LMFDB - Embedded newform 1127.1.x.a.802.1

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			802.1
Root			$0.235759 - 0.971812i$ of defining polynomial
Character	$\chi$	$=$	1127.802
Dual form			1127.1.x.a.912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50842 + 1.18624i) q^{2} +(0.632425 - 2.60689i) q^{4} +(1.34125 + 2.93694i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})$	$q+(-1.50842 + 1.18624i) q^{2} +(0.632425 - 2.60689i) q^{4} +(1.34125 + 2.93694i) q^{8} +(0.327068 - 0.945001i) q^{9} +(-1.12459 + 1.43004i) q^{11} +(-3.12278 - 1.60990i) q^{16} +(0.627639 + 1.81344i) q^{18} -3.49114i q^{22} +(0.981929 + 0.189251i) q^{23} +(0.928368 + 0.371662i) q^{25} +(0.273100 + 0.0801894i) q^{29} +(3.44985 - 0.664903i) q^{32} +(-2.25667 - 1.45027i) q^{36} +(1.02181 + 0.353653i) q^{37} +(1.37491 + 0.627899i) q^{43} +(3.01673 + 3.83609i) q^{44} +(-1.70566 + 0.879330i) q^{46} +(-1.84125 + 0.540641i) q^{50} +(1.97740 - 0.0941952i) q^{53} +(-0.507075 + 0.203002i) q^{58} +(-2.11435 + 2.44009i) q^{64} +(-0.209419 + 0.523103i) q^{67} +(-0.118239 - 0.822373i) q^{71} +(3.21409 - 0.306908i) q^{72} +(-1.96084 + 0.678654i) q^{74} +(-1.97740 - 0.0941952i) q^{79} +(-0.786053 - 0.618159i) q^{81} +(-2.81878 + 0.683830i) q^{86} +(-5.70830 - 1.38482i) q^{88} +(1.11435 - 2.44009i) q^{92} +(0.983568 + 1.53046i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times$.

$n$	$346$	$442$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{22}\right)$

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 $

$2$	−1.50842	+	1.18624i	−1.50842	+	1.18624i	−0.580057	+	0.814576i	$0.696970\pi$
$2$	−1.50842	+	1.18624i	−1.50842	+	1.18624i	−0.928368	+	0.371662i	$0.878788\pi$
$3$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$3$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$4$	0.632425	−	2.60689i	0.632425	−	2.60689i
$5$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$5$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.34125	+	2.93694i	1.34125	+	2.93694i
$9$	0.327068	−	0.945001i	0.327068	−	0.945001i
$10$		0			0
$11$	−1.12459	+	1.43004i	−1.12459	+	1.43004i	−0.235759	+	0.971812i	$0.575758\pi$
$11$	−1.12459	+	1.43004i	−1.12459	+	1.43004i	−0.888835	+	0.458227i	$0.848485\pi$
$12$		0			0
$13$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$13$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$14$		0			0
$15$		0			0
$16$	−3.12278	−	1.60990i	−3.12278	−	1.60990i
$17$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$17$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$18$	0.627639	+	1.81344i	0.627639	+	1.81344i
$19$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$19$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$20$		0			0
$21$		0			0
$22$		−	3.49114i		−	3.49114i
$23$	0.981929	+	0.189251i	0.981929	+	0.189251i
$23$	0.981929	+	0.189251i	0.981929	+	0.189251i
$24$		0			0
$25$	0.928368	+	0.371662i	0.928368	+	0.371662i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$29$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$30$		0			0
$31$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$31$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$32$	3.44985	−	0.664903i	3.44985	−	0.664903i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.25667	−	1.45027i	−2.25667	−	1.45027i
$37$	1.02181	+	0.353653i	1.02181	+	0.353653i	0.786053	−	0.618159i	$-0.212121\pi$
$37$	1.02181	+	0.353653i	1.02181	+	0.353653i	0.235759	+	0.971812i	$0.424242\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$41$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$42$		0			0
$43$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$43$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$44$	3.01673	+	3.83609i	3.01673	+	3.83609i
$45$		0			0
$46$	−1.70566	+	0.879330i	−1.70566	+	0.879330i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$	−1.84125	+	0.540641i	−1.84125	+	0.540641i
$51$		0			0
$52$		0			0
$53$	1.97740	−	0.0941952i	1.97740	−	0.0941952i	0.981929	−	0.189251i	$-0.0606061\pi$
$53$	1.97740	−	0.0941952i	1.97740	−	0.0941952i	0.995472	+	0.0950560i	$0.0303030\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.507075	+	0.203002i	−0.507075	+	0.203002i
$59$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$59$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$60$		0			0
$61$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$61$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$62$		0			0
$63$		0			0
$64$	−2.11435	+	2.44009i	−2.11435	+	2.44009i
$65$		0			0
$66$		0			0
$67$	−0.209419	+	0.523103i	−0.209419	+	0.523103i	−0.995472	−	0.0950560i	$-0.969697\pi$
$67$	−0.209419	+	0.523103i	−0.209419	+	0.523103i	0.786053	+	0.618159i	$0.212121\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$71$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$72$	3.21409	−	0.306908i	3.21409	−	0.306908i
