Embedded newform 130.3.t.b.19.6

• Label: 130.3.t.b.19.6
• 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			19.6
Character	\(\chi\)	\(=\)	130.19
Dual form			130.3.t.b.89.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(3.21490 + 1.85613i) q^{3} +(1.73205 - 1.00000i) q^{4} +(4.67137 + 1.78279i) q^{5} +(5.07103 + 1.35878i) q^{6} +(-7.00719 - 1.87757i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.39040 + 4.14030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	3.21490	+	1.85613i	1.07163	+	0.618708i	0.928628	−	0.371013i	\(-0.120990\pi\)
0.143007	+	0.989722i	\(0.454323\pi\)
\(5\)	4.67137	+	1.78279i	0.934273	+	0.356558i
							\(7\)	−7.00719	−	1.87757i	−1.00103	−	0.268225i	−0.279152	−	0.960247i	\(-0.590053\pi\)
−0.721876	+	0.692022i	\(0.756720\pi\)
\(11\)	−20.2175	+	5.41727i	−1.83796	+	0.492479i	−0.998686	−	0.0512377i	\(-0.983683\pi\)
							−0.839269	+	0.543716i	\(0.817017\pi\)
\(13\)	8.06486	−	10.1960i	0.620374	−	0.784306i
							\(17\)	−0.232239	−	0.402250i	−0.0136611	−	0.0236618i	0.859114	−	0.511784i	\(-0.171015\pi\)
−0.872775	+	0.488122i	\(0.837682\pi\)
\(19\)	4.54585	+	1.21806i	0.239255	+	0.0641083i	0.376454	−	0.926435i	\(-0.377143\pi\)
							−0.137199	+	0.990544i	\(0.543810\pi\)
\(23\)	6.94259	−	12.0249i	0.301852	−	0.522822i	−0.674704	−	0.738089i	\(-0.735729\pi\)
							0.976555	+	0.215266i	\(0.0690620\pi\)
\(29\)	0.160077	−	0.277261i	0.00551988	−	0.00956072i	−0.863252	−	0.504773i	\(-0.831576\pi\)
							0.868772	+	0.495212i	\(0.164910\pi\)
\(31\)	−13.7184	+	13.7184i	−0.442529	+	0.442529i	−0.892861	−	0.450332i	\(-0.851306\pi\)
							0.450332	+	0.892861i	\(0.351306\pi\)
\(37\)	14.0167	+	52.3111i	0.378830	+	1.41381i	0.847667	+	0.530529i	\(0.178007\pi\)
							−0.468837	+	0.883285i	\(0.655327\pi\)
\(41\)	−5.64591	−	21.0708i	−0.137705	−	0.513923i	−0.999972	−	0.00746853i	\(-0.997623\pi\)
							0.862267	−	0.506454i	\(-0.169044\pi\)
\(43\)	−15.9803	−	27.6787i	−0.371634	−	0.643690i	0.618183	−	0.786034i	\(-0.287869\pi\)
							−0.989817	+	0.142345i	\(0.954536\pi\)
\(47\)	3.48187	−	3.48187i	0.0740823	−	0.0740823i	−0.669095	−	0.743177i	\(-0.733318\pi\)
							0.743177	+	0.669095i	\(0.233318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.19.6	yes	28
5.4	even	2		130.3.t.a.19.2	✓	28
13.11	odd	12		130.3.t.a.89.2	yes	28
65.24	odd	12	inner	130.3.t.b.89.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.19.2	✓	28	5.4	even	2
130.3.t.a.89.2	yes	28	13.11	odd	12
130.3.t.b.19.6	yes	28	1.1	even	1	trivial
130.3.t.b.89.6	yes	28	65.24	odd	12	inner