Embedded newform 12.20.b.a.11.7

• Label: 12.20.b.a.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	12.11
Dual form			12.20.b.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-627.004 - 362.153i) q^{2} +(8684.36 + 32967.3i) q^{3} +(261979. + 454142. i) q^{4} -6.03352e6i q^{5} +(6.49408e6 - 2.38157e7i) q^{6} +9.28886e7i q^{7} +(207167. - 3.79625e8i) q^{8} +(-1.01143e9 + 5.72600e8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−627.004	−	362.153i	−0.865934	−	0.500158i
							\(3\)	8684.36	+	32967.3i	0.254733	+	0.967011i
\(5\)		−	6.03352e6i		−	1.38151i								−0.723087	−	0.690757i	\(-0.757277\pi\)
							0.723087	−	0.690757i	\(-0.242723\pi\)
\(7\)			9.28886e7i			0.870023i	0.900425	+	0.435012i	\(0.143256\pi\)
							−0.900425	+	0.435012i	\(0.856744\pi\)
\(11\)	−6.13492e9			−0.784473			−0.392237	−	0.919864i	\(-0.628299\pi\)
							−0.392237	+	0.919864i	\(0.628299\pi\)
\(13\)	4.99409e10			1.30615			0.653077	−	0.757291i	\(-0.273478\pi\)
							0.653077	+	0.757291i	\(0.273478\pi\)
\(17\)			2.01629e11i			0.412372i	0.978513	+	0.206186i	\(0.0661051\pi\)
							−0.978513	+	0.206186i	\(0.933895\pi\)
\(19\)		−	4.67496e11i		−	0.332368i	−0.986095	−	0.166184i	\(-0.946856\pi\)
							0.986095	−	0.166184i	\(-0.0531445\pi\)
\(23\)	−8.46617e12			−0.980105			−0.490052	−	0.871693i	\(-0.663022\pi\)
							−0.490052	+	0.871693i	\(0.663022\pi\)
\(29\)			2.39076e13i			0.306024i	0.988224	+	0.153012i	\(0.0488973\pi\)
							−0.988224	+	0.153012i	\(0.951103\pi\)
\(31\)		−	1.41145e14i		−	0.958801i	−0.877596	−	0.479400i	\(-0.840854\pi\)
							0.877596	−	0.479400i	\(-0.159146\pi\)
\(37\)	−1.37248e15			−1.73616			−0.868082	−	0.496421i	\(-0.834647\pi\)
							−0.868082	+	0.496421i	\(0.834647\pi\)
\(41\)		−	3.22320e15i		−	1.53759i	−0.639496	−	0.768794i	\(-0.720857\pi\)
							0.639496	−	0.768794i	\(-0.279143\pi\)
\(43\)		−	5.84454e15i		−	1.77337i	−0.462371	−	0.886687i	\(-0.653001\pi\)
							0.462371	−	0.886687i	\(-0.346999\pi\)
\(47\)	−1.11649e16			−1.45521			−0.727603	−	0.685998i	\(-0.759366\pi\)
							−0.727603	+	0.685998i	\(0.759366\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.20.b.a.11.7	✓	36
3.2	odd	2	inner	12.20.b.a.11.30	yes	36
4.3	odd	2	inner	12.20.b.a.11.29	yes	36
12.11	even	2	inner	12.20.b.a.11.8	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.20.b.a.11.7	✓	36	1.1	even	1	trivial
12.20.b.a.11.8	yes	36	12.11	even	2	inner
12.20.b.a.11.29	yes	36	4.3	odd	2	inner
12.20.b.a.11.30	yes	36	3.2	odd	2	inner