LMFDB - Embedded newform 1911.1.ep.a.500.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,1,Mod(59,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, base_ring=CyclotomicField(84))

chi = DirichletCharacter(H, H._module([42, 26, 77]))

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

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

chi := DirichletCharacter("1911.59");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1911.ep (of order $84$, degree $24$, 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:	$0.953713239142$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{84})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 $ x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{84}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{84} - \cdots)$

Embedding invariants

Embedding label			500.1
Root			$-0.563320 + 0.826239i$ of defining polynomial
Character	$\chi$	$=$	1911.500
Dual form			1911.1.ep.a.1181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.997204 - 0.0747301i) q^{3} +(-0.974928 + 0.222521i) q^{4} +(0.149042 - 0.988831i) q^{7} +(0.988831 - 0.149042i) q^{9} +O(q^{10})$	\(q+(0.997204 - 0.0747301i) q^{3} +(-0.974928 + 0.222521i) q^{4} +(0.149042 - 0.988831i) q^{7} +(0.988831 - 0.149042i) q^{9} +(-0.955573 + 0.294755i) q^{12} +(-0.680173 + 0.733052i) q^{13} +(0.900969 - 0.433884i) q^{16} +(1.14717 - 0.307384i) q^{19} +(0.0747301 - 0.997204i) q^{21} +(-0.149042 - 0.988831i) q^{25} +(0.974928 - 0.222521i) q^{27} +(0.0747301 + 0.997204i) q^{28} +(1.63575 - 0.438297i) q^{31} +(-0.930874 + 0.365341i) q^{36} +(-0.0633201 + 0.0397866i) q^{37} +(-0.623490 + 0.781831i) q^{39} +(0.167917 - 0.246289i) q^{43} +(0.866025 - 0.500000i) q^{48} +(-0.955573 - 0.294755i) q^{49} +(0.500000 - 0.866025i) q^{52} +(1.12099 - 0.392253i) q^{57} +(-1.29991 + 1.40097i) q^{61} -1.00000i q^{63} +(-0.781831 + 0.623490i) q^{64} +(-1.82344 - 0.488590i) q^{67} +(0.554947 - 0.751927i) q^{73} +(-0.222521 - 0.974928i) q^{75} +(-1.05001 + 0.554947i) q^{76} +(0.974928 + 1.68862i) q^{79} +(0.955573 - 0.294755i) q^{81} +(0.149042 + 0.988831i) q^{84} +(0.623490 + 0.781831i) q^{91} +(1.59842 - 0.559311i) q^{93} +(0.416490 - 1.55436i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{9}+O(q^{10}) $ 24 * q - 2 * q^9	$ 24 q - 2 q^{9} - 2 q^{12} + 4 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{28} + 2 q^{31} + 12 q^{37} + 4 q^{39} + 6 q^{43} - 2 q^{49} + 12 q^{52} + 2 q^{57} - 2 q^{67} - 2 q^{73} - 4 q^{75} - 4 q^{76} + 2 q^{81} - 4 q^{91} - 4 q^{93} - 2 q^{97}+O(q^{100}) $ 24 * q - 2 * q^9 - 2 * q^12 + 4 * q^16 + 2 * q^19 + 2 * q^21 + 2 * q^28 + 2 * q^31 + 12 * q^37 + 4 * q^39 + 6 * q^43 - 2 * q^49 + 12 * q^52 + 2 * q^57 - 2 * q^67 - 2 * q^73 - 4 * q^75 - 4 * q^76 + 2 * q^81 - 4 * q^91 - 4 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$.

$n$	$638$	$1471$	$1522$
$\chi(n)$	$-1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{13}{42}\right)$

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		−0.111964	−	0.993712i	$-0.535714\pi$
$2$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$3$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$3$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$4$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$5$		0			0		0.652287	−	0.757972i	$-0.273810\pi$
$5$		0			0		−0.652287	+	0.757972i	$0.726190\pi$
$6$		0			0
$7$	0.149042	−	0.988831i	0.149042	−	0.988831i
$7$	0.149042	−	0.988831i	0.149042	−	0.988831i
$8$		0			0
$9$	0.988831	−	0.149042i	0.988831	−	0.149042i
$10$		0			0
$11$		0			0		−0.593820	−	0.804598i	$-0.702381\pi$
$11$		0			0		0.593820	+	0.804598i	$0.297619\pi$
$12$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$13$	−0.680173	+	0.733052i	−0.680173	+	0.733052i
$13$	−0.680173	+	0.733052i	−0.680173	+	0.733052i
$14$		0			0
$15$		0			0
$16$	0.900969	−	0.433884i	0.900969	−	0.433884i
$17$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$17$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$18$		0			0
$19$	1.14717	−	0.307384i	1.14717	−	0.307384i	0.365341	−	0.930874i	$-0.380952\pi$
$19$	1.14717	−	0.307384i	1.14717	−	0.307384i	0.781831	+	0.623490i	$0.214286\pi$
$20$		0			0
$21$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$22$		0			0
$23$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$23$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$24$		0			0
$25$	−0.149042	−	0.988831i	−0.149042	−	0.988831i
$26$		0			0
$27$	0.974928	−	0.222521i	0.974928	−	0.222521i
$28$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$29$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$29$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$30$		0			0
$31$	1.63575	−	0.438297i	1.63575	−	0.438297i	0.680173	−	0.733052i	$-0.261905\pi$
$31$	1.63575	−	0.438297i	1.63575	−	0.438297i	0.955573	+	0.294755i	$0.0952381\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.930874	+	0.365341i	−0.930874	+	0.365341i
$37$	−0.0633201	+	0.0397866i	−0.0633201	+	0.0397866i	−0.563320	−	0.826239i	$-0.690476\pi$
$37$	−0.0633201	+	0.0397866i	−0.0633201	+	0.0397866i	0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$40$		0			0
$41$		0			0		−0.982566	−	0.185912i	$-0.940476\pi$
$41$		0			0		0.982566	+	0.185912i	$0.0595238\pi$
$42$		0			0
$43$	0.167917	−	0.246289i	0.167917	−	0.246289i	−0.733052	−	0.680173i	$-0.761905\pi$
