Embedded newform 260.2.bg.c.23.1

• Label: 260.2.bg.c.23.1
• Level: $260$
• Weight: $2$
• Character: 260.23
• Analytic conductor: $2.076$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07611045255\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.1
Character	\(\chi\)	\(=\)	260.23
Dual form			260.2.bg.c.147.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39962 - 0.202672i) q^{2} +(-0.654675 - 0.175420i) q^{3} +(1.91785 + 0.567327i) q^{4} +(1.99368 + 1.01254i) q^{5} +(0.880741 + 0.378205i) q^{6} +(0.970879 + 3.62337i) q^{7} +(-2.56927 - 1.18273i) q^{8} +(-2.20025 - 1.27031i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 6 q^{2} - 12 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\)	\(41\)	\(131\)	\(157\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39962	−	0.202672i	−0.989678	−	0.143311i
							\(3\)	−0.654675	−	0.175420i	−0.377977	−	0.101279i	0.0648283	−	0.997896i	\(-0.479350\pi\)
−0.442805	+	0.896618i	\(0.646017\pi\)
\(5\)	1.99368	+	1.01254i	0.891601	+	0.452822i
							\(7\)	0.970879	+	3.62337i	0.366958	+	1.36951i	0.864747	+	0.502208i	\(0.167479\pi\)
−0.497789	+	0.867298i	\(0.665855\pi\)
\(11\)	0.0663882	+	0.114988i	0.0200168	+	0.0346701i	0.875860	−	0.482565i	\(-0.160295\pi\)
							−0.855843	+	0.517235i	\(0.826961\pi\)
\(13\)	−3.41580	+	1.15426i	−0.947372	+	0.320135i
							\(17\)	0.914819	+	3.41415i	0.221876	+	0.828054i	0.983632	+	0.180189i	\(0.0576710\pi\)
−0.761756	+	0.647864i	\(0.775662\pi\)
\(19\)	6.18368	+	3.57015i	1.41863	+	0.819048i	0.996179	−	0.0873365i	\(-0.0278355\pi\)
							0.422454	+	0.906384i	\(0.361169\pi\)
\(23\)	−1.83343	−	0.491267i	−0.382297	−	0.102436i	0.0625521	−	0.998042i	\(-0.480076\pi\)
							−0.444849	+	0.895606i	\(0.646743\pi\)
\(29\)	1.50675	−	0.869924i	0.279797	−	0.161541i	−0.353534	−	0.935421i	\(-0.615020\pi\)
							0.633332	+	0.773881i	\(0.281687\pi\)
\(31\)	−3.93079			−0.705991			−0.352996	−	0.935625i	\(-0.614837\pi\)
							−0.352996	+	0.935625i	\(0.614837\pi\)
\(37\)	−0.800494	+	2.98748i	−0.131600	+	0.491139i	−0.999989	−	0.00474547i	\(-0.998489\pi\)
							0.868388	+	0.495885i	\(0.165156\pi\)
\(41\)	8.41553	−	4.85871i	1.31428	−	0.758803i	0.331482	−	0.943462i	\(-0.392451\pi\)
							0.982803	+	0.184659i	\(0.0591180\pi\)
\(43\)	1.11184	+	4.14943i	0.169553	+	0.632781i	0.997415	+	0.0718499i	\(0.0228902\pi\)
							−0.827862	+	0.560932i	\(0.810443\pi\)
\(47\)	6.53390	−	6.53390i	0.953068	−	0.953068i	−0.0458791	−	0.998947i	\(-0.514609\pi\)
							0.998947	+	0.0458791i	\(0.0146089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.2.bg.c.23.1	✓	144
4.3	odd	2	inner	260.2.bg.c.23.32	yes	144
5.2	odd	4	inner	260.2.bg.c.127.21	yes	144
13.4	even	6	inner	260.2.bg.c.43.14	yes	144
20.7	even	4	inner	260.2.bg.c.127.14	yes	144
52.43	odd	6	inner	260.2.bg.c.43.21	yes	144
65.17	odd	12	inner	260.2.bg.c.147.32	yes	144
260.147	even	12	inner	260.2.bg.c.147.1	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.bg.c.23.1	✓	144	1.1	even	1	trivial
260.2.bg.c.23.32	yes	144	4.3	odd	2	inner
260.2.bg.c.43.14	yes	144	13.4	even	6	inner
260.2.bg.c.43.21	yes	144	52.43	odd	6	inner
260.2.bg.c.127.14	yes	144	20.7	even	4	inner
260.2.bg.c.127.21	yes	144	5.2	odd	4	inner
260.2.bg.c.147.1	yes	144	260.147	even	12	inner
260.2.bg.c.147.32	yes	144	65.17	odd	12	inner