Embedded newform 130.3.t.b.19.7

• Label: 130.3.t.b.19.7
• 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			19.7
Character	\(\chi\)	\(=\)	130.19
Dual form			130.3.t.b.89.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(3.51172 + 2.02749i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-3.38785 + 3.67729i) q^{5} +(5.53921 + 1.48423i) q^{6} +(0.953023 + 0.255362i) q^{7} +(2.00000 - 2.00000i) q^{8} +(3.72144 + 6.44572i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	3.51172	+	2.02749i	1.17057	+	0.675830i	0.953815	−	0.300395i	\(-0.0971186\pi\)
0.216758	+	0.976225i	\(0.430452\pi\)
\(5\)	−3.38785	+	3.67729i	−0.677570	+	0.735458i
							\(7\)	0.953023	+	0.255362i	0.136146	+	0.0364802i	0.326249	−	0.945284i	\(-0.394215\pi\)
−0.190102	+	0.981764i	\(0.560882\pi\)
\(11\)	11.6927	−	3.13305i	1.06297	−	0.284823i	0.315371	−	0.948969i	\(-0.397871\pi\)
							0.747603	+	0.664145i	\(0.231204\pi\)
\(13\)	−12.9905	−	0.497205i	−0.999268	−	0.0382466i
							\(17\)	−12.3157	−	21.3314i	−0.724453	−	1.25479i	−0.959199	−	0.282733i	\(-0.908759\pi\)
0.234746	−	0.972057i	\(-0.424574\pi\)
\(19\)	8.03819	+	2.15383i	0.423063	+	0.113359i	0.464068	−	0.885800i	\(-0.346389\pi\)
							−0.0410052	+	0.999159i	\(0.513056\pi\)
\(23\)	−0.823656	+	1.42661i	−0.0358111	+	0.0620267i	−0.883375	−	0.468666i	\(-0.844735\pi\)
							0.847564	+	0.530693i	\(0.178068\pi\)
\(29\)	−22.0352	+	38.1661i	−0.759835	+	1.31607i	0.183099	+	0.983095i	\(0.441387\pi\)
							−0.942934	+	0.332979i	\(0.891946\pi\)
\(31\)	28.0596	−	28.0596i	0.905149	−	0.905149i	−0.0907271	−	0.995876i	\(-0.528919\pi\)
							0.995876	+	0.0907271i	\(0.0289191\pi\)
\(37\)	2.55491	+	9.53506i	0.0690516	+	0.257704i	0.991819	−	0.127654i	\(-0.0407446\pi\)
							−0.922767	+	0.385358i	\(0.874078\pi\)
\(41\)	5.33284	+	19.9024i	0.130069	+	0.485425i	0.999970	−	0.00780243i	\(-0.00248361\pi\)
							−0.869900	+	0.493228i	\(0.835817\pi\)
\(43\)	−17.8830	−	30.9743i	−0.415885	−	0.720333i	0.579636	−	0.814875i	\(-0.303195\pi\)
							−0.995521	+	0.0945422i	\(0.969861\pi\)
\(47\)	−50.2935	+	50.2935i	−1.07007	+	1.07007i	−0.0727228	+	0.997352i	\(0.523169\pi\)
							−0.997352	+	0.0727228i	\(0.976831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.19.7	yes	28
5.4	even	2		130.3.t.a.19.1	✓	28
13.11	odd	12		130.3.t.a.89.1	yes	28
65.24	odd	12	inner	130.3.t.b.89.7	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.19.1	✓	28	5.4	even	2
130.3.t.a.89.1	yes	28	13.11	odd	12
130.3.t.b.19.7	yes	28	1.1	even	1	trivial
130.3.t.b.89.7	yes	28	65.24	odd	12	inner