Embedded newform 148.2.l.b.119.3

• Label: 148.2.l.b.119.3
• 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			119.3
Character	\(\chi\)	\(=\)	148.119
Dual form			148.2.l.b.51.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24598 + 0.668988i) q^{2} +(-0.922196 + 1.59729i) q^{3} +(1.10491 - 1.66709i) q^{4} +(3.28039 - 0.878978i) q^{5} +(0.0804659 - 2.60712i) q^{6} +(0.449139 + 0.259311i) q^{7} +(-0.261432 + 2.81632i) q^{8} +(-0.200891 - 0.347953i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{2} - 4 q^{5} + 8 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24598	+	0.668988i	−0.881038	+	0.473046i
							\(3\)	−0.922196	+	1.59729i	−0.532430	+	0.922196i	0.466853	+	0.884335i	\(0.345388\pi\)
−0.999283	+	0.0378610i	\(0.987946\pi\)
\(5\)	3.28039	−	0.878978i	1.46704	−	0.393091i	0.565122	−	0.825007i	\(-0.308829\pi\)
							0.901914	+	0.431916i	\(0.142162\pi\)
\(7\)	0.449139	+	0.259311i	0.169759	+	0.0980102i	0.582472	−	0.812851i	\(-0.302086\pi\)
							−0.412713	+	0.910861i	\(0.635419\pi\)
\(11\)	1.72253			0.519364			0.259682	−	0.965694i	\(-0.416382\pi\)
							0.259682	+	0.965694i	\(0.416382\pi\)
\(13\)	−3.41625	+	0.915381i	−0.947497	+	0.253881i	−0.699300	−	0.714829i	\(-0.746505\pi\)
							−0.248197	+	0.968710i	\(0.579838\pi\)
\(17\)	−1.08378	+	4.04473i	−0.262856	+	0.980991i	0.700694	+	0.713462i	\(0.252874\pi\)
							−0.963550	+	0.267529i	\(0.913793\pi\)
\(19\)	1.08285	+	4.04126i	0.248423	+	0.927128i	0.971632	+	0.236499i	\(0.0759998\pi\)
							−0.723209	+	0.690630i	\(0.757333\pi\)
\(23\)	5.14972	−	5.14972i	1.07379	−	1.07379i	0.0767388	−	0.997051i	\(-0.475549\pi\)
							0.997051	−	0.0767388i	\(-0.0244508\pi\)
\(29\)	5.49297	−	5.49297i	1.02002	−	1.02002i	0.0202240	−	0.999795i	\(-0.493562\pi\)
							0.999795	−	0.0202240i	\(-0.00643792\pi\)
\(31\)	−5.00608	−	5.00608i	−0.899118	−	0.899118i	0.0962402	−	0.995358i	\(-0.469318\pi\)
							−0.995358	+	0.0962402i	\(0.969318\pi\)
\(37\)	−2.34822	+	5.61122i	−0.386046	+	0.922480i
							\(41\)	−1.42619	−	0.823413i	−0.222734	−	0.128596i	0.384482	−	0.923133i	\(-0.374380\pi\)
−0.607216	+	0.794537i	\(0.707714\pi\)
\(43\)	−3.14276	+	3.14276i	−0.479266	+	0.479266i	−0.904897	−	0.425631i	\(-0.860052\pi\)
							0.425631	+	0.904897i	\(0.360052\pi\)
\(47\)		−	6.54927i		−	0.955310i	−0.878548	−	0.477655i	\(-0.841487\pi\)
							0.878548	−	0.477655i	\(-0.158513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.l.b.119.3	yes	64
4.3	odd	2	inner	148.2.l.b.119.6	yes	64
37.14	odd	12	inner	148.2.l.b.51.6	yes	64
148.51	even	12	inner	148.2.l.b.51.3	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.l.b.51.3	✓	64	148.51	even	12	inner
148.2.l.b.51.6	yes	64	37.14	odd	12	inner
148.2.l.b.119.3	yes	64	1.1	even	1	trivial
148.2.l.b.119.6	yes	64	4.3	odd	2	inner