Embedded newform 124.2.p.a.79.13

• Label: 124.2.p.a.79.13
• Level: $124$
• Weight: $2$
• Character: 124.79
• Analytic conductor: $0.990$
• Analytic rank: $0$
• Dimension: $112$
• 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			79.13
Character	\(\chi\)	\(=\)	124.79
Dual form			124.2.p.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25257 + 0.656548i) q^{2} +(-1.36017 + 1.51062i) q^{3} +(1.13789 + 1.64475i) q^{4} +(0.360804 - 0.624930i) q^{5} +(-2.69551 + 0.999150i) q^{6} +(-0.886730 - 0.0931991i) q^{7} +(0.345431 + 2.80725i) q^{8} +(-0.118331 - 1.12584i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.25257	+	0.656548i	0.885704	+	0.464250i
							\(3\)	−1.36017	+	1.51062i	−0.785295	+	0.872158i	−0.994394	−	0.105736i	\(-0.966280\pi\)
0.209099	+	0.977894i	\(0.432947\pi\)
\(5\)	0.360804	−	0.624930i	0.161356	−	0.279477i	−0.773999	−	0.633187i	\(-0.781746\pi\)
							0.935355	+	0.353709i	\(0.115080\pi\)
\(7\)	−0.886730	−	0.0931991i	−0.335152	−	0.0352259i	−0.0645423	−	0.997915i	\(-0.520559\pi\)
							−0.270610	+	0.962689i	\(0.587225\pi\)
\(11\)	1.42417	−	0.634079i	0.429402	−	0.191182i	−0.180643	−	0.983549i	\(-0.557818\pi\)
							0.610045	+	0.792367i	\(0.291151\pi\)
\(13\)	0.856313	−	4.02863i	0.237498	−	1.11734i	−0.684158	−	0.729334i	\(-0.739830\pi\)
							0.921656	−	0.388008i	\(-0.126837\pi\)
\(17\)	0.814056	−	1.82840i	0.197438	−	0.443452i	−0.787511	−	0.616301i	\(-0.788630\pi\)
							0.984948	+	0.172849i	\(0.0552972\pi\)
\(19\)	−0.301308	−	1.41754i	−0.0691248	−	0.325207i	0.929975	−	0.367622i	\(-0.119828\pi\)
							−0.999100	+	0.0424154i	\(0.986495\pi\)
\(23\)	4.76944	−	3.46520i	0.994497	−	0.722544i	0.0335956	−	0.999436i	\(-0.489304\pi\)
							0.960901	+	0.276891i	\(0.0893042\pi\)
\(29\)	−4.54048	−	1.47529i	−0.843146	−	0.273955i	−0.144574	−	0.989494i	\(-0.546181\pi\)
							−0.698573	+	0.715539i	\(0.746181\pi\)
\(31\)	1.78020	−	5.27550i	0.319733	−	0.947508i
							\(37\)	−9.05018	+	5.22512i	−1.48784	+	0.859005i	−0.999904	−	0.0138738i	\(-0.995584\pi\)
−0.487937	+	0.872879i	\(0.662250\pi\)
\(41\)	6.44773	+	7.16093i	1.00697	+	1.11835i	0.992960	+	0.118448i	\(0.0377918\pi\)
							0.0140063	+	0.999902i	\(0.495541\pi\)
\(43\)	−10.1182	+	2.15069i	−1.54301	+	0.327978i	−0.899315	−	0.437301i	\(-0.855935\pi\)
							−0.643699	+	0.765279i	\(0.722601\pi\)
\(47\)	−4.63935	+	1.50742i	−0.676719	+	0.219879i	−0.627158	−	0.778892i	\(-0.715782\pi\)
							−0.0495607	+	0.998771i	\(0.515782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.2.p.a.79.13	yes	112
4.3	odd	2	inner	124.2.p.a.79.1	yes	112
31.11	odd	30	inner	124.2.p.a.11.1	✓	112
124.11	even	30	inner	124.2.p.a.11.13	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.p.a.11.1	✓	112	31.11	odd	30	inner
124.2.p.a.11.13	yes	112	124.11	even	30	inner
124.2.p.a.79.1	yes	112	4.3	odd	2	inner
124.2.p.a.79.13	yes	112	1.1	even	1	trivial