Embedded newform 12.11.d.a.7.1

• Label: 12.11.d.a.7.1
• Level: $12$
• Weight: $11$
• Character: 12.7
• Analytic conductor: $7.624$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	12.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.62428703208\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 2*x^9 + 198*x^8 - 1066*x^7 + 30608*x^6 - 147714*x^5 + 2092829*x^4 - 9587938*x^3 + 101463432*x^2 - 344403128*x + 1648684816
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{39}\cdot 3^{18} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7.1
Root			\(4.69893 - 8.13879i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.11.d.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-30.5317 - 9.58205i) q^{2} -140.296i q^{3} +(840.369 + 585.112i) q^{4} +2598.88 q^{5} +(-1344.32 + 4283.48i) q^{6} -4231.42i q^{7} +(-20051.3 - 25916.9i) q^{8} -19683.0 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 22 q^{2} - 644 q^{4} + 3116 q^{5} + 2430 q^{6} + 20176 q^{8} - 196830 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−30.5317	−	9.58205i	−0.954115	−	0.299439i
							\(3\)		−	140.296i		−	0.577350i
\(5\)	2598.88			0.831641										0.415820	−	0.909447i	\(-0.363495\pi\)
							0.415820	+	0.909447i	\(0.363495\pi\)
\(7\)		−	4231.42i		−	0.251765i	−0.992045	−	0.125883i	\(-0.959824\pi\)
							0.992045	−	0.125883i	\(-0.0401763\pi\)
\(11\)		−	275746.i		−	1.71217i	−0.516838	−	0.856083i	\(-0.672891\pi\)
							0.516838	−	0.856083i	\(-0.327109\pi\)
\(13\)	−503390.			−1.35577			−0.677887	−	0.735166i	\(-0.737104\pi\)
							−0.677887	+	0.735166i	\(0.737104\pi\)
\(17\)	796250.			0.560796			0.280398	−	0.959884i	\(-0.409534\pi\)
							0.280398	+	0.959884i	\(0.409534\pi\)
\(19\)		−	659638.i		−	0.266402i	−0.991089	−	0.133201i	\(-0.957474\pi\)
							0.991089	−	0.133201i	\(-0.0425256\pi\)
\(23\)		−	9.73033e6i		−	1.51178i	−0.654699	−	0.755890i	\(-0.727205\pi\)
							0.654699	−	0.755890i	\(-0.272795\pi\)
\(29\)	1.57660e6			0.0768657			0.0384329	−	0.999261i	\(-0.487763\pi\)
							0.0384329	+	0.999261i	\(0.487763\pi\)
\(31\)		−	2.50977e7i		−	0.876647i	−0.898817	−	0.438323i	\(-0.855572\pi\)
							0.898817	−	0.438323i	\(-0.144428\pi\)
\(37\)	1.92346e7			0.277380			0.138690	−	0.990336i	\(-0.455711\pi\)
							0.138690	+	0.990336i	\(0.455711\pi\)
\(41\)	1.55004e8			1.33790			0.668951	−	0.743307i	\(-0.266744\pi\)
							0.668951	+	0.743307i	\(0.266744\pi\)
\(43\)			2.62293e8i			1.78420i	0.451836	+	0.892101i	\(0.350769\pi\)
							−0.451836	+	0.892101i	\(0.649231\pi\)
\(47\)			1.93022e8i			0.841624i	0.907148	+	0.420812i	\(0.138255\pi\)
							−0.907148	+	0.420812i	\(0.861745\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.11.d.a.7.1	✓	10
3.2	odd	2		36.11.d.e.19.10		10
4.3	odd	2	inner	12.11.d.a.7.2	yes	10
8.3	odd	2		192.11.g.e.127.2		10
8.5	even	2		192.11.g.e.127.7		10
12.11	even	2		36.11.d.e.19.9		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.11.d.a.7.1	✓	10	1.1	even	1	trivial
12.11.d.a.7.2	yes	10	4.3	odd	2	inner
36.11.d.e.19.9		10	12.11	even	2
36.11.d.e.19.10		10	3.2	odd	2
192.11.g.e.127.2		10	8.3	odd	2
192.11.g.e.127.7		10	8.5	even	2