$43$	0.167917	−	0.246289i	0.167917	−	0.246289i	0.900969	+	0.433884i	$0.142857\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.804598	−	0.593820i	$-0.202381\pi$
$47$		0			0		−0.804598	+	0.593820i	$0.797619\pi$
$48$	0.866025	−	0.500000i	0.866025	−	0.500000i
$49$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$50$		0			0
$51$		0			0
$52$	0.500000	−	0.866025i	0.500000	−	0.866025i
$53$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$53$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.12099	−	0.392253i	1.12099	−	0.392253i
$58$		0			0
$59$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$59$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$60$		0			0
$61$	−1.29991	+	1.40097i	−1.29991	+	1.40097i	−0.433884	+	0.900969i	$0.642857\pi$
$61$	−1.29991	+	1.40097i	−1.29991	+	1.40097i	−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		−	1.00000i		−	1.00000i
$64$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$65$		0			0
$66$		0			0
$67$	−1.82344	−	0.488590i	−1.82344	−	0.488590i	−0.826239	−	0.563320i	$-0.809524\pi$
$67$	−1.82344	−	0.488590i	−1.82344	−	0.488590i	−0.997204	+	0.0747301i	$0.976190\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.0373912	−	0.999301i	$-0.488095\pi$
$71$		0			0		−0.0373912	+	0.999301i	$0.511905\pi$
$72$		0			0
$73$	0.554947	−	0.751927i	0.554947	−	0.751927i	−0.433884	−	0.900969i	$-0.642857\pi$
$73$	0.554947	−	0.751927i	0.554947	−	0.751927i	0.988831	+	0.149042i	$0.0476190\pi$
$74$		0			0
$75$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$76$	−1.05001	+	0.554947i	−1.05001	+	0.554947i
$77$		0			0
$78$		0			0
$79$	0.974928	+	1.68862i	0.974928	+	1.68862i	0.680173	+	0.733052i	$0.261905\pi$
$79$	0.974928	+	1.68862i	0.974928	+	1.68862i	0.294755	+	0.955573i	$0.404762\pi$
$80$		0			0
$81$	0.955573	−	0.294755i	0.955573	−	0.294755i
$82$		0			0
$83$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$83$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$84$	0.149042	+	0.988831i	0.149042	+	0.988831i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$89$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$90$		0			0
$91$	0.623490	+	0.781831i	0.623490	+	0.781831i
$92$		0			0
$93$	1.59842	−	0.559311i	1.59842	−	0.559311i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.416490	−	1.55436i	0.416490	−	1.55436i	−0.365341	−	0.930874i	$-0.619048\pi$
$97$	0.416490	−	1.55436i	0.416490	−	1.55436i	0.781831	−	0.623490i	$-0.214286\pi$
$98$		0			0
$99$		0			0
$100$	0.365341	+	0.930874i	0.365341	+	0.930874i
$101$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$101$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$102$		0			0
$103$	0.147791	+	1.97213i	0.147791	+	1.97213i	0.222521	+	0.974928i	$0.428571\pi$
$103$	0.147791	+	1.97213i	0.147791	+	1.97213i	−0.0747301	+	0.997204i	$0.523810\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$107$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$108$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$109$	0.340799	+	0.148689i	0.340799	+	0.148689i	0.563320	−	0.826239i	$-0.309524\pi$
$109$	0.340799	+	0.148689i	0.340799	+	0.148689i	−0.222521	+	0.974928i	$0.571429\pi$
$110$		0			0
$111$	−0.0601697	+	0.0444073i	−0.0601697	+	0.0444073i
$112$	−0.294755	−	0.955573i	−0.294755	−	0.955573i
$113$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$113$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.563320	+	0.826239i	−0.563320	+	0.826239i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.294755	+	0.955573i	−0.294755	+	0.955573i
$122$		0			0
$123$		0			0
$124$	−1.49720	+	0.791295i	−1.49720	+	0.791295i
$125$		0			0
$126$		0			0
$127$	−0.496990	+	0.535628i	−0.496990	+	0.535628i	−0.930874	−	0.365341i	$-0.880952\pi$
$127$	−0.496990	+	0.535628i	−0.496990	+	0.535628i	0.433884	+	0.900969i	$0.357143\pi$
$128$		0			0
$129$	0.149042	−	0.258149i	0.149042	−	0.258149i
$130$		0			0
$131$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$131$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$132$		0			0
$133$	−0.132974	−	1.18017i	−0.132974	−	1.18017i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$137$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$138$		0			0
$139$	−0.865341	+	0.0648483i	−0.865341	+	0.0648483i	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−0.865341	+	0.0648483i	−0.865341	+	0.0648483i	−0.365341	+	0.930874i	$0.619048\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.826239	−	0.563320i	0.826239	−	0.563320i
$145$		0			0
$146$		0			0
$147$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$148$	0.0528791	−	0.0528791i	0.0528791	−	0.0528791i
$149$		0			0		−0.916562	−	0.399892i	$-0.869048\pi$
$149$		0			0		0.916562	+	0.399892i	$0.130952\pi$
$150$		0			0
$151$	0.104635	+	0.197979i	0.104635	+	0.197979i	0.930874	−	0.365341i	$-0.119048\pi$
$151$	0.104635	+	0.197979i	0.104635	+	0.197979i	−0.826239	+	0.563320i	$0.809524\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.433884	−	0.900969i	0.433884	−	0.900969i
$157$	−1.98883	−	0.149042i	−1.98883	−	0.149042i	−0.988831	−	0.149042i	$-0.952381\pi$
$157$	−1.98883	−	0.149042i	−1.98883	−	0.149042i	−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.988831	+	0.850958i	−0.988831	+	0.850958i	−0.988831	−	0.149042i	$-0.952381\pi$
