Embedded newform 12.22.b.a.11.20

• Label: 12.22.b.a.11.20
• Level: $12$
• Weight: $22$
• Character: 12.11
• Analytic conductor: $33.537$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.5372813144\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.20
Character	\(\chi\)	\(=\)	12.11
Dual form			12.22.b.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-219.802 + 1431.38i) q^{2} +(-101700. + 10835.4i) q^{3} +(-2.00053e6 - 629239. i) q^{4} +7.08257e6i q^{5} +(6.84438e6 - 1.47953e8i) q^{6} +8.25046e8i q^{7} +(1.34040e9 - 2.72520e9i) q^{8} +(1.02255e10 - 2.20393e9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 1212040 q^{4} - 73477608 q^{6} + 4862757768 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−219.802	+	1431.38i	−0.151781	+	0.988414i
							\(3\)	−101700.	+	10835.4i	−0.994372	+	0.105943i
\(5\)			7.08257e6i			0.324344i								0.986763	+	0.162172i	\(0.0518499\pi\)
							−0.986763	+	0.162172i	\(0.948150\pi\)
\(7\)			8.25046e8i			1.10395i	0.833861	+	0.551974i	\(0.186125\pi\)
							−0.833861	+	0.551974i	\(0.813875\pi\)
\(11\)	5.64925e10			0.656700			0.328350	−	0.944556i	\(-0.393507\pi\)
							0.328350	+	0.944556i	\(0.393507\pi\)
\(13\)	7.77040e11			1.56328			0.781642	−	0.623727i	\(-0.214382\pi\)
							0.781642	+	0.623727i	\(0.214382\pi\)
\(17\)			2.14638e12i			0.258222i	0.991630	+	0.129111i	\(0.0412123\pi\)
							−0.991630	+	0.129111i	\(0.958788\pi\)
\(19\)			3.98877e13i			1.49254i	0.665644	+	0.746270i	\(0.268157\pi\)
							−0.665644	+	0.746270i	\(0.731843\pi\)
\(23\)	−1.55135e13			−0.0780850			−0.0390425	−	0.999238i	\(-0.512431\pi\)
							−0.0390425	+	0.999238i	\(0.512431\pi\)
\(29\)		−	2.71758e15i		−	1.19951i	−0.800185	−	0.599754i	\(-0.795265\pi\)
							0.800185	−	0.599754i	\(-0.204735\pi\)
\(31\)		−	4.65708e15i		−	1.02051i	−0.860024	−	0.510253i	\(-0.829552\pi\)
							0.860024	−	0.510253i	\(-0.170448\pi\)
\(37\)	9.19330e15			0.314307			0.157153	−	0.987574i	\(-0.449768\pi\)
							0.157153	+	0.987574i	\(0.449768\pi\)
\(41\)			1.28399e17i			1.49393i	0.664862	+	0.746967i	\(0.268491\pi\)
							−0.664862	+	0.746967i	\(0.731509\pi\)
\(43\)			6.02134e16i			0.424888i	0.977173	+	0.212444i	\(0.0681423\pi\)
							−0.977173	+	0.212444i	\(0.931858\pi\)
\(47\)	−3.86477e17			−1.07176			−0.535878	−	0.844296i	\(-0.680019\pi\)
							−0.535878	+	0.844296i	\(0.680019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.22.b.a.11.20	yes	40
3.2	odd	2	inner	12.22.b.a.11.21	yes	40
4.3	odd	2	inner	12.22.b.a.11.22	yes	40
12.11	even	2	inner	12.22.b.a.11.19	✓	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.22.b.a.11.19	✓	40	12.11	even	2	inner
12.22.b.a.11.20	yes	40	1.1	even	1	trivial
12.22.b.a.11.21	yes	40	3.2	odd	2	inner
12.22.b.a.11.22	yes	40	4.3	odd	2	inner