Embedded newform 124.4.p.a.3.13

• Label: 124.4.p.a.3.13
• Level: $124$
• Weight: $4$
• Character: 124.3
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			3.13
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81114 + 2.17250i) q^{2} +(0.804966 - 7.65874i) q^{3} +(-1.43954 - 7.86942i) q^{4} +(-9.96666 + 17.2628i) q^{5} +(15.1807 + 15.6198i) q^{6} +(-3.05831 - 14.3882i) q^{7} +(19.7036 + 11.1252i) q^{8} +(-31.5984 - 6.71644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81114	+	2.17250i	−0.640335	+	0.768096i
							\(3\)	0.804966	−	7.65874i	0.154916	−	1.47393i	−0.590344	−	0.807152i	\(-0.701008\pi\)
0.745260	−	0.666774i	\(-0.232325\pi\)
\(5\)	−9.96666	+	17.2628i	−0.891445	+	1.54403i	−0.0533018	+	0.998578i	\(0.516975\pi\)
							−0.838143	+	0.545450i	\(0.816359\pi\)
\(7\)	−3.05831	−	14.3882i	−0.165133	−	0.776890i	−0.980277	−	0.197631i	\(-0.936675\pi\)
							0.815144	−	0.579259i	\(-0.196658\pi\)
\(11\)	−7.09113	−	7.87550i	−0.194369	−	0.215868i	0.638081	−	0.769970i	\(-0.279729\pi\)
							−0.832449	+	0.554101i	\(0.813062\pi\)
\(13\)	25.0832	+	56.3379i	0.535141	+	1.20195i	0.955609	+	0.294639i	\(0.0951995\pi\)
							−0.420467	+	0.907308i	\(0.638134\pi\)
\(17\)	58.5553	+	52.7234i	0.835397	+	0.752195i	0.971130	−	0.238551i	\(-0.0766723\pi\)
							−0.135733	+	0.990745i	\(0.543339\pi\)
\(19\)	−48.3221	+	108.533i	−0.583466	+	1.31049i	0.344838	+	0.938662i	\(0.387934\pi\)
							−0.928304	+	0.371823i	\(0.878733\pi\)
\(23\)	33.8758	+	104.259i	0.307113	+	0.945196i	0.978880	+	0.204434i	\(0.0655354\pi\)
							−0.671768	+	0.740762i	\(0.734465\pi\)
\(29\)	45.8323	+	63.0828i	0.293478	+	0.403937i	0.930140	−	0.367206i	\(-0.119685\pi\)
							−0.636662	+	0.771143i	\(0.719685\pi\)
\(31\)	−78.3887	+	153.773i	−0.454162	+	0.890919i
							\(37\)	−38.6658	+	22.3237i	−0.171800	+	0.0991890i	−0.583435	−	0.812160i	\(-0.698292\pi\)
0.411634	+	0.911349i	\(0.364958\pi\)
\(41\)	16.9355	+	161.130i	0.0645093	+	0.613765i	0.978245	+	0.207455i	\(0.0665180\pi\)
							−0.913735	+	0.406310i	\(0.866815\pi\)
\(43\)	−387.092	−	172.344i	−1.37281	−	0.611215i	−0.418006	−	0.908444i	\(-0.637271\pi\)
							−0.954806	+	0.297229i	\(0.903937\pi\)
\(47\)	91.2944	−	125.656i	0.283333	−	0.389974i	−0.643501	−	0.765445i	\(-0.722519\pi\)
							0.926834	+	0.375471i	\(0.122519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.13	yes	368
4.3	odd	2	inner	124.4.p.a.3.6	✓	368
31.21	odd	30	inner	124.4.p.a.83.6	yes	368
124.83	even	30	inner	124.4.p.a.83.13	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.6	✓	368	4.3	odd	2	inner
124.4.p.a.3.13	yes	368	1.1	even	1	trivial
124.4.p.a.83.6	yes	368	31.21	odd	30	inner
124.4.p.a.83.13	yes	368	124.83	even	30	inner