Embedded newform 148.2.l.b.103.16

• Label: 148.2.l.b.103.16
• Level: $148$
• Weight: $2$
• Character: 148.103
• 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			103.16
Character	\(\chi\)	\(=\)	148.103
Dual form			148.2.l.b.23.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41173 - 0.0837117i) q^{2} +(-0.854373 + 1.47982i) q^{3} +(1.98598 - 0.236357i) q^{4} +(0.704542 + 2.62939i) q^{5} +(-1.08227 + 2.16063i) q^{6} +(-4.12904 - 2.38390i) q^{7} +(2.78390 - 0.499924i) q^{8} +(0.0400922 + 0.0694418i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{2} - 4 q^{5} + 8 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(75\)	\(113\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41173	−	0.0837117i	0.998247	−	0.0591931i
							\(3\)	−0.854373	+	1.47982i	−0.493273	+	0.854373i	−0.999970	−	0.00775067i	\(-0.997533\pi\)
0.506697	+	0.862124i	\(0.330866\pi\)
\(5\)	0.704542	+	2.62939i	0.315081	+	1.17590i	0.923914	+	0.382600i	\(0.124971\pi\)
							−0.608833	+	0.793298i	\(0.708362\pi\)
\(7\)	−4.12904	−	2.38390i	−1.56063	−	0.901030i	−0.997193	−	0.0748711i	\(-0.976145\pi\)
							−0.563437	−	0.826159i	\(-0.690521\pi\)
\(11\)	3.94243			1.18869			0.594344	−	0.804211i	\(-0.297412\pi\)
							0.594344	+	0.804211i	\(0.297412\pi\)
\(13\)	−1.57873	−	5.89192i	−0.437862	−	1.63412i	−0.734123	−	0.679017i	\(-0.762406\pi\)
							0.296261	−	0.955107i	\(-0.404260\pi\)
\(17\)	−0.620874	−	0.166363i	−0.150584	−	0.0403489i	0.182739	−	0.983161i	\(-0.441504\pi\)
							−0.333324	+	0.942812i	\(0.608170\pi\)
\(19\)	1.18325	−	0.317052i	0.271457	−	0.0727367i	−0.120523	−	0.992711i	\(-0.538457\pi\)
							0.391980	+	0.919974i	\(0.371790\pi\)
\(23\)	0.300590	+	0.300590i	0.0626773	+	0.0626773i	0.737751	−	0.675073i	\(-0.235888\pi\)
							−0.675073	+	0.737751i	\(0.735888\pi\)
\(29\)	−1.31547	−	1.31547i	−0.244277	−	0.244277i	0.574340	−	0.818617i	\(-0.305259\pi\)
							−0.818617	+	0.574340i	\(0.805259\pi\)
\(31\)	−2.75326	+	2.75326i	−0.494500	+	0.494500i	−0.909721	−	0.415221i	\(-0.863704\pi\)
							0.415221	+	0.909721i	\(0.363704\pi\)
\(37\)	−6.00374	+	0.977290i	−0.987009	+	0.160665i
							\(41\)	−9.12544	−	5.26857i	−1.42515	−	0.822813i	−0.428421	−	0.903579i	\(-0.640930\pi\)
−0.996733	+	0.0807661i	\(0.974263\pi\)
\(43\)	2.47578	+	2.47578i	0.377552	+	0.377552i	0.870218	−	0.492666i	\(-0.163978\pi\)
							−0.492666	+	0.870218i	\(0.663978\pi\)
\(47\)			2.74081i			0.399788i	0.979818	+	0.199894i	\(0.0640598\pi\)
							−0.979818	+	0.199894i	\(0.935940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.l.b.103.16	yes	64
4.3	odd	2	inner	148.2.l.b.103.4	yes	64
37.23	odd	12	inner	148.2.l.b.23.4	✓	64
148.23	even	12	inner	148.2.l.b.23.16	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.l.b.23.4	✓	64	37.23	odd	12	inner
148.2.l.b.23.16	yes	64	148.23	even	12	inner
148.2.l.b.103.4	yes	64	4.3	odd	2	inner
148.2.l.b.103.16	yes	64	1.1	even	1	trivial