Embedded newform 130.3.t.b.19.5

• Label: 130.3.t.b.19.5
• Level: $130$
• Weight: $3$
• Character: 130.19
• Analytic conductor: $3.542$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• 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			19.5
Character	\(\chi\)	\(=\)	130.19
Dual form			130.3.t.b.89.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(2.81963 + 1.62791i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-1.01577 - 4.89573i) q^{5} +(4.44754 + 1.19171i) q^{6} +(3.91361 + 1.04865i) q^{7} +(2.00000 - 2.00000i) q^{8} +(0.800200 + 1.38599i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	2.81963	+	1.62791i	0.939876	+	0.542638i	0.889921	−	0.456114i	\(-0.150759\pi\)
0.0499545	+	0.998751i	\(0.484092\pi\)
\(5\)	−1.01577	−	4.89573i	−0.203155	−	0.979147i
							\(7\)	3.91361	+	1.04865i	0.559087	+	0.149807i	0.527287	−	0.849687i	\(-0.323209\pi\)
0.0318004	+	0.999494i	\(0.489876\pi\)
\(11\)	−3.41804	+	0.915862i	−0.310731	+	0.0832602i	−0.410815	−	0.911719i	\(-0.634756\pi\)
							0.100083	+	0.994979i	\(0.468089\pi\)
\(13\)	−1.34302	+	12.9304i	−0.103309	+	0.994649i
							\(17\)	14.4164	+	24.9700i	0.848024	+	1.46882i	0.882969	+	0.469432i	\(0.155541\pi\)
−0.0349446	+	0.999389i	\(0.511125\pi\)
\(19\)	−35.3880	−	9.48220i	−1.86253	−	0.499063i	−0.862553	−	0.505967i	\(-0.831136\pi\)
							−0.999976	+	0.00690400i	\(0.997802\pi\)
\(23\)	−2.41694	+	4.18626i	−0.105084	+	0.182011i	−0.913773	−	0.406226i	\(-0.866845\pi\)
							0.808688	+	0.588237i	\(0.200178\pi\)
\(29\)	16.0797	−	27.8508i	0.554472	−	0.960374i	−0.443472	−	0.896288i	\(-0.646253\pi\)
							0.997944	−	0.0640859i	\(-0.0204132\pi\)
\(31\)	−5.98260	+	5.98260i	−0.192987	+	0.192987i	−0.796986	−	0.603998i	\(-0.793573\pi\)
							0.603998	+	0.796986i	\(0.293573\pi\)
\(37\)	−10.3267	−	38.5399i	−0.279101	−	1.04162i	−0.953043	−	0.302835i	\(-0.902067\pi\)
							0.673942	−	0.738784i	\(-0.264600\pi\)
\(41\)	4.98568	+	18.6068i	0.121602	+	0.453825i	0.999696	−	0.0246638i	\(-0.00785152\pi\)
							−0.878094	+	0.478488i	\(0.841185\pi\)
\(43\)	9.17087	+	15.8844i	0.213276	+	0.369405i	0.952738	−	0.303793i	\(-0.0982533\pi\)
							−0.739462	+	0.673199i	\(0.764920\pi\)
\(47\)	−32.9600	+	32.9600i	−0.701277	+	0.701277i	−0.964685	−	0.263408i	\(-0.915154\pi\)
							0.263408	+	0.964685i	\(0.415154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.19.5	yes	28
5.4	even	2		130.3.t.a.19.3	✓	28
13.11	odd	12		130.3.t.a.89.3	yes	28
65.24	odd	12	inner	130.3.t.b.89.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.19.3	✓	28	5.4	even	2
130.3.t.a.89.3	yes	28	13.11	odd	12
130.3.t.b.19.5	yes	28	1.1	even	1	trivial
130.3.t.b.89.5	yes	28	65.24	odd	12	inner