Embedded newform 12.20.b.a.11.1

• Label: 12.20.b.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	12.11
Dual form			12.20.b.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-723.867 - 17.4378i) q^{2} +(2865.95 - 33971.3i) q^{3} +(523680. + 25245.3i) q^{4} -5.49376e6i q^{5} +(-2.66695e6 + 2.45407e7i) q^{6} +1.05257e8i q^{7} +(-3.78635e8 - 2.74060e7i) q^{8} +(-1.14583e9 - 1.94720e8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−723.867	−	17.4378i	−0.999710	−	0.0240828i
							\(3\)	2865.95	−	33971.3i	0.0840652	−	0.996460i
\(5\)		−	5.49376e6i		−	1.25792i								−0.777436	−	0.628962i	\(-0.783480\pi\)
							0.777436	−	0.628962i	\(-0.216520\pi\)
\(7\)			1.05257e8i			0.985872i	0.870066	+	0.492936i	\(0.164076\pi\)
							−0.870066	+	0.492936i	\(0.835924\pi\)
\(11\)	7.09412e9			0.907127			0.453564	−	0.891224i	\(-0.350152\pi\)
							0.453564	+	0.891224i	\(0.350152\pi\)
\(13\)	−6.63179e10			−1.73448			−0.867240	−	0.497891i	\(-0.834108\pi\)
							−0.867240	+	0.497891i	\(0.834108\pi\)
\(17\)		−	6.50375e11i		−	1.33015i	−0.746779	−	0.665073i	\(-0.768401\pi\)
							0.746779	−	0.665073i	\(-0.231599\pi\)
\(19\)			8.23681e11i			0.585598i	0.956174	+	0.292799i	\(0.0945868\pi\)
							−0.956174	+	0.292799i	\(0.905413\pi\)
\(23\)	−1.08842e13			−1.26003			−0.630015	−	0.776583i	\(-0.716951\pi\)
							−0.630015	+	0.776583i	\(0.716951\pi\)
\(29\)			9.35164e13i			1.19703i	0.801110	+	0.598517i	\(0.204243\pi\)
							−0.801110	+	0.598517i	\(0.795757\pi\)
\(31\)			1.40022e14i			0.951173i	0.879669	+	0.475587i	\(0.157764\pi\)
							−0.879669	+	0.475587i	\(0.842236\pi\)
\(37\)	2.24836e14			0.284413			0.142206	−	0.989837i	\(-0.454580\pi\)
							0.142206	+	0.989837i	\(0.454580\pi\)
\(41\)		−	3.53533e14i		−	0.168649i	−0.996438	−	0.0843245i	\(-0.973127\pi\)
							0.996438	−	0.0843245i	\(-0.0268732\pi\)
\(43\)		−	1.91080e15i		−	0.579783i	−0.957060	−	0.289891i	\(-0.906381\pi\)
							0.957060	−	0.289891i	\(-0.0936192\pi\)
\(47\)	1.54874e15			0.201860			0.100930	−	0.994894i	\(-0.467818\pi\)
							0.100930	+	0.994894i	\(0.467818\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.20.b.a.11.1	✓	36
3.2	odd	2	inner	12.20.b.a.11.36	yes	36
4.3	odd	2	inner	12.20.b.a.11.35	yes	36
12.11	even	2	inner	12.20.b.a.11.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.20.b.a.11.1	✓	36	1.1	even	1	trivial
12.20.b.a.11.2	yes	36	12.11	even	2	inner
12.20.b.a.11.35	yes	36	4.3	odd	2	inner
12.20.b.a.11.36	yes	36	3.2	odd	2	inner