.866025i	$-0.333333\pi$
$331$	0.827068	+	0.0789754i	0.827068	+	0.0789754i	0.327068	+	0.945001i	$0.393939\pi$
$332$		0			0
$333$	−0.106636	+	0.553283i	−0.106636	+	0.553283i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$337$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$338$	−0.205996	+	0.196417i	−0.205996	+	0.196417i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$			0.993683i			0.993683i
$345$		0			0
$346$		0			0
$347$	−0.627639	+	1.81344i	−0.627639	+	1.81344i	−0.0475819	+	0.998867i	$0.515152\pi$
$347$	−0.627639	+	1.81344i	−0.627639	+	1.81344i	−0.580057	+	0.814576i	$0.696970\pi$
$348$		0			0
$349$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$349$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$350$		0			0
$351$		0			0
$352$	0.569716	+	0.597501i	0.569716	+	0.597501i
$353$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$353$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.226900	−	0.0666238i	0.226900	−	0.0666238i
$359$	0.172932	−	1.81103i	0.172932	−	1.81103i	−0.327068	−	0.945001i	$-0.606061\pi$
$359$	0.172932	−	1.81103i	0.172932	−	1.81103i	0.500000	−	0.866025i	$-0.333333\pi$
$360$		0			0
$361$	0.235759	+	0.971812i	0.235759	+	0.971812i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	0.749723	+	0.144497i	0.749723	+	0.144497i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.880786	−	0.627205i	0.880786	−	0.627205i	−0.0475819	−	0.998867i	$-0.515152\pi$
$373$	0.880786	−	0.627205i	0.880786	−	0.627205i	0.928368	+	0.371662i	$0.121212\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.14231	−	0.989821i	−1.14231	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$379$	−1.14231	−	0.989821i	−1.14231	−	0.989821i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	0.130641	+	0.0930289i	0.130641	+	0.0930289i
$383$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$383$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$384$		0			0
$385$		0			0
$386$	0.198939	−	0.435615i	0.198939	−	0.435615i
$387$	1.12459	−	1.43004i	1.12459	−	1.43004i
$388$		0			0
$389$	−1.02181	−	0.353653i	−1.02181	−	0.353653i	−0.235759	−	0.971812i	$-0.575758\pi$
$389$	−1.02181	−	0.353653i	−1.02181	−	0.353653i	−0.786053	+	0.618159i	$0.787879\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−0.210191	+	0.108361i	−0.210191	+	0.108361i
$395$		0			0
$396$	−0.0944555	−	0.989183i	−0.0944555	−	0.989183i
$397$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$397$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$398$		0			0
$399$		0			0
$400$	0.317178	+	0.694523i	0.317178	+	0.694523i
$401$	0.547582	+	0.132842i	0.547582	+	0.132842i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	0.547582	+	0.132842i	0.547582	+	0.132842i	0.0475819	+	0.998867i	$0.484848\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.584585	−	0.171650i	−0.584585	−	0.171650i
$408$		0			0
$409$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$409$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.186393	+	0.215109i	0.186393	+	0.215109i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$419$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$420$		0			0
$421$	−0.512546	+	1.74557i	−0.512546	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$421$	−0.512546	+	1.74557i	−0.512546	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$422$	−0.0227866	−	0.478349i	−0.0227866	−	0.478349i
$423$		0			0
$424$	−0.569716	+	0.597501i	−0.569716	+	0.597501i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.471022	−	0.215109i	0.471022	−	0.215109i
$429$		0			0
$430$		0			0
$431$	−1.12459	−	1.43004i	−1.12459	−	1.43004i	−0.888835	−	0.458227i	$-0.848485\pi$
$431$	−1.12459	−	1.43004i	−1.12459	−	1.43004i	−0.235759	−	0.971812i	$-0.575758\pi$
$432$		0			0
$433$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$433$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$434$		0			0
$435$		0			0
$436$	1.57553	+	0.909632i	1.57553	+	0.909632i
$437$		0			0
$438$		0			0
$439$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$439$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.56199	+	0.625325i	1.56199	+	0.625325i	0.981929	−	0.189251i	$-0.0606061\pi$