$163$	−0.988831	+	0.850958i	−0.988831	+	0.850958i			1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.999301	−	0.0373912i	$-0.0119048\pi$
$167$		0			0		−0.999301	+	0.0373912i	$0.988095\pi$
$168$		0			0
$169$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$170$		0			0
$171$	1.08855	−	0.474928i	1.08855	−	0.474928i
$172$	−0.108903	+	0.277479i	−0.108903	+	0.277479i
$173$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$173$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$174$		0			0
$175$	−1.00000			−1.00000
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$179$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$180$		0			0
$181$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	0.826239	−	0.563320i	$-0.190476\pi$
$181$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$182$		0			0
$183$	−1.19158	+	1.49419i	−1.19158	+	1.49419i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$190$		0			0
$191$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$191$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$192$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$193$	−0.988831	+	1.14904i	−0.988831	+	1.14904i			1.00000i	$0.5\pi$
$193$	−0.988831	+	1.14904i	−0.988831	+	1.14904i	−0.988831	+	0.149042i	$0.952381\pi$
$194$		0			0
$195$		0			0
$196$	0.997204	+	0.0747301i	0.997204	+	0.0747301i
$197$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$197$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$198$		0			0
$199$	−1.67738	−	0.807782i	−1.67738	−	0.807782i	−0.997204	−	0.0747301i	$-0.976190\pi$
$199$	−1.67738	−	0.807782i	−1.67738	−	0.807782i	−0.680173	−	0.733052i	$-0.738095\pi$
$200$		0			0
$201$	−1.85486	−	0.350958i	−1.85486	−	0.350958i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.294755	+	0.955573i	−0.294755	+	0.955573i
$209$		0			0
$210$		0			0
$211$	−1.34515	+	0.202749i	−1.34515	+	0.202749i	−0.781831	−	0.623490i	$-0.785714\pi$
$211$	−1.34515	+	0.202749i	−1.34515	+	0.202749i	−0.563320	+	0.826239i	$0.690476\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.189606	−	1.68280i	−0.189606	−	1.68280i
$218$		0			0
$219$	0.497204	−	0.791295i	0.497204	−	0.791295i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.497204	+	0.940755i	0.497204	+	0.940755i	0.997204	+	0.0747301i	$0.0238095\pi$
$223$	0.497204	+	0.940755i	0.497204	+	0.940755i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−0.294755	−	0.955573i	−0.294755	−	0.955573i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	−1.00560	+	0.631863i	−1.00560	+	0.631863i
$229$	0.785841	−	0.148689i	0.785841	−	0.148689i	0.222521	−	0.974928i	$-0.428571\pi$
$229$	0.785841	−	0.148689i	0.785841	−	0.148689i	0.563320	+	0.826239i	$0.309524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$233$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.09839	+	1.61105i	1.09839	+	1.61105i
$238$		0			0
$239$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$239$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$240$		0			0
$241$	0.559311	+	0.351438i	0.559311	+	0.351438i	0.781831	−	0.623490i	$-0.214286\pi$
$241$	0.559311	+	0.351438i	0.559311	+	0.351438i	−0.222521	+	0.974928i	$0.571429\pi$
$242$		0			0
$243$	0.930874	−	0.365341i	0.930874	−	0.365341i
$244$	0.955573	−	1.65510i	0.955573	−	1.65510i
$245$		0			0
$246$		0			0
$247$	−0.554947	+	1.05001i	−0.554947	+	1.05001i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$251$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$252$	0.222521	+	0.974928i	0.222521	+	0.974928i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.623490	−	0.781831i	0.623490	−	0.781831i
$257$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$257$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$258$		0			0
$259$	0.0299049	+	0.0685427i	0.0299049	+	0.0685427i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.88645	+	0.0705858i	1.88645	+	0.0705858i
$269$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$269$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$270$		0			0
$271$	1.66393	+	1.04551i	1.66393	+	1.04551i	0.930874	+	0.365341i	$0.119048\pi$
$271$	1.66393	+	1.04551i	1.66393	+	1.04551i	0.733052	+	0.680173i	$0.238095\pi$
$272$		0			0
$273$	0.680173	+	0.733052i	0.680173	+	0.733052i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.09839	−	0.250701i	−1.09839	−	0.250701i	−0.365341	−	0.930874i	$-0.619048\pi$
$277$	−1.09839	−	0.250701i	−1.09839	−	0.250701i	−0.733052	+	0.680173i	$0.761905\pi$
$278$		0			0
$279$	1.55215	−	0.677197i	1.55215	−	0.677197i
$280$		0			0
$281$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$281$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$282$		0			0
$283$	0.582926	+	0.0878620i	0.582926	+	0.0878620i	0.433884	−	0.900969i	$-0.357143\pi$
$283$	0.582926	+	0.0878620i	0.582926	+	0.0878620i	0.149042	+	0.988831i	$0.452381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$290$		0			0
$291$	0.299168	−	1.58114i	0.299168	−	1.58114i
$292$	−0.373714	+	0.856562i	−0.373714	+	0.856562i
$293$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$293$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.433884	+	0.900969i	0.433884	+	0.900969i
$301$	−0.218511	−	0.202749i	−0.218511	−	0.202749i
$302$		0			0
$303$		0			0
$304$	0.900198	−	0.774683i	0.900198	−	0.774683i
$305$		0			0
$306$		0			0
$307$	−0.222521	−	1.97493i	−0.222521	−	1.97493i	−0.222521	−	0.974928i	$-0.571429\pi$
