Embedded newform 148.2.l.b.119.4

• Label: 148.2.l.b.119.4
• 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			119.4
Character	\(\chi\)	\(=\)	148.119
Dual form			148.2.l.b.51.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06936 - 0.925461i) q^{2} +(-1.64209 + 2.84419i) q^{3} +(0.287045 + 1.97929i) q^{4} +(-0.134520 + 0.0360445i) q^{5} +(4.38817 - 1.52176i) q^{6} +(-3.74056 - 2.15961i) q^{7} +(1.52481 - 2.38222i) q^{8} +(-3.89294 - 6.74276i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{2} - 4 q^{5} + 8 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06936	−	0.925461i	−0.756149	−	0.654400i
							\(3\)	−1.64209	+	2.84419i	−0.948063	+	1.64209i	−0.198563	+	0.980088i	\(0.563627\pi\)
−0.749500	+	0.662005i	\(0.769706\pi\)
\(5\)	−0.134520	+	0.0360445i	−0.0601591	+	0.0161196i	−0.288773	−	0.957398i	\(-0.593247\pi\)
							0.228614	+	0.973517i	\(0.426581\pi\)
\(7\)	−3.74056	−	2.15961i	−1.41380	−	0.816257i	−0.418054	−	0.908422i	\(-0.637288\pi\)
							−0.995744	+	0.0921653i	\(0.970621\pi\)
\(11\)	1.65631			0.499397			0.249698	−	0.968324i	\(-0.419668\pi\)
							0.249698	+	0.968324i	\(0.419668\pi\)
\(13\)	−2.23698	+	0.599398i	−0.620428	+	0.166243i	−0.555322	−	0.831635i	\(-0.687405\pi\)
							−0.0651055	+	0.997878i	\(0.520738\pi\)
\(17\)	−0.458405	+	1.71079i	−0.111180	+	0.414928i	−0.998973	−	0.0453149i	\(-0.985571\pi\)
							0.887793	+	0.460243i	\(0.152238\pi\)
\(19\)	−0.271752	−	1.01419i	−0.0623441	−	0.232671i	0.927722	−	0.373271i	\(-0.121764\pi\)
							−0.990067	+	0.140599i	\(0.955097\pi\)
\(23\)	−3.39749	+	3.39749i	−0.708427	+	0.708427i	−0.966204	−	0.257778i	\(-0.917010\pi\)
							0.257778	+	0.966204i	\(0.417010\pi\)
\(29\)	−3.19370	+	3.19370i	−0.593056	+	0.593056i	−0.938456	−	0.345400i	\(-0.887743\pi\)
							0.345400	+	0.938456i	\(0.387743\pi\)
\(31\)	−4.53498	−	4.53498i	−0.814506	−	0.814506i	0.170800	−	0.985306i	\(-0.445365\pi\)
							−0.985306	+	0.170800i	\(0.945365\pi\)
\(37\)	−4.78693	−	3.75304i	−0.786966	−	0.616996i
							\(41\)	6.61538	+	3.81939i	1.03315	+	0.596488i	0.917885	−	0.396847i	\(-0.129895\pi\)
0.115263	+	0.993335i	\(0.463229\pi\)
\(43\)	−2.49766	+	2.49766i	−0.380889	+	0.380889i	−0.871422	−	0.490533i	\(-0.836802\pi\)
							0.490533	+	0.871422i	\(0.336802\pi\)
\(47\)		−	0.322236i		−	0.0470030i	−0.999724	−	0.0235015i	\(-0.992519\pi\)
							0.999724	−	0.0235015i	\(-0.00748145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.l.b.119.4	yes	64
4.3	odd	2	inner	148.2.l.b.119.1	yes	64
37.14	odd	12	inner	148.2.l.b.51.1	✓	64
148.51	even	12	inner	148.2.l.b.51.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.l.b.51.1	✓	64	37.14	odd	12	inner
148.2.l.b.51.4	yes	64	148.51	even	12	inner
148.2.l.b.119.1	yes	64	4.3	odd	2	inner
148.2.l.b.119.4	yes	64	1.1	even	1	trivial