Embedded newform 116.2.l.b.27.5

• Label: 116.2.l.b.27.5
• 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			27.5
Character	\(\chi\)	\(=\)	116.27
Dual form			116.2.l.b.43.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.651028 - 1.25545i) q^{2} +(2.37464 + 1.49208i) q^{3} +(-1.15232 + 1.63467i) q^{4} +(-0.476771 + 0.380213i) q^{5} +(0.327284 - 3.95264i) q^{6} +(0.885852 - 0.202190i) q^{7} +(2.80245 + 0.382471i) q^{8} +(2.11094 + 4.38342i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.651028	−	1.25545i	−0.460347	−	0.887739i
							\(3\)	2.37464	+	1.49208i	1.37100	+	0.861455i	0.997800	−	0.0662964i	\(-0.0211183\pi\)
0.373198	+	0.927752i	\(0.378261\pi\)
\(5\)	−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	−0.0518054	−	0.998657i	\(-0.516498\pi\)
							0.386626	+	0.922237i	\(0.373640\pi\)
\(11\)	−3.90678	+	1.36704i	−1.17794	+	0.412179i	−0.847062	−	0.531495i	\(-0.821631\pi\)
							−0.330878	+	0.943674i	\(0.607345\pi\)
\(13\)	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	0.832097	−	0.554630i	\(-0.187140\pi\)
							−0.860738	+	0.509048i	\(0.829997\pi\)
\(23\)	−5.07818	−	4.04972i	−1.05887	−	0.844424i	−0.0706601	−	0.997500i	\(-0.522511\pi\)
							−0.988214	+	0.153076i	\(0.951082\pi\)
\(29\)	1.53869	+	5.16066i	0.285728	+	0.958311i
							\(31\)	−0.697955	−	0.0786406i	−0.125356	−	0.0141243i	0.0490634	−	0.998796i	\(-0.484376\pi\)
−0.174420	+	0.984671i	\(0.555805\pi\)
\(37\)	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	\(0.212584\pi\)
							−0.627660	+	0.778487i	\(0.715987\pi\)
\(47\)	−0.853124	−	2.43809i	−0.124441	−	0.355632i	0.864762	−	0.502181i	\(-0.167469\pi\)
							−0.989203	+	0.146549i	\(0.953183\pi\)

(See \(a_n\) instead)

Twists

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

    

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