$307$	−0.222521	−	1.97493i	−0.222521	−	1.97493i		−	1.00000i	$-0.5\pi$
$308$		0			0
$309$	0.294755	+	1.95557i	0.294755	+	1.95557i
$310$		0			0
$311$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$311$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$312$		0			0
$313$	1.26968	−	0.733052i	1.26968	−	0.733052i	0.294755	−	0.955573i	$-0.404762\pi$
$313$	1.26968	−	0.733052i	1.26968	−	0.733052i	0.974928	+	0.222521i	$0.0714286\pi$
$314$		0			0
$315$		0			0
$316$	−1.32624	−	1.42935i	−1.32624	−	1.42935i
$317$		0			0		0.0373912	−	0.999301i	$-0.488095\pi$
$317$		0			0		−0.0373912	+	0.999301i	$0.511905\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$325$	0.826239	+	0.563320i	0.826239	+	0.563320i
$326$		0			0
$327$	0.350958	+	0.122805i	0.350958	+	0.122805i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.83184	+	0.0685427i	−1.83184	+	0.0685427i	−0.930874	−	0.365341i	$-0.880952\pi$
$331$	−1.83184	+	0.0685427i	−1.83184	+	0.0685427i	−0.900969	+	0.433884i	$0.857143\pi$
$332$		0			0
$333$	−0.0566829	+	0.0487796i	−0.0566829	+	0.0487796i
$334$		0			0
$335$		0			0
$336$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$337$	0.716983	+	1.48883i	0.716983	+	1.48883i	0.866025	+	0.500000i	$0.166667\pi$
$337$	0.716983	+	1.48883i	0.716983	+	1.48883i	−0.149042	+	0.988831i	$0.547619\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$347$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$348$		0			0
$349$	1.19572	+	1.38946i	1.19572	+	1.38946i	0.900969	+	0.433884i	$0.142857\pi$
$349$	1.19572	+	1.38946i	1.19572	+	1.38946i	0.294755	+	0.955573i	$0.404762\pi$
$350$		0			0
$351$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$352$		0			0
$353$		0			0		−0.185912	−	0.982566i	$-0.559524\pi$
$353$		0			0		0.185912	+	0.982566i	$0.440476\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.185912	−	0.982566i	$-0.559524\pi$
$359$		0			0		0.185912	+	0.982566i	$0.440476\pi$
$360$		0			0
$361$	0.355494	−	0.205245i	0.355494	−	0.205245i
$362$		0			0
$363$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$364$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$365$		0			0
$366$		0			0
$367$	−1.04876	−	0.411608i	−1.04876	−	0.411608i	−0.222521	−	0.974928i	$-0.571429\pi$
$367$	−1.04876	−	0.411608i	−1.04876	−	0.411608i	−0.826239	+	0.563320i	$0.809524\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.43388	+	0.900969i	−1.43388	+	0.900969i
$373$	−0.781831	+	1.35417i	−0.781831	+	1.35417i	0.149042	+	0.988831i	$0.452381\pi$
$373$	−0.781831	+	1.35417i	−0.781831	+	1.35417i	−0.930874	+	0.365341i	$0.880952\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.82160	−	0.794755i	−1.82160	−	0.794755i	−0.955573	−	0.294755i	$-0.904762\pi$
$379$	−1.82160	−	0.794755i	−1.82160	−	0.794755i	−0.866025	−	0.500000i	$-0.833333\pi$
$380$		0			0
$381$	−0.455573	+	0.571270i	−0.455573	+	0.571270i
$382$		0			0
$383$		0			0		−0.804598	−	0.593820i	$-0.797619\pi$
$383$		0			0		0.804598	+	0.593820i	$0.202381\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.129334	−	0.268565i	0.129334	−	0.268565i
$388$	−0.0601697	+	1.60807i	−0.0601697	+	1.60807i
$389$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$389$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.328735	−	1.73740i	0.328735	−	1.73740i	−0.294755	−	0.955573i	$-0.595238\pi$
$397$	0.328735	−	1.73740i	0.328735	−	1.73740i	0.623490	−	0.781831i	$-0.285714\pi$
$398$		0			0
$399$	−0.220796	−	1.16694i	−0.220796	−	1.16694i
$400$	−0.563320	−	0.826239i	−0.563320	−	0.826239i
$401$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$401$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$402$		0			0
$403$	−0.791295	+	1.49720i	−0.791295	+	1.49720i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.69226	−	1.06332i	1.69226	−	1.06332i	0.826239	−	0.563320i	$-0.190476\pi$
$409$	1.69226	−	1.06332i	1.69226	−	1.06332i	0.866025	−	0.500000i	$-0.166667\pi$
$410$		0			0
$411$		0			0
$412$	−0.582926	−	1.88980i	−0.582926	−	1.88980i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.858075	+	0.129334i	−0.858075	+	0.129334i
$418$		0			0
$419$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$419$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$420$		0			0
$421$	0.351438	+	0.559311i	0.351438	+	0.559311i	0.974928	−	0.222521i	$-0.0714286\pi$
$421$	0.351438	+	0.559311i	0.351438	+	0.559311i	−0.623490	+	0.781831i	$0.714286\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.19158	+	1.49419i	1.19158	+	1.49419i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.982566	−	0.185912i	$-0.0595238\pi$
$431$		0			0		−0.982566	+	0.185912i	$0.940476\pi$
$432$	0.781831	−	0.623490i	0.781831	−	0.623490i
$433$	0.116853	+	1.55929i	0.116853	+	1.55929i	0.680173	+	0.733052i	$0.261905\pi$
$433$	0.116853	+	1.55929i	0.116853	+	1.55929i	−0.563320	+	0.826239i	$0.690476\pi$
$434$		0			0
$435$		0			0
$436$	−0.365341	−	0.0691263i	−0.365341	−	0.0691263i
$437$		0			0
$438$		0			0
$439$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.0747301	−	0.997204i	$-0.476190\pi$
$439$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.955573	−	0.294755i	$-0.0952381\pi$
$440$		0			0
$441$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$442$		0			0