$73$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$73$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$74$	−1.96084	+	0.678654i	−1.96084	+	0.678654i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.97740	−	0.0941952i	−1.97740	−	0.0941952i	−0.981929	−	0.189251i	$-0.939394\pi$
$79$	−1.97740	−	0.0941952i	−1.97740	−	0.0941952i	−0.995472	+	0.0950560i	$0.969697\pi$
$80$		0			0
$81$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$82$		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
$86$	−2.81878	+	0.683830i	−2.81878	+	0.683830i
$87$		0			0
$88$	−5.70830	−	1.38482i	−5.70830	−	1.38482i
$89$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$89$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$90$		0			0
$91$		0			0
$92$	1.11435	−	2.44009i	1.11435	−	2.44009i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$97$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$98$		0			0
$99$	0.983568	+	1.53046i	0.983568	+	1.53046i
$100$	1.55601	−	2.18511i	1.55601	−	2.18511i
$101$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$101$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$102$		0			0
$103$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$103$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$104$		0			0
$105$		0			0
$106$	−2.87102	+	2.48775i	−2.87102	+	2.48775i
$107$	0.880786	+	0.627205i	0.880786	+	0.627205i	0.928368	−	0.371662i	$-0.121212\pi$
$107$	0.880786	+	0.627205i	0.880786	+	0.627205i	−0.0475819	+	0.998867i	$0.515152\pi$
$108$		0			0
$109$	0.388835	+	0.407799i	0.388835	+	0.407799i	0.888835	−	0.458227i	$-0.151515\pi$
$109$	0.388835	+	0.407799i	0.388835	+	0.407799i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.49611	+	0.215109i	−1.49611	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$113$	−1.49611	+	0.215109i	−1.49611	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$114$		0			0
$115$		0			0
$116$	0.381761	−	0.661229i	0.381761	−	0.661229i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.544537	−	2.24461i	−0.544537	−	2.24461i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	0.654861	+	0.755750i	$0.272727\pi$
$127$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$128$	0.127639	−	2.67947i	0.127639	−	2.67947i
$129$		0			0
$130$		0			0
$131$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$131$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$132$		0			0
$133$		0			0
$134$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$135$		0			0
$136$		0			0
$137$	−0.487975	−	0.281733i	−0.487975	−	0.281733i	0.235759	−	0.971812i	$-0.424242\pi$
$137$	−0.487975	−	0.281733i	−0.487975	−	0.281733i	−0.723734	+	0.690079i	$0.757576\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	1.15389	+	1.10023i	1.15389	+	1.10023i
$143$		0			0
$144$	−2.54272	+	2.42448i	−2.54272	+	2.42448i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.56815	−	2.44009i	1.56815	−	2.44009i
$149$	0.561767	+	1.40323i	0.561767	+	1.40323i	0.888835	+	0.458227i	$0.151515\pi$
$149$	0.561767	+	1.40323i	0.561767	+	1.40323i	−0.327068	+	0.945001i	$0.606061\pi$
$150$		0			0
$151$	−0.252989	+	0.130425i	−0.252989	+	0.130425i	−0.580057	−	0.814576i	$-0.696970\pi$
$151$	−0.252989	+	0.130425i	−0.252989	+	0.130425i	0.327068	+	0.945001i	$0.393939\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$157$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$158$	3.09450	−	2.20358i	3.09450	−	2.20358i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.91899			1.91899
$163$	0.223734	−	0.175946i	0.223734	−	0.175946i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	0.223734	−	0.175946i	0.223734	−	0.175946i	0.723734	+	0.690079i	$0.242424\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$167$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$168$		0			0
$169$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$170$		0			0
$171$		0			0
$172$	2.50639	−	3.18714i	2.50639	−	3.18714i
$173$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$173$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$174$		0			0
$175$		0			0
$176$	5.81408	−	2.65520i	5.81408	−	2.65520i
$177$		0			0
$178$		0			0
$179$	−0.428368	−	1.23769i	−0.428368	−	1.23769i	−0.928368	−	0.371662i	$-0.878788\pi$
$179$	−0.428368	−	1.23769i	−0.428368	−	1.23769i	0.500000	−	0.866025i	$-0.333333\pi$
$180$		0			0
$181$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$181$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$182$		0			0
$183$		0			0
$184$	0.761197	+	3.13770i	0.761197	+	3.13770i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.495472	−	0.961081i	0.495472	−	0.961081i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	0.495472	−	0.961081i	0.495472	−	0.961081i	0.995472	−	0.0950560i	$-0.0303030\pi$
$192$		0			0
