Embedded newform 116.2.l.b.27.7

• Label: 116.2.l.b.27.7
• Level: $116$
• Weight: $2$
• Character: 116.27
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.l (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			27.7
Character	\(\chi\)	\(=\)	116.27
Dual form			116.2.l.b.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.285397 + 1.38512i) q^{2} +(-2.07960 - 1.30670i) q^{3} +(-1.83710 + 0.790618i) q^{4} +(-2.72115 + 2.17005i) q^{5} +(1.21642 - 3.25342i) q^{6} +(0.190870 - 0.0435647i) q^{7} +(-1.61940 - 2.31895i) q^{8} +(1.31562 + 2.73192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{28}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.285397	+	1.38512i	0.201807	+	0.979425i
							\(3\)	−2.07960	−	1.30670i	−1.20066	−	0.754423i	−0.225136	−	0.974327i	\(-0.572283\pi\)
−0.975521	+	0.219904i	\(0.929426\pi\)
\(5\)	−2.72115	+	2.17005i	−1.21694	+	0.970475i	−0.999984	−	0.00570669i	\(-0.998183\pi\)
							−0.216954	+	0.976182i	\(0.569612\pi\)
\(7\)	0.190870	−	0.0435647i	0.0721419	−	0.0164659i	−0.186298	−	0.982493i	\(-0.559649\pi\)
							0.258440	+	0.966027i	\(0.416792\pi\)
\(11\)	−4.11702	+	1.44061i	−1.24133	+	0.434359i	−0.869496	−	0.493941i	\(-0.835556\pi\)
							−0.371832	+	0.928300i	\(0.621270\pi\)
\(13\)	0.215721	−	0.447950i	0.0598303	−	0.124239i	−0.868910	−	0.494971i	\(-0.835179\pi\)
							0.928740	+	0.370732i	\(0.120893\pi\)
\(17\)	3.07771	+	3.07771i	0.746455	+	0.746455i	0.973812	−	0.227357i	\(-0.0730083\pi\)
							−0.227357	+	0.973812i	\(0.573008\pi\)
\(19\)	3.92373	+	6.24458i	0.900165	+	1.43260i	0.901583	+	0.432606i	\(0.142406\pi\)
							−0.00141814	+	0.999999i	\(0.500451\pi\)
\(23\)	−3.53508	−	2.81913i	−0.737114	−	0.587829i	0.181310	−	0.983426i	\(-0.441966\pi\)
							−0.918424	+	0.395597i	\(0.870538\pi\)
\(29\)	−4.83806	−	2.36499i	−0.898406	−	0.439167i
							\(31\)	−5.78825	−	0.652179i	−1.03960	−	0.117135i	−0.424375	−	0.905486i	\(-0.639506\pi\)
−0.615225	+	0.788351i	\(0.710935\pi\)
\(37\)	1.88690	−	5.39245i	0.310204	−	0.886513i	−0.678228	−	0.734851i	\(-0.737252\pi\)
							0.988433	−	0.151661i	\(-0.0484623\pi\)
\(41\)	−4.93420	+	4.93420i	−0.770592	+	0.770592i	−0.978210	−	0.207618i	\(-0.933429\pi\)
							0.207618	+	0.978210i	\(0.433429\pi\)
\(43\)	0.162068	+	1.43840i	0.0247152	+	0.219353i	0.999999	−	0.00160370i	\(-0.000510472\pi\)
							−0.975284	+	0.220957i	\(0.929082\pi\)
\(47\)	1.36704	+	3.90678i	0.199403	+	0.569862i	0.999552	−	0.0299399i	\(-0.00953160\pi\)
							−0.800148	+	0.599802i	\(0.795246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.27.7	yes	144
4.3	odd	2	inner	116.2.l.b.27.6	✓	144
29.14	odd	28	inner	116.2.l.b.43.6	yes	144
116.43	even	28	inner	116.2.l.b.43.7	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.27.6	✓	144	4.3	odd	2	inner
116.2.l.b.27.7	yes	144	1.1	even	1	trivial
116.2.l.b.43.6	yes	144	29.14	odd	28	inner
116.2.l.b.43.7	yes	144	116.43	even	28	inner