Embedded newform 130.3.t.b.89.4

• Label: 130.3.t.b.89.4
• Level: $130$
• Weight: $3$
• Character: 130.89
• 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			89.4
Character	\(\chi\)	\(=\)	130.89
Dual form			130.3.t.b.19.4

$q$-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 \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{12}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.19.4	✓	28	13.6	odd	12
130.3.t.a.89.4	yes	28	5.4	even	2
130.3.t.b.19.4	yes	28	65.19	odd	12	inner
130.3.t.b.89.4	yes	28	1.1	even	1	trivial