Embedded newform 124.3.n.a.7.20

• Label: 124.3.n.a.7.20
• Level: $124$
• Weight: $3$
• Character: 124.7
• Analytic conductor: $3.379$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.37875527807\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			7.20
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.826518 - 1.82123i) q^{2} +(0.272138 + 1.28031i) q^{3} +(-2.63374 - 3.01055i) q^{4} +(2.35737 - 4.08308i) q^{5} +(2.55666 + 0.562572i) q^{6} +(1.03353 - 2.32136i) q^{7} +(-7.65973 + 2.30836i) q^{8} +(6.65678 - 2.96379i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.826518	−	1.82123i	0.413259	−	0.910614i
							\(3\)	0.272138	+	1.28031i	0.0907125	+	0.426769i	0.999945	+	0.0104804i	\(0.00333607\pi\)
−0.909233	+	0.416289i	\(0.863331\pi\)
\(5\)	2.35737	−	4.08308i	0.471474	−	0.816616i	−0.527994	−	0.849248i	\(-0.677056\pi\)
							0.999467	+	0.0326320i	\(0.0103889\pi\)
\(7\)	1.03353	−	2.32136i	0.147648	−	0.331622i	−0.824548	−	0.565792i	\(-0.808570\pi\)
							0.972196	+	0.234170i	\(0.0752371\pi\)
\(11\)	−2.52616	+	0.265510i	−0.229651	+	0.0241373i	−0.218655	−	0.975802i	\(-0.570167\pi\)
							−0.0109961	+	0.999940i	\(0.503500\pi\)
\(13\)	−9.44014	−	10.4843i	−0.726165	−	0.806488i	0.261144	−	0.965300i	\(-0.415900\pi\)
							−0.987309	+	0.158812i	\(0.949234\pi\)
\(17\)	1.65211	−	15.7188i	0.0971830	−	0.924634i	−0.831941	−	0.554865i	\(-0.812770\pi\)
							0.929124	−	0.369769i	\(-0.120563\pi\)
\(19\)	25.0746	+	22.5773i	1.31972	+	1.18828i	0.967647	+	0.252306i	\(0.0811891\pi\)
							0.352071	+	0.935973i	\(0.385478\pi\)
\(23\)	−4.19670	+	5.77627i	−0.182465	+	0.251142i	−0.890445	−	0.455091i	\(-0.849607\pi\)
							0.707980	+	0.706233i	\(0.249607\pi\)
\(29\)	9.31810	+	28.6782i	0.321314	+	0.988902i	0.973077	+	0.230480i	\(0.0740296\pi\)
							−0.651763	+	0.758422i	\(0.725970\pi\)
\(31\)	−29.3237	+	10.0558i	−0.945927	+	0.324381i
							\(37\)	9.60359	+	16.6339i	0.259556	+	0.449565i	0.966123	−	0.258081i	\(-0.0830903\pi\)
−0.706567	+	0.707646i	\(0.749757\pi\)
\(41\)	41.4123	+	8.80247i	1.01006	+	0.214694i	0.683084	−	0.730339i	\(-0.260638\pi\)
							0.326973	+	0.945034i	\(0.393971\pi\)
\(43\)	−14.1821	−	12.7697i	−0.329817	−	0.296969i	0.487539	−	0.873101i	\(-0.337895\pi\)
							−0.817356	+	0.576132i	\(0.804561\pi\)
\(47\)	19.0062	+	6.17550i	0.404388	+	0.131394i	0.504147	−	0.863618i	\(-0.331807\pi\)
							−0.0997588	+	0.995012i	\(0.531807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.20	yes	240
4.3	odd	2	inner	124.3.n.a.7.6	✓	240
31.9	even	15	inner	124.3.n.a.71.6	yes	240
124.71	odd	30	inner	124.3.n.a.71.20	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.6	✓	240	4.3	odd	2	inner
124.3.n.a.7.20	yes	240	1.1	even	1	trivial
124.3.n.a.71.6	yes	240	31.9	even	15	inner
124.3.n.a.71.20	yes	240	124.71	odd	30	inner