$443$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$443$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$444$	0.0487796	−	0.0566829i	0.0487796	−	0.0566829i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$		0			0		0.652287	−	0.757972i	$-0.273810\pi$
$449$		0			0		−0.652287	+	0.757972i	$0.726190\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.119137	+	0.189606i	0.119137	+	0.189606i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.351438	−	1.00435i	0.351438	−	1.00435i	−0.623490	−	0.781831i	$-0.714286\pi$
$457$	0.351438	−	1.00435i	0.351438	−	1.00435i	0.974928	−	0.222521i	$-0.0714286\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.884115	−	0.467269i	$-0.845238\pi$
$461$		0			0		0.884115	+	0.467269i	$0.154762\pi$
$462$		0			0
$463$	1.04551	−	1.66393i	1.04551	−	1.66393i	0.365341	−	0.930874i	$-0.380952\pi$
$463$	1.04551	−	1.66393i	1.04551	−	1.66393i	0.680173	−	0.733052i	$-0.261905\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$467$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$468$	0.365341	−	0.930874i	0.365341	−	0.930874i
$469$	−0.754903	+	1.73026i	−0.754903	+	1.73026i
$470$		0			0
$471$	−1.99441			−1.99441
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.474928	−	1.08855i	−0.474928	−	1.08855i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.804598	−	0.593820i	$-0.202381\pi$
$479$		0			0		−0.804598	+	0.593820i	$0.797619\pi$
$480$		0			0
$481$	0.0139029	−	0.0734787i	0.0139029	−	0.0734787i
$482$		0			0
$483$		0			0
$484$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$485$		0			0
$486$		0			0
$487$	−1.88645	+	0.660096i	−1.88645	+	0.660096i	−0.930874	+	0.365341i	$0.880952\pi$
$487$	−1.88645	+	0.660096i	−1.88645	+	0.660096i	−0.955573	+	0.294755i	$0.904762\pi$
$488$		0			0
$489$	−0.922474	+	0.922474i	−0.922474	+	0.922474i
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.28359	−	1.10462i	1.28359	−	1.10462i
$497$		0			0
$498$		0			0
$499$	0.707101	+	1.62069i	0.707101	+	1.62069i	0.781831	+	0.623490i	$0.214286\pi$
$499$	0.707101	+	1.62069i	0.707101	+	1.62069i	−0.0747301	+	0.997204i	$0.523810\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$503$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.149042	−	0.988831i	−0.149042	−	0.988831i
$508$	0.365341	−	0.632789i	0.365341	−	0.632789i
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$	−0.660818	−	0.660818i	−0.660818	−	0.660818i
$512$		0			0
$513$	1.05001	−	0.554947i	1.05001	−	0.554947i
$514$		0			0
$515$		0			0
$516$	−0.0878620	+	0.284841i	−0.0878620	+	0.284841i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	0.590232	+	1.22563i	0.590232	+	1.22563i	0.955573	+	0.294755i	$0.0952381\pi$
$523$	0.590232	+	1.22563i	0.590232	+	1.22563i	−0.365341	+	0.930874i	$0.619048\pi$
$524$		0			0
$525$	−0.997204	+	0.0747301i	−0.997204	+	0.0747301i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.900969	−	0.433884i	0.900969	−	0.433884i
$530$		0			0
$531$		0			0
$532$	0.392253	+	1.12099i	0.392253	+	1.12099i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.474928	−	0.643504i	−0.474928	−	0.643504i	0.500000	−	0.866025i	$-0.333333\pi$
$541$	−0.474928	−	0.643504i	−0.474928	−	0.643504i	−0.974928	+	0.222521i	$0.928571\pi$
$542$		0			0
$543$	0.101659	+	0.109562i	0.101659	+	0.109562i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$547$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$548$		0			0
$549$	−1.07659	+	1.57906i	−1.07659	+	1.57906i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.81507	−	0.712362i	1.81507	−	0.712362i
$554$		0			0
$555$		0			0
$556$	0.829215	−	0.255779i	0.829215	−	0.255779i
$557$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$557$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$558$		0			0
$559$	0.0663300	+	0.290611i	0.0663300	+	0.290611i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$563$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.149042	−	0.988831i	−0.149042	−	0.988831i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	0.997204	−	1.07473i	0.997204	−	1.07473i		−	1.00000i	$-0.5\pi$
$571$	0.997204	−	1.07473i	0.997204	−	1.07473i	0.997204	−	0.0747301i	$-0.0238095\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.680173	+	0.733052i	−0.680173	+	0.733052i
$577$	0.606214	−	1.38946i	0.606214	−	1.38946i	−0.294755	−	0.955573i	$-0.595238\pi$
$577$	0.606214	−	1.38946i	0.606214	−	1.38946i	0.900969	−	0.433884i	$-0.142857\pi$
$578$		0			0
$579$	−0.900198	+	1.21972i	−0.900198	+	1.21972i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$	1.00000			1.00000
$589$	1.74176	−	1.00560i	1.74176	−	1.00560i
$590$		0			0
$591$		0			0
$592$	−0.0397866	+	0.0633201i	−0.0397866	+	0.0633201i
$593$		0			0		−0.757972	−	0.652287i	$-0.773810\pi$
$593$		0			0		0.757972	+	0.652287i	$0.226190\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.73305	−	0.680173i	−1.73305	−	0.680173i
$598$		0			0
$599$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$599$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$600$		0			0
$601$	0.930874	−	0.365341i	0.930874	−	0.365341i	0.149042	−	0.988831i	$-0.452381\pi$
$601$	0.930874	−	0.365341i	0.930874	−	0.365341i	0.781831	+	0.623490i	$0.214286\pi$
$602$		0			0
$603$	−1.87590	−	0.211363i	−1.87590	−	0.211363i
$604$	−0.146066	−	0.169732i	−0.146066	−	0.169732i
$605$		0			0
$606$		0			0
$607$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$607$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.707101	−	1.62069i	−0.707101	−	1.62069i	−0.781831	−	0.623490i	$-0.785714\pi$
