Embedded newform 12.20.b.a.11.11

• Label: 12.20.b.a.11.11
• 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			11.11
Character	\(\chi\)	\(=\)	12.11
Dual form			12.20.b.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-395.354 - 606.616i) q^{2} +(-27131.3 + 20643.4i) q^{3} +(-211678. + 479656. i) q^{4} +730760. i q^{5} +(2.32491e7 + 8.29684e6i) q^{6} -7.02914e6i q^{7} +(3.74655e8 - 6.12270e7i) q^{8} +(3.09959e8 - 1.12017e9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−395.354	−	606.616i	−0.546011	−	0.837778i
							\(3\)	−27131.3	+	20643.4i	−0.795828	+	0.605522i
\(5\)			730760.i			0.167325i								0.996494	+	0.0836623i	\(0.0266617\pi\)
							−0.996494	+	0.0836623i	\(0.973338\pi\)
\(7\)		−	7.02914e6i		−	0.0658371i	−0.999458	−	0.0329186i	\(-0.989520\pi\)
							0.999458	−	0.0329186i	\(-0.0104802\pi\)
\(11\)	9.08653e9			1.16190			0.580948	−	0.813941i	\(-0.302682\pi\)
							0.580948	+	0.813941i	\(0.302682\pi\)
\(13\)	−2.88811e10			−0.755357			−0.377679	−	0.925937i	\(-0.623278\pi\)
							−0.377679	+	0.925937i	\(0.623278\pi\)
\(17\)		−	3.88234e11i		−	0.794014i	−0.917815	−	0.397007i	\(-0.870049\pi\)
							0.917815	−	0.397007i	\(-0.129951\pi\)
\(19\)			1.03792e12i			0.737910i	0.929447	+	0.368955i	\(0.120284\pi\)
							−0.929447	+	0.368955i	\(0.879716\pi\)
\(23\)	2.94119e12			0.340493			0.170247	−	0.985401i	\(-0.445544\pi\)
							0.170247	+	0.985401i	\(0.445544\pi\)
\(29\)			8.24391e13i			1.05524i	0.849480	+	0.527621i	\(0.176916\pi\)
							−0.849480	+	0.527621i	\(0.823084\pi\)
\(31\)		−	2.06352e12i		−	0.0140175i	−0.999975	−	0.00700877i	\(-0.997769\pi\)
							0.999975	−	0.00700877i	\(-0.00223098\pi\)
\(37\)	−1.42060e15			−1.79703			−0.898517	−	0.438938i	\(-0.855355\pi\)
							−0.898517	+	0.438938i	\(0.855355\pi\)
\(41\)			1.17371e15i			0.559903i	0.960014	+	0.279952i	\(0.0903185\pi\)
							−0.960014	+	0.279952i	\(0.909682\pi\)
\(43\)			3.15853e15i			0.958375i	0.877713	+	0.479187i	\(0.159069\pi\)
							−0.877713	+	0.479187i	\(0.840931\pi\)
\(47\)	−8.86803e15			−1.15584			−0.577920	−	0.816094i	\(-0.696135\pi\)
							−0.577920	+	0.816094i	\(0.696135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.20.b.a.11.11	✓	36
3.2	odd	2	inner	12.20.b.a.11.26	yes	36
4.3	odd	2	inner	12.20.b.a.11.25	yes	36
12.11	even	2	inner	12.20.b.a.11.12	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.20.b.a.11.11	✓	36	1.1	even	1	trivial
12.20.b.a.11.12	yes	36	12.11	even	2	inner
12.20.b.a.11.25	yes	36	4.3	odd	2	inner
12.20.b.a.11.26	yes	36	3.2	odd	2	inner