Embedded newform 148.2.q.b.131.5

• Label: 148.2.q.b.131.5
• 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			131.5
Character	\(\chi\)	\(=\)	148.131
Dual form			148.2.q.b.87.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.965676 + 1.03318i) q^{2} +(-0.539786 + 3.06128i) q^{3} +(-0.134942 - 1.99544i) q^{4} +(-3.16717 + 0.277091i) q^{5} +(-2.64161 - 3.51390i) q^{6} +(0.720212 - 0.858315i) q^{7} +(2.19197 + 1.78753i) q^{8} +(-6.26097 - 2.27881i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{2} - 18 q^{4} - 24 q^{5} - 12 q^{6} - 30 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.965676	+	1.03318i	−0.682836	+	0.730572i
							\(3\)	−0.539786	+	3.06128i	−0.311645	+	1.76743i	0.278798	+	0.960350i	\(0.410064\pi\)
−0.590443	+	0.807079i	\(0.701047\pi\)
\(5\)	−3.16717	+	0.277091i	−1.41640	+	0.123919i	−0.769496	−	0.638651i	\(-0.779493\pi\)
							−0.646905	+	0.762570i	\(0.723937\pi\)
\(7\)	0.720212	−	0.858315i	0.272214	−	0.324412i	−0.612567	−	0.790419i	\(-0.709863\pi\)
							0.884782	+	0.466006i	\(0.154308\pi\)
\(11\)	1.70287	+	2.94945i	0.513434	+	0.889293i	0.999879	+	0.0155819i	\(0.00496008\pi\)
							−0.486445	+	0.873711i	\(0.661707\pi\)
\(13\)	1.06441	−	2.28264i	0.295215	−	0.633090i	−0.701624	−	0.712548i	\(-0.747541\pi\)
							0.996838	+	0.0794579i	\(0.0253189\pi\)
\(17\)	−1.10295	+	0.514313i	−0.267504	+	0.124739i	−0.551743	−	0.834014i	\(-0.686037\pi\)
							0.284239	+	0.958754i	\(0.408259\pi\)
\(19\)	−6.20436	+	4.34434i	−1.42338	+	0.996659i	−0.427686	+	0.903927i	\(0.640671\pi\)
							−0.995691	+	0.0927319i	\(0.970440\pi\)
\(23\)	−3.16367	+	0.847702i	−0.659670	+	0.176758i	−0.573097	−	0.819487i	\(-0.694258\pi\)
							−0.0865731	+	0.996245i	\(0.527592\pi\)
\(29\)	−1.47030	+	5.48725i	−0.273029	+	1.01896i	0.684122	+	0.729367i	\(0.260185\pi\)
							−0.957151	+	0.289590i	\(0.906481\pi\)
\(31\)	−1.90169	+	1.90169i	−0.341554	+	0.341554i	−0.856951	−	0.515398i	\(-0.827644\pi\)
							0.515398	+	0.856951i	\(0.327644\pi\)
\(37\)	5.83795	−	1.70831i	0.959753	−	0.280844i
							\(41\)	−0.469549	−	1.29008i	−0.0733313	−	0.201476i	0.897612	−	0.440787i	\(-0.145301\pi\)
−0.970943	+	0.239311i	\(0.923078\pi\)
\(43\)	6.66047	+	6.66047i	1.01571	+	1.01571i	0.999875	+	0.0158377i	\(0.00504149\pi\)
							0.0158377	+	0.999875i	\(0.494959\pi\)
\(47\)	5.21908	+	3.01324i	0.761281	+	0.439526i	0.829756	−	0.558127i	\(-0.188480\pi\)
							−0.0684744	+	0.997653i	\(0.521813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.q.b.131.5	yes	192
4.3	odd	2	inner	148.2.q.b.131.6	yes	192
37.13	odd	36	inner	148.2.q.b.87.6	yes	192
148.87	even	36	inner	148.2.q.b.87.5	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.q.b.87.5	✓	192	148.87	even	36	inner
148.2.q.b.87.6	yes	192	37.13	odd	36	inner
148.2.q.b.131.5	yes	192	1.1	even	1	trivial
148.2.q.b.131.6	yes	192	4.3	odd	2	inner