Embedded newform 130.3.t.b.19.4

• Label: 130.3.t.b.19.4
• Level: $130$
• Weight: $3$
• Character: 130.19
• Analytic conductor: $3.542$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 130 = 2 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	130.t (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54224343668\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.4
Character	\(\chi\)	\(=\)	130.19
Dual form			130.3.t.b.89.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-1.17643 - 0.679210i) q^{3} +(1.73205 - 1.00000i) q^{4} +(4.29494 - 2.55997i) q^{5} +(-1.85564 - 0.497217i) q^{6} +(-1.13528 - 0.304199i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-3.57735 - 6.19615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	−	0.366025i	0.683013	−	0.183013i
							\(3\)	−1.17643	−	0.679210i	−0.392142	−	0.226403i	0.290946	−	0.956740i	\(-0.406030\pi\)
−0.683088	+	0.730336i	\(0.739363\pi\)
\(5\)	4.29494	−	2.55997i	0.858989	−	0.511995i
							\(7\)	−1.13528	−	0.304199i	−0.162184	−	0.0434570i	0.176814	−	0.984244i	\(-0.443421\pi\)
−0.338997	+	0.940787i	\(0.610088\pi\)
\(11\)	9.64836	−	2.58527i	0.877124	−	0.235025i	0.207958	−	0.978138i	\(-0.433318\pi\)
							0.669166	+	0.743113i	\(0.266652\pi\)
\(13\)	4.68269	+	12.1273i	0.360207	+	0.932872i
							\(17\)	−13.3525	−	23.1272i	−0.785440	−	1.36042i	−0.928736	−	0.370743i	\(-0.879103\pi\)
0.143295	−	0.989680i	\(-0.454230\pi\)
\(19\)	24.1714	+	6.47670i	1.27218	+	0.340879i	0.830865	−	0.556475i	\(-0.187846\pi\)
							0.441313	+	0.897353i	\(0.354513\pi\)
\(23\)	−18.7367	+	32.4529i	−0.814640	+	1.41100i	0.0949463	+	0.995482i	\(0.469732\pi\)
							−0.909586	+	0.415515i	\(0.863601\pi\)
\(29\)	−9.25089	+	16.0230i	−0.318996	+	0.552517i	−0.980279	−	0.197619i	\(-0.936679\pi\)
							0.661283	+	0.750137i	\(0.270012\pi\)
\(31\)	−42.8870	+	42.8870i	−1.38345	+	1.38345i	−0.545044	+	0.838408i	\(0.683487\pi\)
							−0.838408	+	0.545044i	\(0.816513\pi\)
\(37\)	1.82135	+	6.79735i	0.0492255	+	0.183712i	0.986161	−	0.165790i	\(-0.0530174\pi\)
							−0.936936	+	0.349502i	\(0.886351\pi\)
\(41\)	16.2894	+	60.7928i	0.397302	+	1.48275i	0.817824	+	0.575469i	\(0.195180\pi\)
							−0.420522	+	0.907282i	\(0.638153\pi\)
\(43\)	22.1744	+	38.4073i	0.515685	+	0.893192i	0.999834	+	0.0182069i	\(0.00579576\pi\)
							−0.484149	+	0.874985i	\(0.660871\pi\)
\(47\)	39.7607	−	39.7607i	0.845972	−	0.845972i	−0.143656	−	0.989628i	\(-0.545886\pi\)
							0.989628	+	0.143656i	\(0.0458858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.19.4	yes	28
5.4	even	2		130.3.t.a.19.4	✓	28
13.11	odd	12		130.3.t.a.89.4	yes	28
65.24	odd	12	inner	130.3.t.b.89.4	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.19.4	✓	28	5.4	even	2
130.3.t.a.89.4	yes	28	13.11	odd	12
130.3.t.b.19.4	yes	28	1.1	even	1	trivial
130.3.t.b.89.4	yes	28	65.24	odd	12	inner