Embedded newform 124.3.n.a.7.17

• Label: 124.3.n.a.7.17
• 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			7.17
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.595729 + 1.90922i) q^{2} +(-0.631742 - 2.97211i) q^{3} +(-3.29021 + 2.27475i) q^{4} +(1.43502 - 2.48553i) q^{5} +(5.29806 - 2.97670i) q^{6} +(3.93413 - 8.83620i) q^{7} +(-6.30307 - 4.92660i) q^{8} +(-0.212439 + 0.0945840i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.595729	+	1.90922i	0.297864	+	0.954608i
							\(3\)	−0.631742	−	2.97211i	−0.210581	−	0.990704i	−0.948735	−	0.316072i	\(-0.897636\pi\)
0.738155	−	0.674632i	\(-0.235698\pi\)
\(5\)	1.43502	−	2.48553i	0.287004	−	0.497105i	−0.686089	−	0.727517i	\(-0.740674\pi\)
							0.973093	+	0.230412i	\(0.0740074\pi\)
\(7\)	3.93413	−	8.83620i	0.562018	−	1.26231i	−0.379450	−	0.925212i	\(-0.623887\pi\)
							0.941468	−	0.337102i	\(-0.109447\pi\)
\(11\)	1.92885	−	0.202730i	0.175350	−	0.0184300i	−0.0164464	−	0.999865i	\(-0.505235\pi\)
							0.191797	+	0.981435i	\(0.438569\pi\)
\(13\)	5.52553	+	6.13673i	0.425041	+	0.472056i	0.917187	−	0.398457i	\(-0.130454\pi\)
							−0.492146	+	0.870513i	\(0.663787\pi\)
\(17\)	1.19586	−	11.3779i	0.0703450	−	0.669288i	−0.901357	−	0.433077i	\(-0.857428\pi\)
							0.971702	−	0.236211i	\(-0.0759055\pi\)
\(19\)	−2.13513	−	1.92248i	−0.112375	−	0.101183i	0.611028	−	0.791609i	\(-0.290756\pi\)
							−0.723403	+	0.690426i	\(0.757423\pi\)
\(23\)	−4.46861	+	6.15052i	−0.194288	+	0.267414i	−0.895035	−	0.445995i	\(-0.852850\pi\)
							0.700748	+	0.713409i	\(0.252850\pi\)
\(29\)	6.56240	+	20.1970i	0.226290	+	0.696448i	0.998158	+	0.0606655i	\(0.0193223\pi\)
							−0.771869	+	0.635782i	\(0.780678\pi\)
\(31\)	−30.5177	+	5.44709i	−0.984441	+	0.175713i
							\(37\)	9.89774	+	17.1434i	0.267506	+	0.463335i	0.968217	−	0.250111i	\(-0.0804670\pi\)
−0.700711	+	0.713445i	\(0.747134\pi\)
\(41\)	−30.3171	−	6.44411i	−0.739443	−	0.157173i	−0.177234	−	0.984169i	\(-0.556715\pi\)
							−0.562209	+	0.826995i	\(0.690048\pi\)
\(43\)	41.3316	+	37.2152i	0.961201	+	0.865469i	0.990969	−	0.134088i	\(-0.0428104\pi\)
							−0.0297686	+	0.999557i	\(0.509477\pi\)
\(47\)	−54.1269	−	17.5869i	−1.15164	−	0.374190i	−0.329877	−	0.944024i	\(-0.607007\pi\)
							−0.821760	+	0.569834i	\(0.807007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.17	✓	240
4.3	odd	2	inner	124.3.n.a.7.19	yes	240
31.9	even	15	inner	124.3.n.a.71.19	yes	240
124.71	odd	30	inner	124.3.n.a.71.17	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.17	✓	240	1.1	even	1	trivial
124.3.n.a.7.19	yes	240	4.3	odd	2	inner
124.3.n.a.71.17	yes	240	124.71	odd	30	inner
124.3.n.a.71.19	yes	240	31.9	even	15	inner