$613$	−0.707101	−	1.62069i	−0.707101	−	1.62069i	0.0747301	−	0.997204i	$-0.476190\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.467269	−	0.884115i	$-0.345238\pi$
$617$		0			0		−0.467269	+	0.884115i	$0.654762\pi$
$618$		0			0
$619$	0.133975	+	0.500000i	0.133975	+	0.500000i	1.00000			$0$
$619$	0.133975	+	0.500000i	0.133975	+	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$625$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$626$		0			0
$627$		0			0
$628$	1.97213	−	0.297251i	1.97213	−	0.297251i
$629$		0			0
$630$		0			0
$631$	0.371563	+	1.96376i	0.371563	+	1.96376i	0.222521	+	0.974928i	$0.428571\pi$
$631$	0.371563	+	1.96376i	0.371563	+	1.96376i	0.149042	+	0.988831i	$0.452381\pi$
$632$		0			0
$633$	−1.32624	+	0.302705i	−1.32624	+	0.302705i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.866025	−	0.500000i	0.866025	−	0.500000i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$641$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$642$		0			0
$643$	0.242536	+	1.28183i	0.242536	+	1.28183i	0.866025	+	0.500000i	$0.166667\pi$
$643$	0.242536	+	1.28183i	0.242536	+	1.28183i	−0.623490	+	0.781831i	$0.714286\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$647$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.314832	−	1.66393i	−0.314832	−	1.66393i
$652$	0.774683	−	1.04966i	0.774683	−	1.04966i
$653$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$653$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.436680	−	0.826239i	0.436680	−	0.826239i
$658$		0			0
$659$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$659$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$660$		0			0
$661$	−0.565533	−	1.29621i	−0.565533	−	1.29621i	−0.930874	−	0.365341i	$-0.880952\pi$
$661$	−0.565533	−	1.29621i	−0.565533	−	1.29621i	0.365341	−	0.930874i	$-0.380952\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.566116	+	0.900969i	0.566116	+	0.900969i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.85654	+	0.728639i	−1.85654	+	0.728639i	−0.900969	+	0.433884i	$0.857143\pi$
$673$	−1.85654	+	0.728639i	−1.85654	+	0.728639i	−0.955573	+	0.294755i	$0.904762\pi$
$674$		0			0
$675$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$676$	0.294755	+	0.955573i	0.294755	+	0.955573i
$677$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$677$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$678$		0			0
$679$	−1.47493	−	0.643504i	−1.47493	−	0.643504i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$683$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$684$	−0.955573	+	0.705245i	−0.955573	+	0.705245i
$685$		0			0
$686$		0			0
$687$	0.772532	−	0.206999i	0.772532	−	0.206999i
$688$	0.0444272	−	0.294755i	0.0444272	−	0.294755i
$689$		0			0
$690$		0			0
$691$	−0.638050	−	1.82344i	−0.638050	−	1.82344i	−0.563320	−	0.826239i	$-0.690476\pi$
$691$	−0.638050	−	1.82344i	−0.638050	−	1.82344i	−0.0747301	−	0.997204i	$-0.523810\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.974928	−	0.222521i	0.974928	−	0.222521i
$701$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$701$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$702$		0			0
$703$	−0.0604093	+	0.0651057i	−0.0604093	+	0.0651057i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.373714	+	0.707101i	0.373714	+	0.707101i	0.997204	−	0.0747301i	$-0.0238095\pi$
$709$	0.373714	+	0.707101i	0.373714	+	0.707101i	−0.623490	+	0.781831i	$0.714286\pi$
$710$		0			0
$711$	1.21572	+	1.52446i	1.21572	+	1.52446i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$719$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$720$		0			0
$721$	1.97213	+	0.147791i	1.97213	+	0.147791i
$722$		0			0
$723$	0.584010	+	0.308658i	0.584010	+	0.308658i
$724$	−0.116853	−	0.0931869i	−0.116853	−	0.0931869i
$725$		0			0
$726$		0			0
$727$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$727$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$728$		0			0
$729$	0.900969	−	0.433884i	0.900969	−	0.433884i
$730$		0			0
$731$		0			0
$732$	0.829215	−	1.72188i	0.829215	−	1.72188i
$733$	0.132974	+	0.180173i	0.132974	+	0.180173i	0.866025	−	0.500000i	$-0.166667\pi$
$733$	0.132974	+	0.180173i	0.132974	+	0.180173i	−0.733052	+	0.680173i	$0.761905\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.96376	+	0.0734787i	1.96376	+	0.0734787i	0.988831	−	0.149042i	$-0.0476190\pi$
$739$	1.96376	+	0.0734787i	1.96376	+	0.0734787i	0.974928	+	0.222521i	$0.0714286\pi$
$740$		0			0
$741$	−0.474928	+	1.08855i	−0.474928	+	1.08855i
$742$		0			0
$743$		0			0		0.185912	−	0.982566i	$-0.440476\pi$
$743$		0			0		−0.185912	+	0.982566i	$0.559524\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.807782	+	1.67738i	0.807782	+	1.67738i	0.733052	+	0.680173i	$0.238095\pi$
$751$	0.807782	+	1.67738i	0.807782	+	1.67738i	0.0747301	+	0.997204i	$0.476190\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.294755	+	0.955573i	0.294755	+	0.955573i
$757$	−0.829215	−	0.255779i	−0.829215	−	0.255779i	−0.149042	−	0.988831i	$-0.547619\pi$
$757$	−0.829215	−	0.255779i	−0.829215	−	0.255779i	−0.680173	+	0.733052i	$0.738095\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.884115	−	0.467269i	$-0.154762\pi$
$761$		0			0		−0.884115	+	0.467269i	$0.845238\pi$
$762$		0			0