$193$	0.815816	−	0.157236i	0.815816	−	0.157236i	0.235759	−	0.971812i	$-0.424242\pi$
$193$	0.815816	−	0.157236i	0.815816	−	0.157236i	0.580057	+	0.814576i	$0.303030\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$197$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$198$	−3.29913	−	1.14184i	−3.29913	−	1.14184i
$199$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$199$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$200$	0.153628	+	3.22505i	0.153628	+	3.22505i
$201$		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
$211$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$211$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$212$	1.00500	−	5.21444i	1.00500	−	5.21444i
$213$		0			0
$214$	−2.07261	+	0.0987308i	−2.07261	+	0.0987308i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.07028	−	0.153882i	−1.07028	−	0.153882i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$223$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$224$		0			0
$225$	0.654861	−	0.755750i	0.654861	−	0.755750i
$226$	2.00161	−	2.09922i	2.00161	−	2.09922i
$227$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$227$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	0.130785	+	0.909632i	0.130785	+	0.909632i
$233$	−1.67489	+	0.159932i	−1.67489	+	0.159932i	−0.888835	−	0.458227i	$-0.848485\pi$
$233$	−1.67489	+	0.159932i	−1.67489	+	0.159932i	−0.786053	+	0.618159i	$0.787879\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.10181	+	1.27155i	1.10181	+	1.27155i	0.959493	+	0.281733i	$0.0909091\pi$
$239$	1.10181	+	1.27155i	1.10181	+	1.27155i	0.142315	+	0.989821i	$0.454545\pi$
$240$		0			0
$241$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$241$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$242$	3.48404	+	2.73988i	3.48404	+	2.73988i
$243$		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		−0.142315	−	0.989821i	$-0.545455\pi$
$251$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$252$		0			0
$253$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$254$	−1.25667	−	2.17661i	−1.25667	−	2.17661i
$255$		0			0
$256$	1.11312	+	1.56316i	1.11312	+	1.56316i
$257$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$257$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.165101	−	0.231852i	0.165101	−	0.231852i
$262$		0			0
$263$	−0.258195	−	0.500828i	−0.258195	−	0.500828i	0.723734	−	0.690079i	$-0.242424\pi$
$263$	−0.258195	−	0.500828i	−0.258195	−	0.500828i	−0.981929	+	0.189251i	$0.939394\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.23123	+	0.876756i	1.23123	+	0.876756i
$269$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$269$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$270$		0			0
$271$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$271$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$272$		0			0
$273$		0			0
$274$	1.07028	−	0.153882i	1.07028	−	0.153882i
$275$	−1.57553	+	0.909632i	−1.57553	+	0.909632i
$276$		0			0
$277$	0.841254	−	1.45709i	0.841254	−	1.45709i	−0.0475819	−	0.998867i	$-0.515152\pi$
$277$	0.841254	−	1.45709i	0.841254	−	1.45709i	0.888835	−	0.458227i	$-0.151515\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$281$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$282$		0			0
$283$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$283$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$284$	−2.21862	−	0.211852i	−2.21862	−	0.211852i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.500000	−	3.47758i	0.500000	−	3.47758i
$289$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$290$		0			0
$291$		0			0
$292$		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
$295$		0			0
$296$	0.331854	+	3.47533i	0.331854	+	3.47533i
$297$		0			0
$298$	−2.51195	−	1.45027i	−2.51195	−	1.45027i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.226900	−	0.496841i	0.226900	−	0.496841i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$307$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$311$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$312$		0			0
$313$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$313$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$314$		0			0
$315$		0			0
$316$	−1.49611	+	5.09530i	−1.49611	+	5.09530i
$317$	−1.65210	−	0.318417i	−1.65210	−	0.318417i	−0.723734	−	0.690079i	$-0.757576\pi$
$317$	−1.65210	−	0.318417i	−1.65210	−	0.318417i	−0.928368	+	0.371662i	$0.878788\pi$
$318$		0			0
$319$	−0.421801	+	0.300363i	−0.421801	+	0.300363i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.10859	+	1.65822i	−2.10859	+	1.65822i
$325$		0			0
$326$	−0.128772	+	0.530804i	−0.128772	+	0.530804i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.428368	+	1.23769i	−0.428368	+	1.23769i	0.500000	+	0.866025i	$0.333333\pi$
$331$	−0.428368	+	1.23769i	−0.428368	+	1.23769i	−0.928368	+	0.371662i	$0.878788\pi$
