Embedded newform 148.2.q.b.131.11

• Label: 148.2.q.b.131.11
• Level: $148$
• Weight: $2$
• Character: 148.131
• Analytic conductor: $1.182$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 148 = 2^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	148.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.18178594991\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(16\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			131.11
Character	\(\chi\)	\(=\)	148.131
Dual form			148.2.q.b.87.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.612633 - 1.27463i) q^{2} +(-0.242675 + 1.37628i) q^{3} +(-1.24936 - 1.56176i) q^{4} +(1.17953 - 0.103196i) q^{5} +(1.60557 + 1.15247i) q^{6} +(2.52278 - 3.00653i) q^{7} +(-2.75607 + 0.635689i) q^{8} +(0.983834 + 0.358086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{2} - 18 q^{4} - 24 q^{5} - 12 q^{6} - 30 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(75\)	\(113\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{25}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.612633	−	1.27463i	0.433197	−	0.901299i
							\(3\)	−0.242675	+	1.37628i	−0.140108	+	0.794593i	0.831057	+	0.556187i	\(0.187736\pi\)
−0.971165	+	0.238406i	\(0.923375\pi\)
\(5\)	1.17953	−	0.103196i	0.527502	−	0.0461505i	0.179705	−	0.983720i	\(-0.442486\pi\)
							0.347797	+	0.937570i	\(0.386930\pi\)
\(7\)	2.52278	−	3.00653i	0.953522	−	1.13636i	−0.0370423	−	0.999314i	\(-0.511794\pi\)
							0.990564	−	0.137049i	\(-0.0437619\pi\)
\(11\)	−0.220507	−	0.381930i	−0.0664854	−	0.115156i	0.830867	−	0.556472i	\(-0.187845\pi\)
							−0.897352	+	0.441316i	\(0.854512\pi\)
\(13\)	−0.685403	+	1.46985i	−0.190096	+	0.407663i	−0.978137	−	0.207960i	\(-0.933318\pi\)
							0.788041	+	0.615623i	\(0.211096\pi\)
\(17\)	−0.582804	+	0.271766i	−0.141351	+	0.0659130i	−0.492007	−	0.870591i	\(-0.663736\pi\)
							0.350656	+	0.936504i	\(0.385959\pi\)
\(19\)	−3.82792	+	2.68034i	−0.878186	+	0.614912i	−0.923304	−	0.384069i	\(-0.874522\pi\)
							0.0451184	+	0.998982i	\(0.485633\pi\)
\(23\)	−4.39017	+	1.17634i	−0.915415	+	0.245285i	−0.685624	−	0.727956i	\(-0.740471\pi\)
							−0.229790	+	0.973240i	\(0.573804\pi\)
\(29\)	−1.23948	+	4.62579i	−0.230165	+	0.858988i	0.750104	+	0.661320i	\(0.230003\pi\)
							−0.980269	+	0.197668i	\(0.936663\pi\)
\(31\)	5.98281	−	5.98281i	1.07454	−	1.07454i	0.0775563	−	0.996988i	\(-0.475288\pi\)
							0.996988	−	0.0775563i	\(-0.0247117\pi\)
\(37\)	2.00013	−	5.74452i	0.328819	−	0.944393i
							\(41\)	0.867787	+	2.38423i	0.135526	+	0.372354i	0.988828	−	0.149063i	\(-0.0476258\pi\)
−0.853302	+	0.521417i	\(0.825404\pi\)
\(43\)	−2.74009	−	2.74009i	−0.417859	−	0.417859i	0.466606	−	0.884465i	\(-0.345477\pi\)
							−0.884465	+	0.466606i	\(0.845477\pi\)
\(47\)	9.08311	+	5.24414i	1.32491	+	0.764936i	0.984507	−	0.175345i	\(-0.0561039\pi\)
							0.340401	+	0.940280i	\(0.389437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.q.b.131.11	yes	192
4.3	odd	2	inner	148.2.q.b.131.13	yes	192
37.13	odd	36	inner	148.2.q.b.87.13	yes	192
148.87	even	36	inner	148.2.q.b.87.11	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.q.b.87.11	✓	192	148.87	even	36	inner
148.2.q.b.87.13	yes	192	37.13	odd	36	inner
148.2.q.b.131.11	yes	192	1.1	even	1	trivial
148.2.q.b.131.13	yes	192	4.3	odd	2	inner