$443$	1.56199	+	0.625325i	1.56199	+	0.625325i	0.580057	+	0.814576i	$0.303030\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.25667	+	1.45027i	−1.25667	+	1.45027i	−0.415415	+	0.909632i	$0.636364\pi$
$449$	−1.25667	+	1.45027i	−1.25667	+	1.45027i	−0.841254	+	0.540641i	$0.818182\pi$
$450$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i
$451$		0			0
$452$	−0.766099	+	1.48602i	−0.766099	+	1.48602i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.692609	−	1.34347i	−0.692609	−	1.34347i	−0.928368	−	0.371662i	$-0.878788\pi$
$457$	−0.692609	−	1.34347i	−0.692609	−	1.34347i	0.235759	−	0.971812i	$-0.424242\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	0.239446	+	0.153882i	0.239446	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$463$	0.239446	+	0.153882i	0.239446	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$464$	0.786053	+	0.618159i	0.786053	+	0.618159i
$465$		0			0
$466$	0.429342	−	0.337639i	0.429342	−	0.337639i
$467$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$467$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.42368	+	1.35748i	1.42368	+	1.35748i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.49611	−	0.215109i	1.49611	−	0.215109i
$478$	0.316827	+	0.444922i	0.316827	+	0.444922i
$479$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$479$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.155465			0.155465
$485$		0			0
$486$		0			0
$487$	0.771316	−	0.308788i	0.771316	−	0.308788i	0.0475819	−	0.998867i	$-0.484848\pi$
$487$	0.771316	−	0.308788i	0.771316	−	0.308788i	0.723734	+	0.690079i	$0.242424\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$491$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.205996	−	0.196417i	0.205996	−	0.196417i	−0.580057	−	0.814576i	$-0.696970\pi$
$499$	0.205996	−	0.196417i	0.205996	−	0.196417i	0.786053	+	0.618159i	$0.212121\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$503$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$504$		0			0
$505$		0			0
$506$	−0.250698	+	0.178521i	−0.250698	+	0.178521i
$507$		0			0
$508$	−0.249723	+	0.721528i	−0.249723	+	0.721528i
$509$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$509$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$510$		0			0
$511$		0			0
$512$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$513$		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		−0.888835	−	0.458227i	$-0.848485\pi$
$521$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$522$	0.0878875	+	0.362277i	0.0878875	+	0.362277i
$523$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$523$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$524$		0			0
$525$		0			0
$526$	−0.304632	+	0.474017i	−0.304632	+	0.474017i
$527$		0			0
$528$		0			0
$529$	−0.327068	+	0.945001i	−0.327068	+	0.945001i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.05080	−	0.254922i	1.05080	−	0.254922i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.0930932	−	0.268975i	−0.0930932	−	0.268975i	0.888835	−	0.458227i	$-0.151515\pi$
$541$	−0.0930932	−	0.268975i	−0.0930932	−	0.268975i	−0.981929	+	0.189251i	$0.939394\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$547$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$548$	−1.12459	+	1.43004i	−1.12459	+	1.43004i
$549$		0			0
$550$	−0.290838	−	0.100660i	−0.290838	−	0.100660i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.357685	−	0.412791i	−0.357685	−	0.412791i
$555$		0			0
$556$		0			0
$557$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$557$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.418090	+	0.101428i	0.418090	+	0.101428i
$563$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$563$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.881761	+	0.258908i	0.881761	+	0.258908i
$569$	0.209419	−	0.523103i	0.209419	−	0.523103i	−0.786053	−	0.618159i	$-0.787879\pi$
$569$	0.209419	−	0.523103i	0.209419	−	0.523103i	0.995472	+	0.0950560i	$0.0303030\pi$
$570$		0			0
$571$	0.209419	+	0.523103i	0.209419	+	0.523103i	0.995472	−	0.0950560i	$-0.0303030\pi$