$332$		0			0
$333$	0.668404	−	0.849945i	0.668404	−	0.849945i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$337$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−1.00000			$\pi$
$338$	1.70566	+	0.879330i	1.70566	+	0.879330i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$			4.88019i			4.88019i
$345$		0			0
$346$		0			0
$347$	−1.56199	−	0.625325i	−1.56199	−	0.625325i	−0.580057	−	0.814576i	$-0.696970\pi$
$347$	−1.56199	−	0.625325i	−1.56199	−	0.625325i	−0.981929	+	0.189251i	$0.939394\pi$
$348$		0			0
$349$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$349$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$350$		0			0
$351$		0			0
$352$	−2.92884	+	5.68116i	−2.92884	+	5.68116i
$353$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$353$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	2.11435	+	1.35881i	2.11435	+	1.35881i
$359$	1.42837	+	0.494363i	1.42837	+	0.494363i	0.928368	−	0.371662i	$-0.121212\pi$
$359$	1.42837	+	0.494363i	1.42837	+	0.494363i	0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−2.76167	−	2.17180i	−2.76167	−	2.17180i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.344298	+	1.78639i	−0.344298	+	1.78639i	0.235759	+	0.971812i	$0.424242\pi$
$373$	−0.344298	+	1.78639i	−0.344298	+	1.78639i	−0.580057	+	0.814576i	$0.696970\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.95949	−	0.281733i	−1.95949	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$379$	−1.95949	−	0.281733i	−1.95949	−	0.281733i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	0.392690	+	2.03747i	0.392690	+	2.03747i
$383$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$383$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$384$		0			0
$385$		0			0
$386$	−1.04408	+	1.20493i	−1.04408	+	1.20493i
$387$	1.04305	−	1.09392i	1.04305	−	1.09392i
$388$		0			0
$389$	0.676152	−	1.68895i	0.676152	−	1.68895i	−0.0475819	−	0.998867i	$-0.515152\pi$
$389$	0.676152	−	1.68895i	0.676152	−	1.68895i	0.723734	−	0.690079i	$-0.242424\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	2.50196	−	0.238908i	2.50196	−	0.238908i
$395$		0			0
$396$	4.61178	−	1.59615i	4.61178	−	1.59615i
$397$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$397$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$398$		0			0
$399$		0			0
$400$	−2.30075	−	2.65520i	−2.30075	−	2.65520i
$401$	1.08006	+	0.0514495i	1.08006	+	0.0514495i	0.580057	−	0.814576i	$-0.303030\pi$
$401$	1.08006	+	0.0514495i	1.08006	+	0.0514495i	0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.65486	+	1.06351i	−1.65486	+	1.06351i
$408$		0			0
$409$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$409$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.273100	+	1.89945i	0.273100	+	1.89945i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$419$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$420$		0			0
$421$	0.817178	+	1.27155i	0.817178	+	1.27155i	0.959493	+	0.281733i	$0.0909091\pi$
$421$	0.817178	+	1.27155i	0.817178	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$422$	−0.924814	+	1.29872i	−0.924814	+	1.29872i
$423$		0			0
$424$	2.92884	+	5.68116i	2.92884	+	5.68116i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	2.19209	−	1.89945i	2.19209	−	1.89945i
$429$		0			0
$430$		0			0
$431$	−1.04305	−	1.09392i	−1.04305	−	1.09392i	−0.995472	−	0.0950560i	$-0.969697\pi$
$431$	−1.04305	−	1.09392i	−1.04305	−	1.09392i	−0.0475819	−	0.998867i	$-0.515152\pi$
$432$		0			0
$433$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$433$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$434$		0			0
$435$		0			0
$436$	1.30900	−	0.755750i	1.30900	−	0.755750i
$437$		0			0
$438$		0			0
$439$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$439$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.195876	+	0.807410i	0.195876	+	0.807410i	0.981929	+	0.189251i	$0.0606061\pi$
$443$	0.195876	+	0.807410i	0.195876	+	0.807410i	−0.786053	+	0.618159i	$0.787879\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.239446	−	1.66538i	0.239446	−	1.66538i	−0.415415	−	0.909632i	$-0.636364\pi$
$449$	0.239446	−	1.66538i	0.239446	−	1.66538i	0.654861	−	0.755750i	$-0.272727\pi$
$450$	−0.0913090	+	1.91681i	−0.0913090	+	1.91681i
$451$		0			0
$452$	−0.385415	+	4.03625i	−0.385415	+	4.03625i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.188177	−	1.97068i	−0.188177	−	1.97068i	−0.235759	−	0.971812i	$-0.575758\pi$
$457$	−0.188177	−	1.97068i	−0.188177	−	1.97068i	0.0475819	−	0.998867i	$-0.484848\pi$
$458$		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
$463$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$463$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$464$	−0.723734	−	0.690079i	−0.723734	−	0.690079i
$465$		0			0
$466$	2.33673	−	2.22806i	2.33673	−	2.22806i
