Embedded newform 12.6.b.a.11.1

• Label: 12.6.b.a.11.1
• 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			11.1
Root			\(-2.29661 + 1.35416i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.11
Dual form			12.6.b.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.41443 - 1.63829i) q^{2} +(4.17995 + 15.0176i) q^{3} +(26.6320 + 17.7408i) q^{4} +73.1202i q^{5} +(1.97116 - 88.1596i) q^{6} -51.8209i q^{7} +(-115.132 - 139.687i) q^{8} +(-208.056 + 125.545i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.41443	−	1.63829i	−0.957144	−	0.289612i
							\(3\)	4.17995	+	15.0176i	0.268144	+	0.963379i
\(5\)			73.1202i			1.30801i								0.756489	+	0.654007i	\(0.226913\pi\)
							−0.756489	+	0.654007i	\(0.773087\pi\)
\(7\)		−	51.8209i		−	0.399723i	−0.979824	−	0.199862i	\(-0.935951\pi\)
							0.979824	−	0.199862i	\(-0.0640493\pi\)
\(11\)	371.603			0.925971			0.462986	−	0.886366i	\(-0.346778\pi\)
							0.462986	+	0.886366i	\(0.346778\pi\)
\(13\)	424.112			0.696021			0.348011	−	0.937491i	\(-0.386857\pi\)
							0.348011	+	0.937491i	\(0.386857\pi\)
\(17\)		−	1401.69i		−	1.17634i	−0.808739	−	0.588168i	\(-0.799849\pi\)
							0.808739	−	0.588168i	\(-0.200151\pi\)
\(19\)		−	319.814i		−	0.203242i	−0.994823	−	0.101621i	\(-0.967597\pi\)
							0.994823	−	0.101621i	\(-0.0324029\pi\)
\(23\)	2421.07			0.954306			0.477153	−	0.878820i	\(-0.341669\pi\)
							0.477153	+	0.878820i	\(0.341669\pi\)
\(29\)			3392.51i			0.749076i	0.927212	+	0.374538i	\(0.122199\pi\)
							−0.927212	+	0.374538i	\(0.877801\pi\)
\(31\)			4656.09i			0.870197i	0.900383	+	0.435098i	\(0.143286\pi\)
							−0.900383	+	0.435098i	\(0.856714\pi\)
\(37\)	395.888			0.0475409			0.0237705	−	0.999717i	\(-0.492433\pi\)
							0.0237705	+	0.999717i	\(0.492433\pi\)
\(41\)		−	5780.65i		−	0.537053i	−0.963272	−	0.268526i	\(-0.913463\pi\)
							0.963272	−	0.268526i	\(-0.0865367\pi\)
\(43\)		−	16385.7i		−	1.35143i	−0.737164	−	0.675714i	\(-0.763835\pi\)
							0.737164	−	0.675714i	\(-0.236165\pi\)
\(47\)	−23646.2			−1.56141			−0.780706	−	0.624899i	\(-0.785140\pi\)
							−0.780706	+	0.624899i	\(0.785140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.6.b.a.11.1	✓	8
3.2	odd	2	inner	12.6.b.a.11.8	yes	8
4.3	odd	2	inner	12.6.b.a.11.7	yes	8
8.3	odd	2		192.6.c.e.191.6		8
8.5	even	2		192.6.c.e.191.3		8
12.11	even	2	inner	12.6.b.a.11.2	yes	8
24.5	odd	2		192.6.c.e.191.5		8
24.11	even	2		192.6.c.e.191.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.6.b.a.11.1	✓	8	1.1	even	1	trivial
12.6.b.a.11.2	yes	8	12.11	even	2	inner
12.6.b.a.11.7	yes	8	4.3	odd	2	inner
12.6.b.a.11.8	yes	8	3.2	odd	2	inner
192.6.c.e.191.3		8	8.5	even	2
192.6.c.e.191.4		8	24.11	even	2
192.6.c.e.191.5		8	24.5	odd	2
192.6.c.e.191.6		8	8.3	odd	2