Embedded newform 116.2.l.b.19.9

• Label: 116.2.l.b.19.9
• Level: $116$
• Weight: $2$
• Character: 116.19
• 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			19.9
Character	\(\chi\)	\(=\)	116.19
Dual form			116.2.l.b.55.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.810064 + 1.15922i) q^{2} +(-1.24636 - 0.436119i) q^{3} +(-0.687594 + 1.87809i) q^{4} +(1.45778 + 0.332728i) q^{5} +(-0.504069 - 1.79809i) q^{6} +(2.16342 + 4.49238i) q^{7} +(-2.73412 + 0.724296i) q^{8} +(-0.982291 - 0.783351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.810064	+	1.15922i	0.572801	+	0.819694i
							\(3\)	−1.24636	−	0.436119i	−0.719584	−	0.251793i	−0.0544491	−	0.998517i	\(-0.517340\pi\)
−0.665135	+	0.746723i	\(0.731626\pi\)
\(5\)	1.45778	+	0.332728i	0.651937	+	0.148800i	0.535683	−	0.844419i	\(-0.320054\pi\)
							0.116254	+	0.993219i	\(0.462911\pi\)
\(7\)	2.16342	+	4.49238i	0.817694	+	1.69796i	0.710473	+	0.703724i	\(0.248481\pi\)
							0.107221	+	0.994235i	\(0.465805\pi\)
\(11\)	0.429829	−	3.81484i	0.129598	−	1.15022i	−0.745580	−	0.666416i	\(-0.767828\pi\)
							0.875178	−	0.483800i	\(-0.160744\pi\)
\(13\)	0.621110	−	0.495319i	0.172265	−	0.137377i	−0.533562	−	0.845761i	\(-0.679147\pi\)
							0.705827	+	0.708384i	\(0.250576\pi\)
\(17\)	2.22518	−	2.22518i	0.539684	−	0.539684i	−0.383752	−	0.923436i	\(-0.625368\pi\)
							0.923436	+	0.383752i	\(0.125368\pi\)
\(19\)	−1.65837	−	4.73935i	−0.380456	−	1.08728i	−0.961831	−	0.273646i	\(-0.911770\pi\)
							0.581374	−	0.813636i	\(-0.302515\pi\)
\(23\)	3.02104	−	0.689533i	0.629931	−	0.143778i	0.104377	−	0.994538i	\(-0.466715\pi\)
							0.525554	+	0.850760i	\(0.323858\pi\)
\(29\)	3.73708	+	3.87740i	0.693959	+	0.720015i
							\(31\)	−4.84391	+	3.04363i	−0.869992	+	0.546652i	−0.891451	−	0.453117i	\(-0.850312\pi\)
0.0214587	+	0.999770i	\(0.493169\pi\)
\(37\)	−2.57644	+	0.290295i	−0.423564	+	0.0477242i	−0.321174	−	0.947020i	\(-0.604078\pi\)
							−0.102390	+	0.994744i	\(0.532649\pi\)
\(41\)	0.977383	+	0.977383i	0.152642	+	0.152642i	0.779297	−	0.626655i	\(-0.215577\pi\)
							−0.626655	+	0.779297i	\(0.715577\pi\)
\(43\)	5.24606	−	8.34906i	0.800017	−	1.27322i	−0.158266	−	0.987397i	\(-0.550590\pi\)
							0.958282	−	0.285823i	\(-0.0922669\pi\)
\(47\)	−7.11425	−	0.801583i	−1.03772	−	0.116923i	−0.423370	−	0.905957i	\(-0.639153\pi\)
							−0.614350	+	0.789034i	\(0.710582\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.19.9	yes	144
4.3	odd	2	inner	116.2.l.b.19.8	✓	144
29.26	odd	28	inner	116.2.l.b.55.8	yes	144
116.55	even	28	inner	116.2.l.b.55.9	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.19.8	✓	144	4.3	odd	2	inner
116.2.l.b.19.9	yes	144	1.1	even	1	trivial
116.2.l.b.55.8	yes	144	29.26	odd	28	inner
116.2.l.b.55.9	yes	144	116.55	even	28	inner