$467$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$467$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.44413	+	1.26004i	−2.44413	+	1.26004i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.557730	−	1.89945i	0.557730	−	1.89945i
$478$	−3.17036	−	0.611037i	−3.17036	−	0.611037i
$479$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$479$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−6.19584			−6.19584
$485$		0			0
$486$		0			0
$487$	−0.308779	+	1.27280i	−0.308779	+	1.27280i	0.580057	+	0.814576i	$0.303030\pi$
$487$	−0.308779	+	1.27280i	−0.308779	+	1.27280i	−0.888835	+	0.458227i	$0.848485\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$491$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.70566	−	0.879330i	−1.70566	−	0.879330i	−0.981929	−	0.189251i	$-0.939394\pi$
$499$	−1.70566	−	0.879330i	−1.70566	−	0.879330i	−0.723734	−	0.690079i	$-0.757576\pi$
$500$		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
$506$	0.660703	−	3.42805i	0.660703	−	3.42805i
$507$		0			0
$508$	3.26167	+	1.30578i	3.26167	+	1.30578i
$509$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$509$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$510$		0			0
$511$		0			0
$512$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$513$		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		−0.995472	−	0.0950560i	$-0.969697\pi$
$521$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$522$	0.0259893	+	0.545582i	0.0259893	+	0.545582i
$523$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$523$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$524$		0			0
$525$		0			0
$526$	0.983568	+	0.449181i	0.983568	+	0.449181i
$527$		0			0
$528$		0			0
$529$	0.928368	+	0.371662i	0.928368	+	0.371662i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.81720	+	0.0865641i	−1.81720	+	0.0865641i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.78153	−	0.713215i	1.78153	−	0.713215i	0.786053	−	0.618159i	$-0.212121\pi$
$541$	1.78153	−	0.713215i	1.78153	−	0.713215i	0.995472	−	0.0950560i	$-0.0303030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$547$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$548$	−1.04305	+	1.09392i	−1.04305	+	1.09392i
$549$		0			0
$550$	1.29753	−	3.24106i	1.29753	−	3.24106i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.459493	+	3.19584i	0.459493	+	3.19584i
$555$		0			0
$556$		0			0
$557$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$557$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	3.79460	+	0.180759i	3.79460	+	0.180759i
$563$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$563$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	2.25667	−	1.45027i	2.25667	−	1.45027i
$569$	1.05080	−	0.254922i	1.05080	−	0.254922i	0.327068	−	0.945001i	$-0.393939\pi$
$569$	1.05080	−	0.254922i	1.05080	−	0.254922i	0.723734	+	0.690079i	$0.242424\pi$
$570$		0			0
$571$	1.05080	+	0.254922i	1.05080	+	0.254922i	0.723734	−	0.690079i	$-0.242424\pi$
$571$	1.05080	+	0.254922i	1.05080	+	0.254922i	0.327068	+	0.945001i	$0.393939\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.841254	+	0.540641i	0.841254	+	0.540641i
$576$	1.61435	+	2.79614i	1.61435	+	2.79614i
$577$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$577$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$578$	1.11312	+	1.56316i	1.11312	+	1.56316i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.08907	+	2.93369i	−2.08907	+	2.93369i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$587$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−2.62155	−	2.74940i	−2.62155	−	2.74940i
$593$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$593$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$594$		0			0
$595$		0			0
$596$	4.01334	−	0.577031i	4.01334	−	0.577031i
$597$		0			0
$598$		0			0
$599$	−0.959493	+	1.66189i	−0.959493	+	1.66189i	−0.235759	+	0.971812i	$0.575758\pi$
$599$	−0.959493	+	1.66189i	−0.959493	+	1.66189i	−0.723734	+	0.690079i	$0.757576\pi$
$600$		0			0
$601$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$601$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$602$		0			0
$603$	0.425839	+	0.368991i	0.425839	+	0.368991i
$604$	0.180007	+	0.741999i	0.180007	+	0.741999i
$605$		0			0
$606$		0			0
$607$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$607$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.0535608	−	0.560914i	0.0535608	−	0.560914i	−0.928368	−	0.371662i	$-0.878788\pi$
$613$	0.0535608	−	0.560914i	0.0535608	−	0.560914i	0.981929	−	0.189251i	$-0.0606061\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.158746	+	0.540641i	0.158746	+	0.540641i	1.00000			$0$
$617$	0.158746	+	0.540641i	0.158746	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$618$		0			0
