Embedded newform 12.6.b.a.11.3

• Label: 12.6.b.a.11.3
• Level: $12$
• Weight: $6$
• Character: 12.11
• Analytic conductor: $1.925$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.92460583776\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - x^{7} + 7x^{6} + 6x^{5} - 11x^{4} - 73x^{3} + 223x^{2} - 768x + 912 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.3
Root			\(0.713157 - 2.93555i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.11
Dual form			12.6.b.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91937 - 5.32128i) q^{2} +(-14.9174 - 4.52459i) q^{3} +(-24.6320 + 20.4270i) q^{4} -35.2908i q^{5} +(4.55538 + 88.0639i) q^{6} -190.564i q^{7} +(155.976 + 91.8667i) q^{8} +(202.056 + 134.990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91937	−	5.32128i	−0.339300	−	0.940678i
							\(3\)	−14.9174	−	4.52459i	−0.956950	−	0.290253i
\(5\)		−	35.2908i		−	0.631301i								−0.948876	−	0.315650i	\(-0.897777\pi\)
							0.948876	−	0.315650i	\(-0.102223\pi\)
\(7\)		−	190.564i		−	1.46993i	−0.678107	−	0.734963i	\(-0.737199\pi\)
							0.678107	−	0.734963i	\(-0.262801\pi\)
\(11\)	33.3357			0.0830668			0.0415334	−	0.999137i	\(-0.486776\pi\)
							0.0415334	+	0.999137i	\(0.486776\pi\)
\(13\)	−396.112			−0.650070			−0.325035	−	0.945702i	\(-0.605376\pi\)
							−0.325035	+	0.945702i	\(0.605376\pi\)
\(17\)		−	1279.32i		−	1.07364i	−0.843698	−	0.536818i	\(-0.819626\pi\)
							0.843698	−	0.536818i	\(-0.180374\pi\)
\(19\)			1333.42i			0.847387i	0.905806	+	0.423693i	\(0.139267\pi\)
							−0.905806	+	0.423693i	\(0.860733\pi\)
\(23\)	2235.78			0.881270			0.440635	−	0.897686i	\(-0.354753\pi\)
							0.440635	+	0.897686i	\(0.354753\pi\)
\(29\)		−	3984.36i		−	0.879760i	−0.898057	−	0.439880i	\(-0.855021\pi\)
							0.898057	−	0.439880i	\(-0.144979\pi\)
\(31\)		−	3670.76i		−	0.686044i	−0.939327	−	0.343022i	\(-0.888549\pi\)
							0.939327	−	0.343022i	\(-0.111451\pi\)
\(37\)	1216.11			0.146039			0.0730196	−	0.997331i	\(-0.476736\pi\)
							0.0730196	+	0.997331i	\(0.476736\pi\)
\(41\)			9048.65i			0.840667i	0.907370	+	0.420334i	\(0.138087\pi\)
							−0.907370	+	0.420334i	\(0.861913\pi\)
\(43\)			5349.13i			0.441176i	0.975367	+	0.220588i	\(0.0707977\pi\)
							−0.975367	+	0.220588i	\(0.929202\pi\)
\(47\)	16413.1			1.08379			0.541895	−	0.840446i	\(-0.317707\pi\)
							0.541895	+	0.840446i	\(0.317707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.6.b.a.11.3	✓	8
3.2	odd	2	inner	12.6.b.a.11.6	yes	8
4.3	odd	2	inner	12.6.b.a.11.5	yes	8
8.3	odd	2		192.6.c.e.191.1		8
8.5	even	2		192.6.c.e.191.8		8
12.11	even	2	inner	12.6.b.a.11.4	yes	8
24.5	odd	2		192.6.c.e.191.2		8
24.11	even	2		192.6.c.e.191.7		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.6.b.a.11.3	✓	8	1.1	even	1	trivial
12.6.b.a.11.4	yes	8	12.11	even	2	inner
12.6.b.a.11.5	yes	8	4.3	odd	2	inner
12.6.b.a.11.6	yes	8	3.2	odd	2	inner
192.6.c.e.191.1		8	8.3	odd	2
192.6.c.e.191.2		8	24.5	odd	2
192.6.c.e.191.7		8	24.11	even	2
192.6.c.e.191.8		8	8.5	even	2