Embedded newform 124.2.p.a.11.5

• Label: 124.2.p.a.11.5
• Level: $124$
• Weight: $2$
• Character: 124.11
• Analytic conductor: $0.990$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.990144985064\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	124.11
Dual form			124.2.p.a.79.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.683091 - 1.23830i) q^{2} +(2.11915 + 2.35355i) q^{3} +(-1.06677 + 1.69174i) q^{4} +(-0.103778 - 0.179749i) q^{5} +(1.46683 - 4.23183i) q^{6} +(-1.63343 + 0.171681i) q^{7} +(2.82359 + 0.165369i) q^{8} +(-0.734833 + 6.99146i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.683091	−	1.23830i	−0.483018	−	0.875610i
							\(3\)	2.11915	+	2.35355i	1.22349	+	1.35882i	0.912850	+	0.408295i	\(0.133876\pi\)
0.310639	+	0.950528i	\(0.399457\pi\)
\(5\)	−0.103778	−	0.179749i	−0.0464109	−	0.0803861i	0.841887	−	0.539654i	\(-0.181445\pi\)
							−0.888298	+	0.459268i	\(0.848112\pi\)
\(7\)	−1.63343	+	0.171681i	−0.617380	+	0.0648892i	−0.408056	−	0.912957i	\(-0.633793\pi\)
							−0.209324	+	0.977846i	\(0.567126\pi\)
\(11\)	3.58117	+	1.59444i	1.07976	+	0.480742i	0.867992	−	0.496578i	\(-0.165410\pi\)
							0.211771	+	0.977319i	\(0.432077\pi\)
\(13\)	−1.20642	−	5.67576i	−0.334601	−	1.57417i	−0.748048	−	0.663645i	\(-0.769008\pi\)
							0.413447	−	0.910528i	\(-0.364325\pi\)
\(17\)	−0.938658	−	2.10826i	−0.227658	−	0.511328i	0.763212	−	0.646148i	\(-0.223621\pi\)
							−0.990870	+	0.134820i	\(0.956954\pi\)
\(19\)	0.598618	−	2.81628i	0.137332	−	0.646098i	−0.854595	−	0.519294i	\(-0.826195\pi\)
							0.991928	−	0.126804i	\(-0.0404718\pi\)
\(23\)	0.893069	+	0.648852i	0.186218	+	0.135295i	0.676988	−	0.735994i	\(-0.263285\pi\)
							−0.490771	+	0.871289i	\(0.663285\pi\)
\(29\)	−4.16589	+	1.35358i	−0.773587	+	0.251354i	−0.669100	−	0.743173i	\(-0.733320\pi\)
							−0.104487	+	0.994526i	\(0.533320\pi\)
\(31\)	−4.94053	+	2.56732i	−0.887346	+	0.461105i
							\(37\)	3.97640	+	2.29578i	0.653716	+	0.377423i	0.789879	−	0.613263i	\(-0.210144\pi\)
−0.136162	+	0.990687i	\(0.543477\pi\)
\(41\)	4.36364	−	4.84632i	0.681487	−	0.756867i	−0.298829	−	0.954307i	\(-0.596596\pi\)
							0.980315	+	0.197439i	\(0.0632626\pi\)
\(43\)	−8.44441	−	1.79491i	−1.28776	−	0.273722i	−0.487367	−	0.873197i	\(-0.662043\pi\)
							−0.800393	+	0.599475i	\(0.795376\pi\)
\(47\)	4.42905	+	1.43909i	0.646044	+	0.209912i	0.613669	−	0.789563i	\(-0.289693\pi\)
							0.0323748	+	0.999476i	\(0.489693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.2.p.a.11.5	✓	112
4.3	odd	2	inner	124.2.p.a.11.7	yes	112
31.17	odd	30	inner	124.2.p.a.79.7	yes	112
124.79	even	30	inner	124.2.p.a.79.5	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.p.a.11.5	✓	112	1.1	even	1	trivial
124.2.p.a.11.7	yes	112	4.3	odd	2	inner
124.2.p.a.79.5	yes	112	124.79	even	30	inner
124.2.p.a.79.7	yes	112	31.17	odd	30	inner