$619$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$619$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.723734	+	0.690079i	0.723734	+	0.690079i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.817178	−	1.27155i	0.817178	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$631$	0.817178	−	1.27155i	0.817178	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$632$	−2.37555	−	5.93384i	−2.37555	−	5.93384i
$633$		0			0
$634$	2.86979	−	1.47948i	2.86979	−	1.47948i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.279953	−	0.953431i	0.279953	−	0.953431i
$639$	−0.815816	−	0.157236i	−0.815816	−	0.157236i
$640$		0			0
$641$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$641$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$642$		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		0.814576	−	0.580057i	$-0.196970\pi$
$647$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$648$	0.761197	−	3.13770i	0.761197	−	3.13770i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.317178	−	0.694523i	−0.317178	−	0.694523i
$653$	−0.627639	+	1.81344i	−0.627639	+	1.81344i	−0.0475819	+	0.998867i	$0.515152\pi$
$653$	−0.627639	+	1.81344i	−0.627639	+	1.81344i	−0.580057	+	0.814576i	$0.696970\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$659$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$660$		0			0
$661$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$661$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$662$	−0.822032	−	2.37511i	−0.822032	−	2.37511i
$663$		0			0
$664$		0			0
$665$		0			0
$666$			2.07496i			2.07496i
$667$	0.252989	+	0.130425i	0.252989	+	0.130425i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$673$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$674$	1.59973	−	3.10305i	1.59973	−	3.10305i
$675$		0			0
$676$	−2.63403	+	0.507668i	−2.63403	+	0.507668i
$677$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$677$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.0800569	+	1.68060i	0.0800569	+	1.68060i	0.580057	+	0.814576i	$0.303030\pi$
$683$	0.0800569	+	1.68060i	0.0800569	+	1.68060i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−3.28268	−	4.17426i	−3.28268	−	4.17426i
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$	3.09792	−	0.909632i	3.09792	−	0.909632i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$701$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i	−0.415415	+	0.909632i	$0.636364\pi$
$702$		0			0
$703$		0			0
$704$	−1.11164	−	5.76772i	−1.11164	−	5.76772i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.25544	+	1.31666i	−1.25544	+	1.31666i	−0.327068	+	0.945001i	$0.606061\pi$
$709$	−1.25544	+	1.31666i	−1.25544	+	1.31666i	−0.928368	+	0.371662i	$0.878788\pi$
$710$		0			0
$711$	−0.735759	+	1.83784i	−0.735759	+	1.83784i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−3.49743	+	0.333964i	−3.49743	+	0.333964i
$717$		0			0
$718$	−2.74102	+	0.948676i	−2.74102	+	0.948676i
$719$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$719$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$720$		0			0
$721$		0			0
$722$	−1.25667	−	1.45027i	−1.25667	−	1.45027i
$723$		0			0
$724$		0			0
$725$	0.223734	+	0.175946i	0.223734	+	0.175946i
$726$		0			0
$727$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$727$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$728$		0			0
$729$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$733$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$734$		0			0
$735$		0			0
$736$	3.51334			3.51334
$737$	−0.512546	−	0.887756i	−0.512546	−	0.887756i
$738$		0			0
$739$	−1.11312	−	1.56316i	−1.11312	−	1.56316i	−0.786053	−	0.618159i	$-0.787879\pi$
$739$	−1.11312	−	1.56316i	−1.11312	−	1.56316i	−0.327068	−	0.945001i	$-0.606061\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.304632	+	0.474017i	0.304632	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$743$	0.304632	+	0.474017i	0.304632	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$744$		0			0
$745$		0			0
$746$	−1.59973	−	3.10305i	−1.59973	−	3.10305i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.61257	+	1.14831i	1.61257	+	1.14831i	0.888835	+	0.458227i	$0.151515\pi$
$751$	1.61257	+	1.14831i	1.61257	+	1.14831i	0.723734	+	0.690079i	$0.242424\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.80075	−	0.258908i	1.80075	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$757$	1.80075	−	0.258908i	1.80075	−	0.258908i	0.959493	+	0.281733i	$0.0909091\pi$
$758$	3.28995	−	1.89945i	3.28995	−	1.89945i
$759$		0			0
$760$		0			0
$761$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$761$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$762$		0			0
$763$		0			0
$764$	−2.19209	−	1.89945i	−2.19209	−	1.89945i
$765$		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			0
$772$	0.106046	−	2.22618i	0.106046	−	2.22618i