$571$	0.209419	+	0.523103i	0.209419	+	0.523103i	−0.786053	+	0.618159i	$0.787879\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$576$	−0.273100	+	0.473023i	−0.273100	+	0.473023i
$577$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$577$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$578$	0.0135432	−	0.284307i	0.0135432	−	0.284307i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.0777658	+	1.63251i	0.0777658	+	1.63251i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$587$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.265943	+	0.338174i	0.265943	+	0.338174i
$593$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$593$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$594$		0			0
$595$		0			0
$596$	1.26352	−	1.09485i	1.26352	−	1.09485i
$597$		0			0
$598$		0			0
$599$	−0.142315	−	0.246497i	−0.142315	−	0.246497i	0.786053	−	0.618159i	$-0.212121\pi$
$599$	−0.142315	−	0.246497i	−0.142315	−	0.246497i	−0.928368	+	0.371662i	$0.878788\pi$
$600$		0			0
$601$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$601$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$602$		0			0
$603$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$604$	−1.11740	−	0.447339i	−1.11740	−	0.447339i
$605$		0			0
$606$		0			0
$607$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$607$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.907125	−	1.75958i	0.907125	−	1.75958i	0.327068	−	0.945001i	$-0.393939\pi$
$613$	0.907125	−	1.75958i	0.907125	−	1.75958i	0.580057	−	0.814576i	$-0.303030\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.95949	+	0.281733i	1.95949	+	0.281733i	1.00000			$0$
$617$	1.95949	+	0.281733i	1.95949	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$618$		0			0
$619$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$619$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.512546	−	1.74557i	−0.512546	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$631$	−0.512546	−	1.74557i	−0.512546	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$632$	0.780175	−	0.270021i	0.780175	−	0.270021i
$633$		0			0
$634$	0.395304	+	0.376921i	0.395304	+	0.376921i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.398983	+	0.0573652i	−0.398983	+	0.0573652i
$639$	−0.975950	−	1.37053i	−0.975950	−	1.37053i
$640$		0			0
$641$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$641$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$647$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$648$	−0.507075	+	0.203002i	−0.507075	+	0.203002i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.01255	−	0.650724i	1.01255	−	0.650724i
$653$	−0.283341	−	0.0270558i	−0.283341	−	0.0270558i	−0.0475819	−	0.998867i	$-0.515152\pi$
$653$	−0.283341	−	0.0270558i	−0.283341	−	0.0270558i	−0.235759	+	0.971812i	$0.575758\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.817178	+	1.27155i	0.817178	+	1.27155i	0.959493	+	0.281733i	$0.0909091\pi$
$659$	0.817178	+	1.27155i	0.817178	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$660$		0			0
$661$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$661$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$662$	0.235408	−	0.0224787i	0.235408	−	0.0224787i
$663$		0			0
$664$		0			0
$665$		0			0
$666$			0.160379i			0.160379i
$667$	−0.947890	+	0.903811i	−0.947890	+	0.903811i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$673$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$674$	−0.212382	−	0.222740i	−0.212382	−	0.222740i
$675$		0			0
$676$	0.533064	−	0.748584i	0.533064	−	0.748584i
$677$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$677$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.452418	−	1.86489i	−0.452418	−	1.86489i	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−0.452418	−	1.86489i	−0.452418	−	1.86489i	0.0475819	−	0.998867i	$-0.484848\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.262879	−	1.36394i	−0.262879	−	1.36394i
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$	−0.0777324	+	0.540641i	−0.0777324	+	0.540641i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$701$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$702$		0			0
$703$		0			0