$763$	0.197822	−	0.314832i	0.197822	−	0.314832i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.563320	−	0.826239i	0.563320	−	0.826239i
$769$	1.55436	+	1.14717i	1.55436	+	1.14717i	0.930874	+	0.365341i	$0.119048\pi$
$769$	1.55436	+	1.14717i	1.55436	+	1.14717i	0.623490	+	0.781831i	$0.285714\pi$
$770$		0			0
$771$		0			0
$772$	0.708353	−	1.34027i	0.708353	−	1.34027i
$773$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$773$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$774$		0			0
$775$	−0.677197	−	1.55215i	−0.677197	−	1.55215i
$776$		0			0
$777$	0.0349435	+	0.0661163i	0.0349435	+	0.0661163i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$785$		0			0
$786$		0			0
$787$	−1.88645	+	0.660096i	−1.88645	+	0.660096i	−0.930874	+	0.365341i	$0.880952\pi$
$787$	−1.88645	+	0.660096i	−1.88645	+	0.660096i	−0.955573	+	0.294755i	$0.904762\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.142820	−	1.90580i	−0.142820	−	1.90580i
$794$		0			0
$795$		0			0
$796$	1.81507	+	0.414278i	1.81507	+	0.414278i
$797$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$797$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.88645	−	0.0705858i	1.88645	−	0.0705858i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$809$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$810$		0			0
$811$	−0.806531	+	1.28359i	−0.806531	+	1.28359i	0.149042	+	0.988831i	$0.452381\pi$
$811$	−0.806531	+	1.28359i	−0.806531	+	1.28359i	−0.955573	+	0.294755i	$0.904762\pi$
$812$		0			0
$813$	1.73740	+	0.918245i	1.73740	+	0.918245i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.116924	−	0.334151i	0.116924	−	0.334151i
$818$		0			0
$819$	0.733052	+	0.680173i	0.733052	+	0.680173i
$820$		0			0
$821$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$821$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$822$		0			0
$823$	−0.678448	−	1.40881i	−0.678448	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$823$	−0.678448	−	1.40881i	−0.678448	−	1.40881i	0.222521	−	0.974928i	$-0.428571\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$827$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$828$		0			0
$829$	0.997204	−	0.925270i	0.997204	−	0.925270i		−	1.00000i	$-0.5\pi$
$829$	0.997204	−	0.925270i	0.997204	−	0.925270i	0.997204	+	0.0747301i	$0.0238095\pi$
$830$		0			0
$831$	−1.11406	−	0.167917i	−1.11406	−	0.167917i
$832$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.49720	−	0.791295i	1.49720	−	0.791295i
$838$		0			0
$839$		0			0		0.982566	−	0.185912i	$-0.0595238\pi$
$839$		0			0		−0.982566	+	0.185912i	$0.940476\pi$
$840$		0			0
$841$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$842$		0			0
$843$		0			0
$844$	1.26631	−	0.496990i	1.26631	−	0.496990i
$845$		0			0
$846$		0			0
$847$	0.900969	+	0.433884i	0.900969	+	0.433884i
$848$		0			0
$849$	0.587862	+	0.0440542i	0.587862	+	0.0440542i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.119137	+	0.189606i	0.119137	+	0.189606i	0.900969	−	0.433884i	$-0.142857\pi$
$853$	0.119137	+	0.189606i	0.119137	+	0.189606i	−0.781831	+	0.623490i	$0.785714\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$857$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$858$		0			0
$859$	−0.496990	−	0.535628i	−0.496990	−	0.535628i	0.433884	−	0.900969i	$-0.357143\pi$
$859$	−0.496990	−	0.535628i	−0.496990	−	0.535628i	−0.930874	+	0.365341i	$0.880952\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$863$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.930874	−	0.365341i	−0.930874	−	0.365341i
$868$	0.559311	+	1.59842i	0.559311	+	1.59842i
$869$		0			0
$870$		0			0
$871$	1.59842	−	1.00435i	1.59842	−	1.00435i
$872$		0			0
$873$	0.180173	−	1.59908i	0.180173	−	1.59908i
$874$		0			0
$875$		0			0
$876$	−0.308658	+	0.882094i	−0.308658	+	0.882094i
$877$	0.369485	−	1.95278i	0.369485	−	1.95278i	0.0747301	−	0.997204i	$-0.476190\pi$
$877$	0.369485	−	1.95278i	0.369485	−	1.95278i	0.294755	−	0.955573i	$-0.404762\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$			0.149460i			0.149460i	0.997204	+	0.0747301i	$0.0238095\pi$
$883$			0.149460i			0.149460i	−0.997204	+	0.0747301i	$0.976190\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$887$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$888$		0			0
$889$	0.455573	+	0.571270i	0.455573	+	0.571270i
$890$		0			0
$891$		0			0
$892$	−0.694076	−	0.806531i	−0.694076	−	0.806531i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.500000	+	0.866025i	0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−0.233052	−	0.185853i	−0.233052	−	0.185853i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.53825	+	0.603718i	1.53825	+	0.603718i	0.974928	−	0.222521i	$-0.0714286\pi$
$907$	1.53825	+	0.603718i	1.53825	+	0.603718i	0.563320	+	0.826239i	$0.309524\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$911$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$912$	0.839789	−	0.839789i	0.839789	−	0.839789i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.733052	+	0.319827i	−0.733052	+	0.319827i
$917$		0			0
$918$		0			0
$919$	−1.46200	−	1.35654i	−1.46200	−	1.35654i	−0.781831	−	0.623490i	$-0.785714\pi$
$919$	−1.46200	−	1.35654i	−1.46200	−	1.35654i	−0.680173	−	0.733052i	$-0.738095\pi$
$920$		0			0
$921$	−0.369485	−	1.95278i	−0.369485	−	1.95278i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.0487796	+	0.0566829i	0.0487796	+	0.0566829i
