Embedded newform 148.2.l.b.119.10

• Label: 148.2.l.b.119.10
• Level: $148$
• Weight: $2$
• Character: 148.119
• Analytic conductor: $1.182$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 148 = 2^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	148.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.18178594991\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			119.10
Character	\(\chi\)	\(=\)	148.119
Dual form			148.2.l.b.51.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.296252 - 1.38284i) q^{2} +(1.14905 - 1.99021i) q^{3} +(-1.82447 - 0.819337i) q^{4} +(-1.56548 + 0.419468i) q^{5} +(-2.41173 - 2.17855i) q^{6} +(-0.420905 - 0.243010i) q^{7} +(-1.67351 + 2.28021i) q^{8} +(-1.14062 - 1.97562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{2} - 4 q^{5} + 8 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/148\mathbb{Z}\right)^\times\).

\(n\)	\(75\)	\(113\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.296252	−	1.38284i	0.209482	−	0.977812i
							\(3\)	1.14905	−	1.99021i	0.663403	−	1.14905i	−0.316312	−	0.948655i	\(-0.602445\pi\)
0.979716	−	0.200393i	\(-0.0642220\pi\)
\(5\)	−1.56548	+	0.419468i	−0.700103	+	0.187592i	−0.591277	−	0.806469i	\(-0.701376\pi\)
							−0.108826	+	0.994061i	\(0.534709\pi\)
\(7\)	−0.420905	−	0.243010i	−0.159087	−	0.0918491i	0.418343	−	0.908289i	\(-0.362611\pi\)
							−0.577430	+	0.816440i	\(0.695944\pi\)
\(11\)	5.48520			1.65385			0.826926	−	0.562311i	\(-0.190088\pi\)
							0.826926	+	0.562311i	\(0.190088\pi\)
\(13\)	1.44110	−	0.386140i	0.399688	−	0.107096i	−0.0533751	−	0.998575i	\(-0.516998\pi\)
							0.453063	+	0.891478i	\(0.350331\pi\)
\(17\)	−0.0808954	+	0.301906i	−0.0196200	+	0.0732229i	−0.975042	−	0.222022i	\(-0.928734\pi\)
							0.955422	+	0.295245i	\(0.0954011\pi\)
\(19\)	−1.05445	−	3.93525i	−0.241907	−	0.902808i	−0.974913	−	0.222586i	\(-0.928550\pi\)
							0.733006	−	0.680222i	\(-0.238117\pi\)
\(23\)	1.26103	−	1.26103i	0.262943	−	0.262943i	−0.563306	−	0.826248i	\(-0.690471\pi\)
							0.826248	+	0.563306i	\(0.190471\pi\)
\(29\)	−5.30845	+	5.30845i	−0.985754	+	0.985754i	−0.999900	−	0.0141462i	\(-0.995497\pi\)
							0.0141462	+	0.999900i	\(0.495497\pi\)
\(31\)	−3.17364	−	3.17364i	−0.570003	−	0.570003i	0.362126	−	0.932129i	\(-0.382051\pi\)
							−0.932129	+	0.362126i	\(0.882051\pi\)
\(37\)	2.97868	+	5.30353i	0.489692	+	0.871895i
							\(41\)	9.47668	+	5.47137i	1.48001	+	0.854484i	0.999744	−	0.0226395i	\(-0.00720699\pi\)
0.480265	+	0.877123i	\(0.340540\pi\)
\(43\)	−6.00290	+	6.00290i	−0.915434	+	0.915434i	−0.996693	−	0.0812591i	\(-0.974106\pi\)
							0.0812591	+	0.996693i	\(0.474106\pi\)
\(47\)			8.15908i			1.19012i	0.803680	+	0.595062i	\(0.202873\pi\)
							−0.803680	+	0.595062i	\(0.797127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.l.b.119.10	yes	64
4.3	odd	2	inner	148.2.l.b.119.8	yes	64
37.14	odd	12	inner	148.2.l.b.51.8	✓	64
148.51	even	12	inner	148.2.l.b.51.10	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.l.b.51.8	✓	64	37.14	odd	12	inner
148.2.l.b.51.10	yes	64	148.51	even	12	inner
148.2.l.b.119.8	yes	64	4.3	odd	2	inner
148.2.l.b.119.10	yes	64	1.1	even	1	trivial