$704$	−0.481086	−	0.342580i	−0.481086	−	0.342580i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.668404	+	0.849945i	−0.668404	+	0.849945i	−0.995472	−	0.0950560i	$-0.969697\pi$
$709$	−0.668404	+	0.849945i	−0.668404	+	0.849945i	0.327068	+	0.945001i	$0.393939\pi$
$710$		0			0
$711$	−1.42837	−	0.494363i	−1.42837	−	0.494363i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.678645	+	0.349866i	−0.678645	+	0.349866i
$717$		0			0
$718$	−0.0492216	−	0.515472i	−0.0492216	−	0.515472i
$719$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$719$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$720$		0			0
$721$		0			0
$722$	0.118239	+	0.258908i	0.118239	+	0.258908i
$723$		0			0
$724$		0			0
$725$	−1.28605	−	0.247866i	−1.28605	−	0.247866i
$726$		0			0
$727$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$727$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$728$		0			0
$729$	0.959493	+	0.281733i	0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$733$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$734$		0			0
$735$		0			0
$736$	0.763521			0.763521
$737$	1.07028	−	1.85377i	1.07028	−	1.85377i
$738$		0			0
$739$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i	0.981929	+	0.189251i	$0.0606061\pi$
$739$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i	−0.995472	+	0.0950560i	$0.969697\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$743$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$744$		0			0
$745$		0			0
$746$	0.212382	−	0.222740i	0.212382	−	0.222740i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.50979	+	0.0719200i	−1.50979	+	0.0719200i	−0.786053	−	0.618159i	$-0.787879\pi$
$751$	−1.50979	+	0.0719200i	−1.50979	+	0.0719200i	−0.723734	+	0.690079i	$0.757576\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$757$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$758$	−0.372579	−	0.215109i	−0.372579	−	0.215109i
$759$		0			0
$760$		0			0
$761$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$761$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$762$		0			0
$763$		0			0
$764$	−0.471022	−	0.215109i	−0.471022	−	0.215109i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$769$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$770$		0			0
$771$		0			0
$772$	−0.364531	+	1.50262i	−0.364531	+	1.50262i
$773$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$773$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$774$	0.237277	−	0.460254i	0.237277	−	0.460254i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.304632	−	0.0437995i	−0.304632	−	0.0437995i
$779$		0			0
$780$		0			0
$781$	1.57553	−	0.909632i	1.57553	−	0.909632i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$787$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$788$	0.600168	−	0.471977i	0.600168	−	0.471977i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.166390	−	0.566673i	−0.166390	−	0.566673i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$797$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$798$		0			0
$799$		0			0
$800$	0.442886	+	0.621946i	0.442886	+	0.621946i
$801$		0			0
$802$	0.160197	+	0.00763113i	0.160197	+	0.00763113i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.21590	+	0.486774i	−1.21590	+	0.486774i	−0.888835	−	0.458227i	$-0.848485\pi$
$809$	−1.21590	+	0.486774i	−1.21590	+	0.486774i	−0.327068	+	0.945001i	$0.606061\pi$
$810$		0			0
$811$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$811$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$812$		0			0
$813$		0			0
$814$	−0.172630	−	0.0164841i	−0.172630	−	0.0164841i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.21769	+	1.16106i	−1.21769	+	1.16106i	−0.235759	+	0.971812i	$0.575758\pi$
$821$	−1.21769	+	1.16106i	−1.21769	+	1.16106i	−0.981929	+	0.189251i	$0.939394\pi$
$822$		0			0
$823$	1.30379	−	0.124497i	1.30379	−	0.124497i	0.580057	−	0.814576i	$-0.303030\pi$
$823$	1.30379	−	0.124497i	1.30379	−	0.124497i	0.723734	+	0.690079i	$0.242424\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	0.563465i		−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$827$		−	0.563465i		−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$828$	−0.722372	−	0.568079i	−0.722372	−	0.568079i
