Embedded newform 124.2.p.a.11.1

• Label: 124.2.p.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	124.11
Dual form			124.2.p.a.79.1

$q$-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 \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39926	+	0.205086i	−0.989429	+	0.145018i
							\(3\)	1.36017	+	1.51062i	0.785295	+	0.872158i	0.994394	−	0.105736i	\(-0.0337199\pi\)
−0.209099	+	0.977894i	\(0.567053\pi\)
\(5\)	0.360804	+	0.624930i	0.161356	+	0.279477i	0.935355	−	0.353709i	\(-0.115080\pi\)
							−0.773999	+	0.633187i	\(0.781746\pi\)
\(7\)	0.886730	−	0.0931991i	0.335152	−	0.0352259i	0.0645423	−	0.997915i	\(-0.479441\pi\)
							0.270610	+	0.962689i	\(0.412775\pi\)
\(11\)	−1.42417	−	0.634079i	−0.429402	−	0.191182i	0.180643	−	0.983549i	\(-0.442182\pi\)
							−0.610045	+	0.792367i	\(0.708849\pi\)
\(13\)	0.856313	+	4.02863i	0.237498	+	1.11734i	0.921656	+	0.388008i	\(0.126837\pi\)
							−0.684158	+	0.729334i	\(0.739830\pi\)
\(17\)	0.814056	+	1.82840i	0.197438	+	0.443452i	0.984948	−	0.172849i	\(-0.0552972\pi\)
							−0.787511	+	0.616301i	\(0.788630\pi\)
\(19\)	0.301308	−	1.41754i	0.0691248	−	0.325207i	−0.929975	−	0.367622i	\(-0.880172\pi\)
							0.999100	+	0.0424154i	\(0.0135053\pi\)
\(23\)	−4.76944	−	3.46520i	−0.994497	−	0.722544i	−0.0335956	−	0.999436i	\(-0.510696\pi\)
							−0.960901	+	0.276891i	\(0.910696\pi\)
\(29\)	−4.54048	+	1.47529i	−0.843146	+	0.273955i	−0.698573	−	0.715539i	\(-0.746181\pi\)
							−0.144574	+	0.989494i	\(0.546181\pi\)
\(31\)	−1.78020	−	5.27550i	−0.319733	−	0.947508i
							\(37\)	−9.05018	−	5.22512i	−1.48784	−	0.859005i	−0.487937	−	0.872879i	\(-0.662250\pi\)
−0.999904	+	0.0138738i	\(0.995584\pi\)
\(41\)	6.44773	−	7.16093i	1.00697	−	1.11835i	0.0140063	−	0.999902i	\(-0.495541\pi\)
							0.992960	−	0.118448i	\(-0.0377918\pi\)
\(43\)	10.1182	+	2.15069i	1.54301	+	0.327978i	0.899315	−	0.437301i	\(-0.144065\pi\)
							0.643699	+	0.765279i	\(0.277399\pi\)
\(47\)	4.63935	+	1.50742i	0.676719	+	0.219879i	0.627158	−	0.778892i	\(-0.284218\pi\)
							0.0495607	+	0.998771i	\(0.484218\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.p.a.11.1	✓	112	1.1	even	1	trivial
124.2.p.a.11.13	yes	112	4.3	odd	2	inner
124.2.p.a.79.1	yes	112	124.79	even	30	inner
124.2.p.a.79.13	yes	112	31.17	odd	30	inner