Embedded newform 116.2.l.b.19.8

• Label: 116.2.l.b.19.8
• 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			19.8
Character	\(\chi\)	\(=\)	116.19
Dual form			116.2.l.b.55.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.692950 + 1.23281i) q^{2} +(1.24636 + 0.436119i) q^{3} +(-1.03964 + 1.70855i) q^{4} +(1.45778 + 0.332728i) q^{5} +(0.326011 + 1.83873i) q^{6} +(-2.16342 - 4.49238i) q^{7} +(-2.82674 - 0.0977383i) q^{8} +(-0.982291 - 0.783351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.692950	+	1.23281i	0.489990	+	0.871728i
							\(3\)	1.24636	+	0.436119i	0.719584	+	0.251793i	0.665135	−	0.746723i	\(-0.268374\pi\)
0.0544491	+	0.998517i	\(0.482660\pi\)
\(5\)	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	\(-0.751519\pi\)
							−0.107221	−	0.994235i	\(-0.534195\pi\)
\(11\)	−0.429829	+	3.81484i	−0.129598	+	1.15022i	0.745580	+	0.666416i	\(0.232172\pi\)
							−0.875178	+	0.483800i	\(0.839256\pi\)
\(13\)	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	\(0.0882296\pi\)
							−0.581374	+	0.813636i	\(0.697485\pi\)
\(23\)	−3.02104	+	0.689533i	−0.629931	+	0.143778i	−0.525554	−	0.850760i	\(-0.676142\pi\)
							−0.104377	+	0.994538i	\(0.533285\pi\)
\(29\)	3.73708	+	3.87740i	0.693959	+	0.720015i
							\(31\)	4.84391	−	3.04363i	0.869992	−	0.546652i	−0.0214587	−	0.999770i	\(-0.506831\pi\)
0.891451	+	0.453117i	\(0.149688\pi\)
\(37\)	−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	\(0.449410\pi\)
							−0.958282	+	0.285823i	\(0.907733\pi\)
\(47\)	7.11425	+	0.801583i	1.03772	+	0.116923i	0.614350	−	0.789034i	\(-0.289418\pi\)
							0.423370	+	0.905957i	\(0.360847\pi\)

(See \(a_n\) instead)

Twists

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

    

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