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$839$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$840$		0			0
$841$	−0.686393	+	0.201543i	−0.686393	+	0.201543i
$842$	−0.0492216	+	0.515472i	−0.0492216	+	0.515472i
$843$		0			0
$844$	0.364531	+	1.50262i	0.364531	+	1.50262i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	0.623933	−	0.970858i	0.623933	−	0.970858i
$849$		0			0
$850$		0			0
$851$	−0.487975	+	0.281733i	−0.487975	+	0.281733i
$852$		0			0
$853$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$853$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$854$		0			0
$855$		0			0
$856$	0.250698	−	0.178521i	0.250698	−	0.178521i
$857$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$857$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$858$		0			0
$859$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$859$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$860$		0			0
$861$		0			0
$862$	−0.391340	−	0.339098i	−0.391340	−	0.339098i
$863$	−0.550294	−	1.58997i	−0.550294	−	1.58997i	−0.786053	−	0.618159i	$-0.787879\pi$
$863$	−0.550294	−	1.58997i	−0.550294	−	1.58997i	0.235759	−	0.971812i	$-0.424242\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.678936	−	1.48666i	0.678936	−	1.48666i
$870$		0			0
$871$		0			0
$872$	1.02181	+	0.353653i	1.02181	+	0.353653i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.49547	+	0.770969i	−1.49547	+	0.770969i	−0.995472	−	0.0950560i	$-0.969697\pi$
$877$	−1.49547	+	0.770969i	−1.49547	+	0.770969i	−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$881$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$882$		0			0
$883$	0.544078	+	1.19136i	0.544078	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$883$	0.544078	+	1.19136i	0.544078	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$884$		0			0
$885$		0			0
$886$	0.470237	+	0.0906308i	0.470237	+	0.0906308i
$887$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$887$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.401872	+	1.00383i	−0.401872	+	1.00383i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.273100	+	0.473023i	−0.273100	+	0.473023i
$899$		0			0
$900$	0.0437271	−	0.917945i	0.0437271	−	0.917945i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.279953	+	0.953431i	−0.279953	+	0.953431i
$905$		0			0
$906$		0			0
$907$	1.36611	−	1.43273i	1.36611	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$907$	1.36611	−	1.43273i	1.36611	−	1.43273i	0.786053	−	0.618159i	$-0.212121\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$911$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$912$		0			0
$913$		0			0
$914$	−0.265943	−	0.338174i	−0.265943	−	0.338174i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.487975	+	0.281733i	0.487975	+	0.281733i	0.723734	−	0.690079i	$-0.242424\pi$
$919$	0.487975	+	0.281733i	0.487975	+	0.281733i	−0.235759	+	0.971812i	$0.575758\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$926$	0.0752108	+	0.0301099i	0.0752108	+	0.0301099i
$927$		0			0
$928$	0.888835	+	0.458227i	0.888835	+	0.458227i
$929$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$929$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$930$		0			0
$931$		0			0
$932$	−1.15486	+	1.33278i	−1.15486	+	1.33278i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$937$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$941$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.471022	+	0.302708i	0.471022	+	0.302708i
$947$	−1.50842	−	1.18624i	−1.50842	−	1.18624i	−0.928368	−	0.371662i	$-0.878788\pi$
$947$	−1.50842	−	1.18624i	−1.50842	−	1.18624i	−0.580057	−	0.814576i	$-0.696970\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.158746	−	0.540641i	−0.158746	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$953$	−0.158746	−	0.540641i	−0.158746	−	0.540641i	−1.00000			$\pi$
$954$	0.406556	−	0.140710i	0.406556	−	0.140710i
$955$		0			0
$956$	−1.27632	−	1.21697i	−1.27632	−	1.21697i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.580057	−	0.814576i	−0.580057	−	0.814576i
