Embedded newform 12.15.d.a.7.13

• Label: 12.15.d.a.7.13
• 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			7.13
Root			\(-5.72169 + 9.91026i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.15.d.a.7.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(125.864 - 23.2846i) q^{2} +1262.67i q^{3} +(15299.7 - 5861.41i) q^{4} -92279.1 q^{5} +(29400.7 + 158924. i) q^{6} +1.24880e6i q^{7} +(1.78920e6 - 1.09399e6i) q^{8} -1.59432e6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 182 q^{2} + 9308 q^{4} - 16124 q^{5} + 56862 q^{6} + 4352816 q^{8} - 22320522 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	125.864	−	23.2846i	0.983315	−	0.181911i
							\(3\)			1262.67i			0.577350i
\(5\)	−92279.1			−1.18117										−0.590586	−	0.806975i	\(-0.701103\pi\)
							−0.590586	+	0.806975i	\(0.701103\pi\)
\(7\)			1.24880e6i			1.51638i	0.652035	+	0.758189i	\(0.273916\pi\)
							−0.652035	+	0.758189i	\(0.726084\pi\)
\(11\)			1.79657e7i			0.921924i	0.887420	+	0.460962i	\(0.152496\pi\)
							−0.887420	+	0.460962i	\(0.847504\pi\)
\(13\)	−8.85330e7			−1.41092			−0.705459	−	0.708751i	\(-0.749259\pi\)
							−0.705459	+	0.708751i	\(0.749259\pi\)
\(17\)	2.74034e8			0.667823			0.333912	−	0.942604i	\(-0.391631\pi\)
							0.333912	+	0.942604i	\(0.391631\pi\)
\(19\)			1.46846e9i			1.64280i	0.570350	+	0.821402i	\(0.306807\pi\)
							−0.570350	+	0.821402i	\(0.693193\pi\)
\(23\)			3.18853e8i			0.0936473i	0.998903	+	0.0468237i	\(0.0149099\pi\)
							−0.998903	+	0.0468237i	\(0.985090\pi\)
\(29\)	7.35294e9			0.426260			0.213130	−	0.977024i	\(-0.431634\pi\)
							0.213130	+	0.977024i	\(0.431634\pi\)
\(31\)		−	4.19267e10i		−	1.52391i	−0.647630	−	0.761955i	\(-0.724240\pi\)
							0.647630	−	0.761955i	\(-0.275760\pi\)
\(37\)	8.88622e10			0.936063			0.468031	−	0.883712i	\(-0.344963\pi\)
							0.468031	+	0.883712i	\(0.344963\pi\)
\(41\)	3.04838e10			0.156525			0.0782623	−	0.996933i	\(-0.475063\pi\)
							0.0782623	+	0.996933i	\(0.475063\pi\)
\(43\)		−	1.51054e11i		−	0.555716i	−0.960622	−	0.277858i	\(-0.910375\pi\)
							0.960622	−	0.277858i	\(-0.0896245\pi\)
\(47\)		−	8.00575e10i		−	0.158022i	−0.996874	−	0.0790109i	\(-0.974824\pi\)
							0.996874	−	0.0790109i	\(-0.0251762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.15.d.a.7.13	✓	14
3.2	odd	2		36.15.d.e.19.2		14
4.3	odd	2	inner	12.15.d.a.7.14	yes	14
12.11	even	2		36.15.d.e.19.1		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.15.d.a.7.13	✓	14	1.1	even	1	trivial
12.15.d.a.7.14	yes	14	4.3	odd	2	inner
36.15.d.e.19.1		14	12.11	even	2
36.15.d.e.19.2		14	3.2	odd	2