Embedded newform 116.2.l.b.27.9

• Label: 116.2.l.b.27.9
• 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			27.9
Character	\(\chi\)	\(=\)	116.27
Dual form			116.2.l.b.43.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.914070 - 1.07911i) q^{2} +(-2.37464 - 1.49208i) q^{3} +(-0.328951 - 1.97276i) q^{4} +(-0.476771 + 0.380213i) q^{5} +(-3.78071 + 1.19862i) q^{6} +(-0.885852 + 0.202190i) q^{7} +(-2.42951 - 1.44827i) q^{8} +(2.11094 + 4.38342i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.914070	−	1.07911i	0.646345	−	0.763045i
							\(3\)	−2.37464	−	1.49208i	−1.37100	−	0.861455i	−0.373198	−	0.927752i	\(-0.621739\pi\)
−0.997800	+	0.0662964i	\(0.978882\pi\)
\(5\)	−0.476771	+	0.380213i	−0.213219	+	0.170036i	−0.724275	−	0.689511i	\(-0.757826\pi\)
							0.511057	+	0.859547i	\(0.329254\pi\)
\(7\)	−0.885852	+	0.202190i	−0.334821	+	0.0764207i	−0.386626	−	0.922237i	\(-0.626360\pi\)
							0.0518054	+	0.998657i	\(0.483502\pi\)
\(11\)	3.90678	−	1.36704i	1.17794	−	0.412179i	0.330878	−	0.943674i	\(-0.392655\pi\)
							0.847062	+	0.531495i	\(0.178369\pi\)
\(13\)	2.90115	−	6.02429i	0.804633	−	1.67084i	0.0649435	−	0.997889i	\(-0.479313\pi\)
							0.739689	−	0.672948i	\(-0.234972\pi\)
\(17\)	−2.05404	−	2.05404i	−0.498178	−	0.498178i	0.412692	−	0.910871i	\(-0.364589\pi\)
							−0.910871	+	0.412692i	\(0.864589\pi\)
\(19\)	0.124844	+	0.198688i	0.0286411	+	0.0455821i	0.860738	−	0.509048i	\(-0.170003\pi\)
							−0.832097	+	0.554630i	\(0.812860\pi\)
\(23\)	5.07818	+	4.04972i	1.05887	+	0.844424i	0.988214	−	0.153076i	\(-0.0489180\pi\)
							0.0706601	+	0.997500i	\(0.477489\pi\)
\(29\)	1.53869	+	5.16066i	0.285728	+	0.958311i
							\(31\)	0.697955	+	0.0786406i	0.125356	+	0.0141243i	0.174420	−	0.984671i	\(-0.444195\pi\)
−0.0490634	+	0.998796i	\(0.515624\pi\)
\(37\)	0.954745	−	2.72851i	0.156959	−	0.448564i	−0.838370	−	0.545101i	\(-0.816491\pi\)
							0.995329	+	0.0965378i	\(0.0307769\pi\)
\(41\)	2.37072	−	2.37072i	0.370244	−	0.370244i	−0.497322	−	0.867566i	\(-0.665683\pi\)
							0.867566	+	0.497322i	\(0.165683\pi\)
\(43\)	−1.03275	−	9.16591i	−0.157493	−	1.39779i	−0.785153	−	0.619301i	\(-0.787416\pi\)
							0.627660	−	0.778487i	\(-0.284013\pi\)
\(47\)	0.853124	+	2.43809i	0.124441	+	0.355632i	0.989203	−	0.146549i	\(-0.0468167\pi\)
							−0.864762	+	0.502181i	\(0.832531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.27.9	yes	144
4.3	odd	2	inner	116.2.l.b.27.5	✓	144
29.14	odd	28	inner	116.2.l.b.43.5	yes	144
116.43	even	28	inner	116.2.l.b.43.9	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.27.5	✓	144	4.3	odd	2	inner
116.2.l.b.27.9	yes	144	1.1	even	1	trivial
116.2.l.b.43.5	yes	144	29.14	odd	28	inner
116.2.l.b.43.9	yes	144	116.43	even	28	inner