$962$		0			0
$963$	−0.562827	−	0.0268107i	−0.562827	−	0.0268107i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$967$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$968$	0.0907309	−	0.0174869i	0.0907309	−	0.0174869i
$969$		0			0
$970$		0			0
$971$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$971$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$972$		0			0
$973$		0			0
$974$	0.198939	−	0.127850i	0.198939	−	0.127850i
$975$		0			0
$976$		0			0
$977$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$977$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$982$	−0.0586326	+	0.0559061i	−0.0586326	+	0.0559061i
$983$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$983$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.81720	−	0.0865641i	1.81720	−	0.0865641i
$990$		0			0
$991$	0.0930932	−	0.268975i	0.0930932	−	0.268975i	−0.888835	−	0.458227i	$-0.848485\pi$
$991$	0.0930932	−	0.268975i	0.0930932	−	0.268975i	0.981929	+	0.189251i	$0.0606061\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$997$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$998$	0.0469928	−	0.0659921i	0.0469928	−	0.0659921i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.1.x.a.79.1		20
7.2	even	3		1127.1.o.a.148.1	yes	10
7.3	odd	6	inner	1127.1.x.a.263.1		20
7.4	even	3	inner	1127.1.x.a.263.1		20
7.5	odd	6		1127.1.o.a.148.1	yes	10
7.6	odd	2	CM	1127.1.x.a.79.1		20
23.7	odd	22	inner	1127.1.x.a.30.1		20
161.30	odd	66		1127.1.o.a.99.1	✓	10
161.53	odd	66	inner	1127.1.x.a.214.1		20
161.76	even	22	inner	1127.1.x.a.30.1		20
161.122	even	66	inner	1127.1.x.a.214.1		20
161.145	even	66		1127.1.o.a.99.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1127.1.o.a.99.1	✓	10	161.30	odd	66
1127.1.o.a.99.1	✓	10	161.145	even	66
1127.1.o.a.148.1	yes	10	7.2	even	3
1127.1.o.a.148.1	yes	10	7.5	odd	6
1127.1.x.a.30.1		20	23.7	odd	22	inner
1127.1.x.a.30.1		20	161.76	even	22	inner
1127.1.x.a.79.1		20	1.1	even	1	trivial
1127.1.x.a.79.1		20	7.6	odd	2	CM
1127.1.x.a.214.1		20	161.53	odd	66	inner
1127.1.x.a.214.1		20	161.122	even	66	inner
1127.1.x.a.263.1		20	7.3	odd	6	inner
1127.1.x.a.263.1		20	7.4	even	3	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:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1127.x (of order \(66\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.279486 - 0.0538665i) q^{2} +(-0.853157 + 0.341553i) q^{4} +(-0.459493 + 0.295298i) q^{8} +(0.995472 + 0.0950560i) q^{9} +O(q^{10})\)	\(q+(0.279486 - 0.0538665i) q^{2} +(-0.853157 + 0.341553i) q^{4} +(-0.459493 + 0.295298i) q^{8} +(0.995472 + 0.0950560i) q^{9} +(-0.204634 + 1.06174i) q^{11} +(0.552586 - 0.526890i) q^{16} +(0.283341 - 0.0270558i) q^{18} +0.307765i q^{22} +(0.580057 + 0.814576i) q^{23} +(-0.327068 + 0.945001i) q^{25} +(0.186393 + 1.29639i) q^{29} +(0.442886 - 0.621946i) q^{32} +(-0.881761 + 0.258908i) q^{36} +(-0.0535608 + 0.560914i) q^{37} +(0.983568 - 1.53046i) q^{43} +(-0.188056 - 0.975725i) q^{44} +(0.205996 + 0.196417i) q^{46} +(-0.0405070 + 0.281733i) q^{50} +(1.46889 - 0.356349i) q^{53} +(0.121926 + 0.352283i) q^{58} +(-0.226900 + 0.496841i) q^{64} +(-1.87076 - 0.647478i) q^{67} +(-1.10181 - 1.27155i) q^{71} +(-0.485482 + 0.250283i) q^{72} +(0.0152450 + 0.159653i) q^{74} +(-1.46889 - 0.356349i) q^{79} +(0.981929 + 0.189251i) q^{81} +(0.192453 - 0.480724i) q^{86} +(-0.219503 - 0.548291i) q^{88} +(-0.773100 - 0.496841i) q^{92} +(-0.304632 + 1.03748i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 3 q^{4} + 8 q^{8} - q^{9}+O(q^{10}) \) 20 * q - 2 * q^2 + 3 * q^4 + 8 * q^8 - q^9	\( 20 q - 2 q^{2} + 3 q^{4} + 8 q^{8} - q^{9} - 6 q^{16} + 2 q^{18} + q^{23} + q^{25} - 4 q^{29} + 5 q^{32} - 16 q^{36} + 11 q^{44} - 2 q^{46} - 18 q^{50} + 7 q^{58} - 14 q^{64} - 4 q^{71} + 4 q^{72} - 11 q^{74} + q^{81} - 11 q^{88} - 6 q^{92}+O(q^{100}) \) 20 * q - 2 * q^2 + 3 * q^4 + 8 * q^8 - q^9 - 6 * q^16 + 2 * q^18 + q^23 + q^25 - 4 * q^29 + 5 * q^32 - 16 * q^36 + 11 * q^44 - 2 * q^46 - 18 * q^50 + 7 * q^58 - 14 * q^64 - 4 * q^71 + 4 * q^72 - 11 * q^74 + q^81 - 11 * q^88 - 6 * q^92

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