Embedded newform 12.15.d.a.7.4

• Label: 12.15.d.a.7.4
• Level: $12$
• Weight: $15$
• Character: 12.7
• Analytic conductor: $14.919$
• Analytic rank: $0$
• Dimension: $14$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	12.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9194761782\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	x^14 - x^13 + 9158*x^12 + 65217*x^11 + 61148515*x^10 + 439019974*x^9 + 189458968156*x^8 + 1788546506656*x^7 + 430738312102192*x^6 + 3436431888153888*x^5 + 232550799864813632*x^4 + 2976833070807329792*x^3 + 104055634594691484928*x^2 + 901843441512469175808*x + 8926647999096479376384
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{81}\cdot 3^{41} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7.4
Root			\(29.9153 - 51.8148i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.15.d.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-79.6923 + 100.166i) q^{2} +1262.67i q^{3} +(-3682.27 - 15964.8i) q^{4} +53637.2 q^{5} +(-126476. - 100625. i) q^{6} +1.43231e6i q^{7} +(1.89258e6 + 903439. i) q^{8} -1.59432e6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 182 q^{2} + 9308 q^{4} - 16124 q^{5} + 56862 q^{6} + 4352816 q^{8} - 22320522 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−79.6923	+	100.166i	−0.622596	+	0.782543i
							\(3\)			1262.67i			0.577350i
\(5\)	53637.2			0.686556										0.343278	−	0.939234i	\(-0.388463\pi\)
							0.343278	+	0.939234i	\(0.388463\pi\)
\(7\)			1.43231e6i			1.73920i	0.493754	+	0.869602i	\(0.335624\pi\)
							−0.493754	+	0.869602i	\(0.664376\pi\)
\(11\)			7.27275e6i			0.373207i	0.982435	+	0.186603i	\(0.0597480\pi\)
							−0.982435	+	0.186603i	\(0.940252\pi\)
\(13\)	4.85094e7			0.773077			0.386538	−	0.922273i	\(-0.373671\pi\)
							0.386538	+	0.922273i	\(0.373671\pi\)
\(17\)	−6.93883e8			−1.69100			−0.845500	−	0.533975i	\(-0.820698\pi\)
							−0.845500	+	0.533975i	\(0.820698\pi\)
\(19\)		−	1.07271e9i		−	1.20007i	−0.799972	−	0.600037i	\(-0.795152\pi\)
							0.799972	−	0.600037i	\(-0.204848\pi\)
\(23\)			3.05663e9i			0.897734i	0.893598	+	0.448867i	\(0.148172\pi\)
							−0.893598	+	0.448867i	\(0.851828\pi\)
\(29\)	−4.50702e9			−0.261278			−0.130639	−	0.991430i	\(-0.541703\pi\)
							−0.130639	+	0.991430i	\(0.541703\pi\)
\(31\)			1.56430e10i			0.568574i	0.958739	+	0.284287i	\(0.0917568\pi\)
							−0.958739	+	0.284287i	\(0.908243\pi\)
\(37\)	5.26900e10			0.555030			0.277515	−	0.960721i	\(-0.410489\pi\)
							0.277515	+	0.960721i	\(0.410489\pi\)
\(41\)	−3.51680e11			−1.80576			−0.902882	−	0.429889i	\(-0.858553\pi\)
							−0.902882	+	0.429889i	\(0.858553\pi\)
\(43\)			3.97069e10i			0.146079i	0.997329	+	0.0730393i	\(0.0232699\pi\)
							−0.997329	+	0.0730393i	\(0.976730\pi\)
\(47\)			5.14416e11i			1.01538i	0.861539	+	0.507691i	\(0.169501\pi\)
							−0.861539	+	0.507691i	\(0.830499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.15.d.a.7.4	yes	14
3.2	odd	2		36.15.d.e.19.11		14
4.3	odd	2	inner	12.15.d.a.7.3	✓	14
12.11	even	2		36.15.d.e.19.12		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.15.d.a.7.3	✓	14	4.3	odd	2	inner
12.15.d.a.7.4	yes	14	1.1	even	1	trivial
36.15.d.e.19.11		14	3.2	odd	2
36.15.d.e.19.12		14	12.11	even	2