Embedded newform 12.13.d.a.7.10

• Label: 12.13.d.a.7.10
• 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			7.10
Root			\(-2.17114 + 3.76053i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.13.d.a.7.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(29.4812 + 56.8054i) q^{2} +420.888i q^{3} +(-2357.72 + 3349.39i) q^{4} +28943.6 q^{5} +(-23908.8 + 12408.3i) q^{6} +157943. i q^{7} +(-259772. - 35187.3i) q^{8} -177147. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	29.4812	+	56.8054i	0.460644	+	0.887585i
							\(3\)			420.888i			0.577350i
\(5\)	28943.6			1.85239										0.926194	−	0.377047i	\(-0.123061\pi\)
							0.926194	+	0.377047i	\(0.123061\pi\)
\(7\)			157943.i			1.34249i	0.741236	+	0.671245i	\(0.234240\pi\)
							−0.741236	+	0.671245i	\(0.765760\pi\)
\(11\)		−	1.55775e6i		−	0.879311i	−0.898167	−	0.439655i	\(-0.855101\pi\)
							0.898167	−	0.439655i	\(-0.144899\pi\)
\(13\)	−4.64352e6			−0.962027			−0.481013	−	0.876713i	\(-0.659731\pi\)
							−0.481013	+	0.876713i	\(0.659731\pi\)
\(17\)	9.00152e6			0.372926			0.186463	−	0.982462i	\(-0.440298\pi\)
							0.186463	+	0.982462i	\(0.440298\pi\)
\(19\)			4.57988e7i			0.973493i	0.873543	+	0.486747i	\(0.161816\pi\)
							−0.873543	+	0.486747i	\(0.838184\pi\)
\(23\)		−	7.97384e6i		−	0.0538643i	−0.999637	−	0.0269321i	\(-0.991426\pi\)
							0.999637	−	0.0269321i	\(-0.00857380\pi\)
\(29\)	8.09215e8			1.36043			0.680214	−	0.733013i	\(-0.261887\pi\)
							0.680214	+	0.733013i	\(0.261887\pi\)
\(31\)		−	4.31584e8i		−	0.486290i	−0.969990	−	0.243145i	\(-0.921821\pi\)
							0.969990	−	0.243145i	\(-0.0781790\pi\)
\(37\)	−1.23351e9			−0.480764			−0.240382	−	0.970678i	\(-0.577273\pi\)
							−0.240382	+	0.970678i	\(0.577273\pi\)
\(41\)	3.60463e9			0.758854			0.379427	−	0.925222i	\(-0.376121\pi\)
							0.379427	+	0.925222i	\(0.376121\pi\)
\(43\)			1.36544e8i			0.0216005i	0.999942	+	0.0108002i	\(0.00343789\pi\)
							−0.999942	+	0.0108002i	\(0.996562\pi\)
\(47\)		−	5.87982e9i		−	0.545478i	−0.962088	−	0.272739i	\(-0.912070\pi\)
							0.962088	−	0.272739i	\(-0.0879295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.13.d.a.7.10	yes	12
3.2	odd	2		36.13.d.d.19.3		12
4.3	odd	2	inner	12.13.d.a.7.9	✓	12
8.3	odd	2		192.13.g.e.127.7		12
8.5	even	2		192.13.g.e.127.1		12
12.11	even	2		36.13.d.d.19.4		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.13.d.a.7.9	✓	12	4.3	odd	2	inner
12.13.d.a.7.10	yes	12	1.1	even	1	trivial
36.13.d.d.19.3		12	3.2	odd	2
36.13.d.d.19.4		12	12.11	even	2
192.13.g.e.127.1		12	8.5	even	2
192.13.g.e.127.7		12	8.3	odd	2