Embedded newform 12.20.b.a.11.13

• Label: 12.20.b.a.11.13
• Level: $12$
• Weight: $20$
• Character: 12.11
• Analytic conductor: $27.458$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(27.4580035868\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.13
Character	\(\chi\)	\(=\)	12.11
Dual form			12.20.b.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-246.303 - 680.898i) q^{2} +(12845.9 - 31579.2i) q^{3} +(-402957. + 335415. i) q^{4} +225554. i q^{5} +(-2.46662e7 - 968699. i) q^{6} +9.84546e7i q^{7} +(3.27633e8 + 1.91759e8i) q^{8} +(-8.32227e8 - 8.11326e8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−246.303	−	680.898i	−0.340161	−	0.940367i
							\(3\)	12845.9	−	31579.2i	0.376801	−	0.926294i
\(5\)			225554.i			0.0516459i								0.999667	+	0.0258229i	\(0.00822061\pi\)
							−0.999667	+	0.0258229i	\(0.991779\pi\)
\(7\)			9.84546e7i			0.922157i	0.887359	+	0.461078i	\(0.152537\pi\)
							−0.887359	+	0.461078i	\(0.847463\pi\)
\(11\)	6.04702e9			0.773234			0.386617	−	0.922240i	\(-0.373644\pi\)
							0.386617	+	0.922240i	\(0.373644\pi\)
\(13\)	4.74938e10			1.24215			0.621076	−	0.783750i	\(-0.286696\pi\)
							0.621076	+	0.783750i	\(0.286696\pi\)
\(17\)			8.55357e11i			1.74937i	0.484689	+	0.874687i	\(0.338933\pi\)
							−0.484689	+	0.874687i	\(0.661067\pi\)
\(19\)		−	3.94199e11i		−	0.280257i	−0.990133	−	0.140129i	\(-0.955248\pi\)
							0.990133	−	0.140129i	\(-0.0447516\pi\)
\(23\)	−5.41125e10			−0.00626446			−0.00313223	−	0.999995i	\(-0.500997\pi\)
							−0.00313223	+	0.999995i	\(0.500997\pi\)
\(29\)		−	3.42747e12i		−	0.0438725i	−0.999759	−	0.0219362i	\(-0.993017\pi\)
							0.999759	−	0.0219362i	\(-0.00698308\pi\)
\(31\)		−	2.38228e14i		−	1.61829i	−0.587607	−	0.809146i	\(-0.699930\pi\)
							0.587607	−	0.809146i	\(-0.300070\pi\)
\(37\)	4.65196e14			0.588464			0.294232	−	0.955734i	\(-0.404936\pi\)
							0.294232	+	0.955734i	\(0.404936\pi\)
\(41\)		−	2.62403e14i		−	0.125176i	−0.998039	−	0.0625881i	\(-0.980065\pi\)
							0.998039	−	0.0625881i	\(-0.0199354\pi\)
\(43\)			2.87474e13i			0.00872265i	0.999990	+	0.00436132i	\(0.00138826\pi\)
							−0.999990	+	0.00436132i	\(0.998612\pi\)
\(47\)	−5.76501e15			−0.751398			−0.375699	−	0.926742i	\(-0.622597\pi\)
							−0.375699	+	0.926742i	\(0.622597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.20.b.a.11.13	✓	36
3.2	odd	2	inner	12.20.b.a.11.24	yes	36
4.3	odd	2	inner	12.20.b.a.11.23	yes	36
12.11	even	2	inner	12.20.b.a.11.14	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.20.b.a.11.13	✓	36	1.1	even	1	trivial
12.20.b.a.11.14	yes	36	12.11	even	2	inner
12.20.b.a.11.23	yes	36	4.3	odd	2	inner
12.20.b.a.11.24	yes	36	3.2	odd	2	inner