Embedded newform 148.2.l.b.103.10

• Label: 148.2.l.b.103.10
• 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			103.10
Character	\(\chi\)	\(=\)	148.103
Dual form			148.2.l.b.23.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.148406 + 1.40641i) q^{2} +(-0.403769 + 0.699348i) q^{3} +(-1.95595 + 0.417438i) q^{4} +(0.498662 + 1.86103i) q^{5} +(-1.04349 - 0.464075i) q^{6} +(-0.164777 - 0.0951340i) q^{7} +(-0.877362 - 2.68891i) q^{8} +(1.17394 + 2.03333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{2} - 4 q^{5} + 8 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.148406	+	1.40641i	0.104939	+	0.994479i
							\(3\)	−0.403769	+	0.699348i	−0.233116	+	0.403769i	−0.958723	−	0.284340i	\(-0.908226\pi\)
0.725607	+	0.688109i	\(0.241559\pi\)
\(5\)	0.498662	+	1.86103i	0.223009	+	0.832279i	0.983193	+	0.182571i	\(0.0584421\pi\)
							−0.760184	+	0.649708i	\(0.774891\pi\)
\(7\)	−0.164777	−	0.0951340i	−0.0622798	−	0.0359573i	0.468537	−	0.883444i	\(-0.344781\pi\)
							−0.530817	+	0.847487i	\(0.678115\pi\)
\(11\)	−4.50587			−1.35857			−0.679286	−	0.733874i	\(-0.737710\pi\)
							−0.679286	+	0.733874i	\(0.737710\pi\)
\(13\)	−0.0401953	−	0.150011i	−0.0111482	−	0.0416055i	0.960128	−	0.279562i	\(-0.0901892\pi\)
							−0.971276	+	0.237956i	\(0.923523\pi\)
\(17\)	2.04346	+	0.547544i	0.495613	+	0.132799i	0.497963	−	0.867198i	\(-0.334082\pi\)
							−0.00235044	+	0.999997i	\(0.500748\pi\)
\(19\)	6.13947	−	1.64507i	1.40849	−	0.377404i	0.527105	−	0.849800i	\(-0.323278\pi\)
							0.881386	+	0.472396i	\(0.156611\pi\)
\(23\)	5.48553	+	5.48553i	1.14381	+	1.14381i	0.987747	+	0.156066i	\(0.0498812\pi\)
							0.156066	+	0.987747i	\(0.450119\pi\)
\(29\)	−1.52531	−	1.52531i	−0.283243	−	0.283243i	0.551158	−	0.834401i	\(-0.314186\pi\)
							−0.834401	+	0.551158i	\(0.814186\pi\)
\(31\)	0.399087	−	0.399087i	0.0716781	−	0.0716781i	−0.670359	−	0.742037i	\(-0.733860\pi\)
							0.742037	+	0.670359i	\(0.233860\pi\)
\(37\)	−0.904241	−	6.01518i	−0.148656	−	0.988889i
							\(41\)	4.12634	+	2.38234i	0.644426	+	0.372060i	0.786317	−	0.617823i	\(-0.211985\pi\)
−0.141891	+	0.989882i	\(0.545318\pi\)
\(43\)	3.01262	+	3.01262i	0.459420	+	0.459420i	0.898465	−	0.439045i	\(-0.144683\pi\)
							−0.439045	+	0.898465i	\(0.644683\pi\)
\(47\)			0.677579i			0.0988351i	0.998778	+	0.0494175i	\(0.0157365\pi\)
							−0.998778	+	0.0494175i	\(0.984263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.l.b.103.10	yes	64
4.3	odd	2	inner	148.2.l.b.103.6	yes	64
37.23	odd	12	inner	148.2.l.b.23.6	✓	64
148.23	even	12	inner	148.2.l.b.23.10	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.l.b.23.6	✓	64	37.23	odd	12	inner
148.2.l.b.23.10	yes	64	148.23	even	12	inner
148.2.l.b.103.6	yes	64	4.3	odd	2	inner
148.2.l.b.103.10	yes	64	1.1	even	1	trivial