Embedded newform 12.15.d.a.7.11

• Label: 12.15.d.a.7.11
• 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			7.11
Root			\(12.2149 + 21.1568i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.15.d.a.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(121.962 - 38.8495i) q^{2} -1262.67i q^{3} +(13365.4 - 9476.31i) q^{4} +122615. q^{5} +(-49053.8 - 153997. i) q^{6} +1.05178e6i q^{7} +(1.26193e6 - 1.67499e6i) q^{8} -1.59432e6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 182 q^{2} + 9308 q^{4} - 16124 q^{5} + 56862 q^{6} + 4352816 q^{8} - 22320522 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	121.962	−	38.8495i	0.952828	−	0.303511i
							\(3\)		−	1262.67i		−	0.577350i
\(5\)	122615.			1.56947										0.784736	−	0.619830i	\(-0.212799\pi\)
							0.784736	+	0.619830i	\(0.212799\pi\)
\(7\)			1.05178e6i			1.27713i	0.769566	+	0.638567i	\(0.220473\pi\)
							−0.769566	+	0.638567i	\(0.779527\pi\)
\(11\)		−	2.20758e7i		−	1.13284i	−0.824117	−	0.566419i	\(-0.808328\pi\)
							0.824117	−	0.566419i	\(-0.191672\pi\)
\(13\)	3.10236e7			0.494412			0.247206	−	0.968963i	\(-0.420488\pi\)
							0.247206	+	0.968963i	\(0.420488\pi\)
\(17\)	−2.03936e8			−0.496995			−0.248497	−	0.968633i	\(-0.579937\pi\)
							−0.248497	+	0.968633i	\(0.579937\pi\)
\(19\)			4.23046e8i			0.473274i	0.971598	+	0.236637i	\(0.0760452\pi\)
							−0.971598	+	0.236637i	\(0.923955\pi\)
\(23\)		−	3.68228e9i		−	1.08149i	−0.841187	−	0.540744i	\(-0.818143\pi\)
							0.841187	−	0.540744i	\(-0.181857\pi\)
\(29\)	−2.80865e10			−1.62822			−0.814109	−	0.580713i	\(-0.802774\pi\)
							−0.814109	+	0.580713i	\(0.802774\pi\)
\(31\)			4.74123e10i			1.72329i	0.507508	+	0.861647i	\(0.330567\pi\)
							−0.507508	+	0.861647i	\(0.669433\pi\)
\(37\)	−1.08565e11			−1.14361			−0.571804	−	0.820390i	\(-0.693756\pi\)
							−0.571804	+	0.820390i	\(0.693756\pi\)
\(41\)	2.80861e11			1.44213			0.721065	−	0.692867i	\(-0.243653\pi\)
							0.721065	+	0.692867i	\(0.243653\pi\)
\(43\)			2.68648e11i			0.988335i	0.869367	+	0.494168i	\(0.164527\pi\)
							−0.869367	+	0.494168i	\(0.835473\pi\)
\(47\)			4.43027e11i			0.874471i	0.899347	+	0.437236i	\(0.144042\pi\)
							−0.899347	+	0.437236i	\(0.855958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.15.d.a.7.11	✓	14
3.2	odd	2		36.15.d.e.19.4		14
4.3	odd	2	inner	12.15.d.a.7.12	yes	14
12.11	even	2		36.15.d.e.19.3		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.15.d.a.7.11	✓	14	1.1	even	1	trivial
12.15.d.a.7.12	yes	14	4.3	odd	2	inner
36.15.d.e.19.3		14	12.11	even	2
36.15.d.e.19.4		14	3.2	odd	2