$773$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$773$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$774$	−0.275714	+	2.88741i	−0.275714	+	2.88741i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.983568	+	3.34973i	0.983568	+	3.34973i
$779$		0			0
$780$		0			0
$781$	1.30900	+	0.755750i	1.30900	+	0.755750i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$787$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$788$	−2.54272	+	2.42448i	−2.54272	+	2.42448i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−3.17565	+	4.94141i	−3.17565	+	4.94141i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$797$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$798$		0			0
$799$		0			0
$800$	3.44985	+	0.664903i	3.44985	+	0.664903i
$801$		0			0
$802$	−1.69022	+	1.20360i	−1.69022	+	1.20360i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i	−0.995472	−	0.0950560i	$-0.969697\pi$
$809$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i	0.928368	+	0.371662i	$0.121212\pi$
$810$		0			0
$811$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$811$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$812$		0			0
$813$		0			0
$814$	1.23465	−	3.56729i	1.23465	−	3.56729i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.738471	+	0.380708i	0.738471	+	0.380708i	0.786053	−	0.618159i	$-0.212121\pi$
$821$	0.738471	+	0.380708i	0.738471	+	0.380708i	−0.0475819	+	0.998867i	$0.515152\pi$
$822$		0			0
$823$	0.0930932	+	0.268975i	0.0930932	+	0.268975i	0.981929	−	0.189251i	$-0.0606061\pi$
$823$	0.0930932	+	0.268975i	0.0930932	+	0.268975i	−0.888835	+	0.458227i	$0.848485\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.08128i		−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$827$		−	1.08128i		−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$828$	−1.94142	−	1.85114i	−1.94142	−	1.85114i
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		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		0.142315	−	0.989821i	$-0.454545\pi$
$839$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$840$		0			0
$841$	−0.773100	−	0.496841i	−0.773100	−	0.496841i
$842$	−2.74102	−	0.948676i	−2.74102	−	0.948676i
$843$		0			0
$844$	−0.106046	−	2.22618i	−0.106046	−	2.22618i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−6.32663	−	2.88927i	−6.32663	−	2.88927i
$849$		0			0
$850$		0			0
$851$	0.936417	+	0.540641i	0.936417	+	0.540641i
$852$		0			0
$853$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$853$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$854$		0			0
$855$		0			0
$856$	−0.660703	+	3.42805i	−0.660703	+	3.42805i
$857$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$857$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$858$		0			0
$859$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$859$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$860$		0			0
$861$		0			0
$862$	2.87102	+	0.412791i	2.87102	+	0.412791i
$863$	0.771316	−	0.308788i	0.771316	−	0.308788i	0.0475819	−	0.998867i	$-0.484848\pi$
$863$	0.771316	−	0.308788i	0.771316	−	0.308788i	0.723734	+	0.690079i	$0.242424\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.35848	−	2.72183i	2.35848	−	2.72183i
$870$		0			0
$871$		0			0
$872$	−0.676152	+	1.68895i	−0.676152	+	1.68895i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i	−0.327068	+	0.945001i	$0.606061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$881$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$882$		0			0
$883$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$883$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$884$		0			0
$885$		0			0
$886$	−1.25324	−	0.985562i	−1.25324	−	0.985562i
$887$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$887$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.76798	−	0.428908i	1.76798	−	0.428908i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.61435	+	2.79614i	1.61435	+	2.79614i
$899$		0			0
$900$	−1.55601	−	2.18511i	−1.55601	−	2.18511i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−2.63843	−	4.10548i	−2.63843	−	4.10548i
$905$		0			0
$906$		0			0
$907$	0.258195	+	0.500828i	0.258195	+	0.500828i	0.981929	−	0.189251i	$-0.0606061\pi$
$907$	0.258195	+	0.500828i	0.258195	+	0.500828i	−0.723734	+	0.690079i	$0.757576\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.425839	−	0.368991i	0.425839	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$911$	0.425839	−	0.368991i	0.425839	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$912$		0			0
$913$		0			0
$914$	2.62155	+	2.74940i	2.62155	+	2.74940i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.936417	+	0.540641i	−0.936417	+	0.540641i	−0.888835	−	0.458227i	$-0.848485\pi$
$919$	−0.936417	+	0.540641i	−0.936417	+	0.540641i	−0.0475819	+	0.998867i	$0.515152\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.817178	+	0.708089i	0.817178	+	0.708089i
