Embedded newform 124.2.p.a.79.1

• Label: 124.2.p.a.79.1
• 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			79.1
Character	\(\chi\)	\(=\)	124.79
Dual form			124.2.p.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39926 - 0.205086i) q^{2} +(1.36017 - 1.51062i) q^{3} +(1.91588 + 0.573940i) q^{4} +(0.360804 - 0.624930i) q^{5} +(-2.21305 + 1.83481i) q^{6} +(0.886730 + 0.0931991i) q^{7} +(-2.56311 - 1.19601i) q^{8} +(-0.118331 - 1.12584i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39926	−	0.205086i	−0.989429	−	0.145018i
							\(3\)	1.36017	−	1.51062i	0.785295	−	0.872158i	−0.209099	−	0.977894i	\(-0.567053\pi\)
0.994394	+	0.105736i	\(0.0337199\pi\)
\(5\)	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	0.270610	−	0.962689i	\(-0.412775\pi\)
							0.0645423	+	0.997915i	\(0.479441\pi\)
\(11\)	−1.42417	+	0.634079i	−0.429402	+	0.191182i	−0.610045	−	0.792367i	\(-0.708849\pi\)
							0.180643	+	0.983549i	\(0.442182\pi\)
\(13\)	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	0.999100	−	0.0424154i	\(-0.0135053\pi\)
							−0.929975	+	0.367622i	\(0.880172\pi\)
\(23\)	−4.76944	+	3.46520i	−0.994497	+	0.722544i	−0.960901	−	0.276891i	\(-0.910696\pi\)
							−0.0335956	+	0.999436i	\(0.510696\pi\)
\(29\)	−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	0.643699	−	0.765279i	\(-0.277399\pi\)
							0.899315	+	0.437301i	\(0.144065\pi\)
\(47\)	4.63935	−	1.50742i	0.676719	−	0.219879i	0.0495607	−	0.998771i	\(-0.484218\pi\)
							0.627158	+	0.778892i	\(0.284218\pi\)

(See \(a_n\) instead)

Twists

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

    

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