$926$		0			0
$927$	0.440071	+	1.92808i	0.440071	+	1.92808i
$928$		0			0
$929$		0			0		0.399892	−	0.916562i	$-0.369048\pi$
$929$		0			0		−0.399892	+	0.916562i	$0.630952\pi$
$930$		0			0
$931$	−1.18681	−	0.0444073i	−1.18681	−	0.0444073i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.317031	−	0.658322i	−0.317031	−	0.658322i	0.680173	−	0.733052i	$-0.261905\pi$
$937$	−0.317031	−	0.658322i	−0.317031	−	0.658322i	−0.997204	+	0.0747301i	$0.976190\pi$
$938$		0			0
$939$	1.21135	−	0.825886i	1.21135	−	0.825886i
$940$		0			0
$941$		0			0		0.757972	−	0.652287i	$-0.226190\pi$
$941$		0			0		−0.757972	+	0.652287i	$0.773810\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$947$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$948$	−1.42935	−	1.32624i	−1.42935	−	1.32624i
$949$	0.173741	+	0.918245i	0.173741	+	0.918245i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$953$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.61753	−	0.933884i	1.61753	−	0.933884i
$962$		0			0
$963$		0			0
$964$	−0.623490	−	0.218169i	−0.623490	−	0.218169i
$965$		0			0
$966$		0			0
$967$	0.205245	+	1.82160i	0.205245	+	1.82160i	0.500000	+	0.866025i	$0.333333\pi$
$967$	0.205245	+	1.82160i	0.205245	+	1.82160i	−0.294755	+	0.955573i	$0.595238\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$971$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$972$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$973$	−0.0648483	+	0.865341i	−0.0648483	+	0.865341i
$974$		0			0
$975$	0.866025	+	0.500000i	0.866025	+	0.500000i
$976$	−0.563320	+	1.82624i	−0.563320	+	1.82624i
$977$		0			0		−0.757972	−	0.652287i	$-0.773810\pi$
$977$		0			0		0.757972	+	0.652287i	$0.226190\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.359154	+	0.0962349i	0.359154	+	0.0962349i
$982$		0			0
$983$		0			0		0.185912	−	0.982566i	$-0.440476\pi$
$983$		0			0		−0.185912	+	0.982566i	$0.559524\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.307384	−	1.14717i	0.307384	−	1.14717i
$989$		0			0
$990$		0			0
$991$	−1.95557	−	0.294755i	−1.95557	−	0.294755i	−0.955573	−	0.294755i	$-0.904762\pi$
$991$	−1.95557	−	0.294755i	−1.95557	−	0.294755i	−1.00000			$\pi$
$992$		0			0
$993$	−1.82160	+	0.205245i	−1.82160	+	0.205245i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.68862	+	0.385418i	1.68862	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$997$	1.68862	+	0.385418i	1.68862	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$998$		0			0
$999$	−0.0528791	+	0.0528791i	−0.0528791	+	0.0528791i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.1.ep.a.500.1	yes	24
3.2	odd	2	CM	1911.1.ep.a.500.1	yes	24
13.11	odd	12		1911.1.ea.a.206.1	✓	24
39.11	even	12		1911.1.ea.a.206.1	✓	24
49.5	odd	42		1911.1.ea.a.1475.1	yes	24
147.5	even	42		1911.1.ea.a.1475.1	yes	24
637.544	even	84	inner	1911.1.ep.a.1181.1	yes	24
1911.1181	odd	84	inner	1911.1.ep.a.1181.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1911.1.ea.a.206.1	✓	24	13.11	odd	12
1911.1.ea.a.206.1	✓	24	39.11	even	12
1911.1.ea.a.1475.1	yes	24	49.5	odd	42
1911.1.ea.a.1475.1	yes	24	147.5	even	42
1911.1.ep.a.500.1	yes	24	1.1	even	1	trivial
1911.1.ep.a.500.1	yes	24	3.2	odd	2	CM
1911.1.ep.a.1181.1	yes	24	637.544	even	84	inner
1911.1.ep.a.1181.1	yes	24	1911.1181	odd	84	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:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1911.ep (of order \(84\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(0.997204 - 0.0747301i) q^{3} +(-0.974928 + 0.222521i) q^{4} +(0.149042 - 0.988831i) q^{7} +(0.988831 - 0.149042i) q^{9} +O(q^{10})\)	\(q+(0.997204 - 0.0747301i) q^{3} +(-0.974928 + 0.222521i) q^{4} +(0.149042 - 0.988831i) q^{7} +(0.988831 - 0.149042i) q^{9} +(-0.955573 + 0.294755i) q^{12} +(-0.680173 + 0.733052i) q^{13} +(0.900969 - 0.433884i) q^{16} +(1.14717 - 0.307384i) q^{19} +(0.0747301 - 0.997204i) q^{21} +(-0.149042 - 0.988831i) q^{25} +(0.974928 - 0.222521i) q^{27} +(0.0747301 + 0.997204i) q^{28} +(1.63575 - 0.438297i) q^{31} +(-0.930874 + 0.365341i) q^{36} +(-0.0633201 + 0.0397866i) q^{37} +(-0.623490 + 0.781831i) q^{39} +(0.167917 - 0.246289i) q^{43} +(0.866025 - 0.500000i) q^{48} +(-0.955573 - 0.294755i) q^{49} +(0.500000 - 0.866025i) q^{52} +(1.12099 - 0.392253i) q^{57} +(-1.29991 + 1.40097i) q^{61} -1.00000i q^{63} +(-0.781831 + 0.623490i) q^{64} +(-1.82344 - 0.488590i) q^{67} +(0.554947 - 0.751927i) q^{73} +(-0.222521 - 0.974928i) q^{75} +(-1.05001 + 0.554947i) q^{76} +(0.974928 + 1.68862i) q^{79} +(0.955573 - 0.294755i) q^{81} +(0.149042 + 0.988831i) q^{84} +(0.623490 + 0.781831i) q^{91} +(1.59842 - 0.559311i) q^{93} +(0.416490 - 1.55436i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9}+O(q^{10}) \) 24 * q - 2 * q^9	\( 24 q - 2 q^{9} - 2 q^{12} + 4 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{28} + 2 q^{31} + 12 q^{37} + 4 q^{39} + 6 q^{43} - 2 q^{49} + 12 q^{52} + 2 q^{57} - 2 q^{67} - 2 q^{73} - 4 q^{75} - 4 q^{76} + 2 q^{81} - 4 q^{91} - 4 q^{93} - 2 q^{97}+O(q^{100}) \) 24 * q - 2 * q^9 - 2 * q^12 + 4 * q^16 + 2 * q^19 + 2 * q^21 + 2 * q^28 + 2 * q^31 + 12 * q^37 + 4 * q^39 + 6 * q^43 - 2 * q^49 + 12 * q^52 + 2 * q^57 - 2 * q^67 - 2 * q^73 - 4 * q^75 - 4 * q^76 + 2 * q^81 - 4 * q^91 - 4 * q^93 - 2 * q^97

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