$926$	0.868184	+	3.57870i	0.868184	+	3.57870i
$927$		0			0
$928$	0.995472	+	0.0950560i	0.995472	+	0.0950560i
$929$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$929$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$930$		0			0
$931$		0			0
$932$	−0.642315	+	4.46740i	−0.642315	+	4.46740i
$933$		0			0
$934$		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			0
$940$		0			0
$941$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$941$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	2.19209	−	4.80000i	2.19209	−	4.80000i
$947$	−1.21769	−	1.16106i	−1.21769	−	1.16106i	−0.981929	−	0.189251i	$-0.939394\pi$
$947$	−1.21769	−	1.16106i	−1.21769	−	1.16106i	−0.235759	−	0.971812i	$-0.575758\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.584585	+	0.909632i	−0.584585	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$953$	−0.584585	+	0.909632i	−0.584585	+	0.909632i	−1.00000			$1.00000\pi$
$954$	1.41191	+	3.52678i	1.41191	+	3.52678i
$955$		0			0
$956$	4.01161	−	2.06813i	4.01161	−	2.06813i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.981929	−	0.189251i	−0.981929	−	0.189251i
$962$		0			0
$963$	0.880786	−	0.627205i	0.880786	−	0.627205i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$967$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$968$	5.86192	−	4.60987i	5.86192	−	4.60987i
$969$		0			0
$970$		0			0
$971$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$971$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$972$		0			0
$973$		0			0
$974$	−1.04408	−	2.28621i	−1.04408	−	2.28621i
$975$		0			0
$976$		0			0
$977$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$977$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.512546	−	0.234072i	0.512546	−	0.234072i
$982$	3.27314	+	1.68742i	3.27314	+	1.68742i
$983$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$983$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.23123	+	0.876756i	1.23123	+	0.876756i
$990$		0			0
$991$	−1.78153	−	0.713215i	−1.78153	−	0.713215i	−0.995472	−	0.0950560i	$-0.969697\pi$
$991$	−1.78153	−	0.713215i	−1.78153	−	0.713215i	−0.786053	−	0.618159i	$-0.787879\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$997$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$998$	3.61596	−	0.696919i	3.61596	−	0.696919i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.1.x.a.802.1		20
7.2	even	3	inner	1127.1.x.a.618.1		20
7.3	odd	6		1127.1.o.a.687.1	✓	10
7.4	even	3		1127.1.o.a.687.1	✓	10
7.5	odd	6	inner	1127.1.x.a.618.1		20
7.6	odd	2	CM	1127.1.x.a.802.1		20
23.15	odd	22	inner	1127.1.x.a.1096.1		20
161.38	even	66		1127.1.o.a.981.1	yes	10
161.61	even	66	inner	1127.1.x.a.912.1		20
161.107	odd	66	inner	1127.1.x.a.912.1		20
161.130	odd	66		1127.1.o.a.981.1	yes	10
161.153	even	22	inner	1127.1.x.a.1096.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1127.1.o.a.687.1	✓	10	7.3	odd	6
1127.1.o.a.687.1	✓	10	7.4	even	3
1127.1.o.a.981.1	yes	10	161.38	even	66
1127.1.o.a.981.1	yes	10	161.130	odd	66
1127.1.x.a.618.1		20	7.2	even	3	inner
1127.1.x.a.618.1		20	7.5	odd	6	inner
1127.1.x.a.802.1		20	1.1	even	1	trivial
1127.1.x.a.802.1		20	7.6	odd	2	CM
1127.1.x.a.912.1		20	161.61	even	66	inner
1127.1.x.a.912.1		20	161.107	odd	66	inner
1127.1.x.a.1096.1		20	23.15	odd	22	inner
1127.1.x.a.1096.1		20	161.153	even	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:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1127.x (of order \(66\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50842 + 1.18624i) q^{2} +(0.632425 - 2.60689i) q^{4} +(1.34125 + 2.93694i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)	\(q+(-1.50842 + 1.18624i) q^{2} +(0.632425 - 2.60689i) q^{4} +(1.34125 + 2.93694i) q^{8} +(0.327068 - 0.945001i) q^{9} +(-1.12459 + 1.43004i) q^{11} +(-3.12278 - 1.60990i) q^{16} +(0.627639 + 1.81344i) q^{18} -3.49114i q^{22} +(0.981929 + 0.189251i) q^{23} +(0.928368 + 0.371662i) q^{25} +(0.273100 + 0.0801894i) q^{29} +(3.44985 - 0.664903i) q^{32} +(-2.25667 - 1.45027i) q^{36} +(1.02181 + 0.353653i) q^{37} +(1.37491 + 0.627899i) q^{43} +(3.01673 + 3.83609i) q^{44} +(-1.70566 + 0.879330i) q^{46} +(-1.84125 + 0.540641i) q^{50} +(1.97740 - 0.0941952i) q^{53} +(-0.507075 + 0.203002i) q^{58} +(-2.11435 + 2.44009i) q^{64} +(-0.209419 + 0.523103i) q^{67} +(-0.118239 - 0.822373i) q^{71} +(3.21409 - 0.306908i) q^{72} +(-1.96084 + 0.678654i) q^{74} +(-1.97740 - 0.0941952i) q^{79} +(-0.786053 - 0.618159i) q^{81} +(-2.81878 + 0.683830i) q^{86} +(-5.70830 - 1.38482i) q^{88} +(1.11435 - 2.44009i) q^{92} +(0.983568 + 1.53046i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 3 q^{4} + 8 q^{8} - q^{9}+O(q^{10}) \) 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

Introduction

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