Embedded newform 12.16.b.a.11.18

• Label: 12.16.b.a.11.18
• Level: $12$
• Weight: $16$
• Character: 12.11
• Analytic conductor: $17.123$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1232206120\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.18
Character	\(\chi\)	\(=\)	12.11
Dual form			12.16.b.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(80.5973 + 162.087i) q^{2} +(2767.36 + 2586.62i) q^{3} +(-19776.2 + 26127.5i) q^{4} +332581. i q^{5} +(-196215. + 657027. i) q^{6} -2.13481e6i q^{7} +(-5.82882e6 - 1.09965e6i) q^{8} +(967701. + 1.43162e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 26968 q^{4} + 823656 q^{6} - 3812052 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	80.5973	+	162.087i	0.445241	+	0.895411i
							\(3\)	2767.36	+	2586.62i	0.730562	+	0.682847i
\(5\)			332581.i			1.90380i								0.306404	+	0.951902i	\(0.400874\pi\)
							−0.306404	+	0.951902i	\(0.599126\pi\)
\(7\)		−	2.13481e6i		−	0.979769i	−0.871787	−	0.489884i	\(-0.837039\pi\)
							0.871787	−	0.489884i	\(-0.162961\pi\)
\(11\)	7.24202e7			1.12051			0.560253	−	0.828321i	\(-0.310704\pi\)
							0.560253	+	0.828321i	\(0.310704\pi\)
\(13\)	5.97405e7			0.264055			0.132027	−	0.991246i	\(-0.457851\pi\)
							0.132027	+	0.991246i	\(0.457851\pi\)
\(17\)		−	9.88823e8i		−	0.584456i	−0.956349	−	0.292228i	\(-0.905603\pi\)
							0.956349	−	0.292228i	\(-0.0943966\pi\)
\(19\)			6.22690e7i			0.0159816i	0.999968	+	0.00799079i	\(0.00254357\pi\)
							−0.999968	+	0.00799079i	\(0.997456\pi\)
\(23\)	2.88688e9			0.176795			0.0883976	−	0.996085i	\(-0.471825\pi\)
							0.0883976	+	0.996085i	\(0.471825\pi\)
\(29\)		−	3.09233e10i		−	0.332890i	−0.986051	−	0.166445i	\(-0.946771\pi\)
							0.986051	−	0.166445i	\(-0.0532289\pi\)
\(31\)			4.97193e10i			0.324573i	0.986744	+	0.162286i	\(0.0518868\pi\)
							−0.986744	+	0.162286i	\(0.948113\pi\)
\(37\)	2.95345e11			0.511466			0.255733	−	0.966747i	\(-0.417683\pi\)
							0.255733	+	0.966747i	\(0.417683\pi\)
\(41\)			1.49988e12i			1.20276i	0.798964	+	0.601379i	\(0.205382\pi\)
							−0.798964	+	0.601379i	\(0.794618\pi\)
\(43\)			2.69129e12i			1.50990i	0.655785	+	0.754948i	\(0.272338\pi\)
							−0.655785	+	0.754948i	\(0.727662\pi\)
\(47\)	4.45404e12			1.28239			0.641194	−	0.767378i	\(-0.278439\pi\)
							0.641194	+	0.767378i	\(0.278439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.16.b.a.11.18	yes	28
3.2	odd	2	inner	12.16.b.a.11.11	✓	28
4.3	odd	2	inner	12.16.b.a.11.12	yes	28
12.11	even	2	inner	12.16.b.a.11.17	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.16.b.a.11.11	✓	28	3.2	odd	2	inner
12.16.b.a.11.12	yes	28	4.3	odd	2	inner
12.16.b.a.11.17	yes	28	12.11	even	2	inner
12.16.b.a.11.18	yes	28	1.1	even	1	trivial