Embedded newform 12.13.d.a.7.12

• Label: 12.13.d.a.7.12
• Level: $12$
• Weight: $13$
• Character: 12.7
• Analytic conductor: $10.968$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	12.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9679258073\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 3*x^11 + 1570*x^10 - 4077*x^9 + 1884069*x^8 - 3551868*x^7 + 881574992*x^6 + 2167220880*x^5 + 304613421264*x^4 + 27792635520*x^3 + 5806366144000*x^2 + 1120796928000*x + 104882177440000
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{57}\cdot 3^{25} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7.12
Root			\(-10.3112 + 17.8596i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.13.d.a.7.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(51.3214 + 38.2376i) q^{2} +420.888i q^{3} +(1171.77 + 3924.81i) q^{4} -24558.7 q^{5} +(-16093.8 + 21600.6i) q^{6} -96476.2i q^{7} +(-89938.5 + 246233. i) q^{8} -177147. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	51.3214	+	38.2376i	0.801897	+	0.597462i
							\(3\)			420.888i			0.577350i
\(5\)	−24558.7			−1.57176										−0.785879	−	0.618380i	\(-0.787789\pi\)
							−0.785879	+	0.618380i	\(0.787789\pi\)
\(7\)		−	96476.2i		−	0.820034i	−0.912078	−	0.410017i	\(-0.865523\pi\)
							0.912078	−	0.410017i	\(-0.134477\pi\)
\(11\)			582563.i			0.328842i	0.986390	+	0.164421i	\(0.0525755\pi\)
							−0.986390	+	0.164421i	\(0.947424\pi\)
\(13\)	890008.			0.184389			0.0921943	−	0.995741i	\(-0.470612\pi\)
							0.0921943	+	0.995741i	\(0.470612\pi\)
\(17\)	−4.03434e7			−1.67139			−0.835697	−	0.549190i	\(-0.814936\pi\)
							−0.835697	+	0.549190i	\(0.814936\pi\)
\(19\)			4.37876e7i			0.930742i	0.885116	+	0.465371i	\(0.154079\pi\)
							−0.885116	+	0.465371i	\(0.845921\pi\)
\(23\)			1.62216e8i			1.09579i	0.836548	+	0.547894i	\(0.184570\pi\)
							−0.836548	+	0.547894i	\(0.815430\pi\)
\(29\)	9.58630e8			1.61162			0.805811	−	0.592173i	\(-0.201730\pi\)
							0.805811	+	0.592173i	\(0.201730\pi\)
\(31\)			1.21198e9i			1.36561i	0.730600	+	0.682806i	\(0.239240\pi\)
							−0.730600	+	0.682806i	\(0.760760\pi\)
\(37\)	8.12465e7			0.0316661			0.0158330	−	0.999875i	\(-0.494960\pi\)
							0.0158330	+	0.999875i	\(0.494960\pi\)
\(41\)	−4.04840e9			−0.852277			−0.426138	−	0.904658i	\(-0.640126\pi\)
							−0.426138	+	0.904658i	\(0.640126\pi\)
\(43\)		−	1.25030e10i		−	1.97789i	−0.148288	−	0.988944i	\(-0.547376\pi\)
							0.148288	−	0.988944i	\(-0.452624\pi\)
\(47\)			1.73724e10i			1.61166i	0.592147	+	0.805830i	\(0.298281\pi\)
							−0.592147	+	0.805830i	\(0.701719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.13.d.a.7.12	yes	12
3.2	odd	2		36.13.d.d.19.1		12
4.3	odd	2	inner	12.13.d.a.7.11	✓	12
8.3	odd	2		192.13.g.e.127.12		12
8.5	even	2		192.13.g.e.127.6		12
12.11	even	2		36.13.d.d.19.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.13.d.a.7.11	✓	12	4.3	odd	2	inner
12.13.d.a.7.12	yes	12	1.1	even	1	trivial
36.13.d.d.19.1		12	3.2	odd	2
36.13.d.d.19.2		12	12.11	even	2
192.13.g.e.127.6		12	8.5	even	2
192.13.g.e.127.12		12	8.3	odd	2