Embedded newform 38.8.c.a.7.5

• Label: 38.8.c.a.7.5
• Level: $38$
• Weight: $8$
• Character: 38.7
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8706309684\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	x^12 + 10165*x^10 - 228496*x^9 + 79937698*x^8 - 1362528496*x^7 + 222057453949*x^6 - 1418140039384*x^5 + 423936076682002*x^4 - 1421321152056072*x^3 + 376701330885210661*x^2 + 2892184322477465880*x + 206636219229734030025
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.5
Root			\(25.8545 + 44.7813i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.8.c.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(26.8545 + 46.5134i) q^{3} +(-32.0000 + 55.4256i) q^{4} +(253.607 + 439.261i) q^{5} +(214.836 - 372.107i) q^{6} -1033.15 q^{7} +512.000 q^{8} +(-348.830 + 604.192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{2} + 12 q^{3} - 384 q^{4} + 124 q^{5} + 96 q^{6} - 1036 q^{7} + 6144 q^{8} - 7232 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.00000	−	6.92820i	−0.353553	−	0.612372i
							\(3\)	26.8545	+	46.5134i	0.574239	+	0.994612i	0.996124	+	0.0879621i	\(0.0280355\pi\)
−0.421884	+	0.906650i	\(0.638631\pi\)
\(5\)	253.607	+	439.261i	0.907333	+	1.57155i	0.817755	+	0.575567i	\(0.195219\pi\)
							0.0895785	+	0.995980i	\(0.471448\pi\)
\(7\)	−1033.15			−1.13846			−0.569231	−	0.822178i	\(-0.692759\pi\)
							−0.569231	+	0.822178i	\(0.692759\pi\)
\(11\)	−7906.62			−1.79109			−0.895543	−	0.444975i	\(-0.853212\pi\)
							−0.895543	+	0.444975i	\(0.853212\pi\)
\(13\)	3118.88	−	5402.06i	0.393729	−	0.681958i	−0.599209	−	0.800592i	\(-0.704518\pi\)
							0.992938	+	0.118634i	\(0.0378516\pi\)
\(17\)	−2603.67	−	4509.69i	−0.128533	−	0.222626i	0.794575	−	0.607166i	\(-0.207694\pi\)
							−0.923108	+	0.384540i	\(0.874360\pi\)
\(19\)	23952.4	+	17892.9i	0.801145	+	0.598470i
							\(23\)	−48488.7	+	83984.8i	−0.830984	+	1.43931i	0.0662740	+	0.997801i	\(0.478889\pi\)
−0.897258	+	0.441506i	\(0.854444\pi\)
\(29\)	52156.4	−	90337.6i	0.397114	−	0.687821i	−0.596255	−	0.802795i	\(-0.703345\pi\)
							0.993368	+	0.114974i	\(0.0366785\pi\)
\(31\)	−108673.			−0.655174			−0.327587	−	0.944821i	\(-0.606235\pi\)
							−0.327587	+	0.944821i	\(0.606235\pi\)
\(37\)	324853.			1.05434			0.527171	−	0.849759i	\(-0.323253\pi\)
							0.527171	+	0.849759i	\(0.323253\pi\)
\(41\)	103728.	+	179663.i	0.235047	+	0.407113i	0.959286	−	0.282436i	\(-0.0911423\pi\)
							−0.724239	+	0.689549i	\(0.757809\pi\)
\(43\)	67213.7	+	116417.i	0.128919	+	0.223295i	0.923258	−	0.384180i	\(-0.125516\pi\)
							−0.794339	+	0.607475i	\(0.792183\pi\)
\(47\)	−401378.	+	695208.i	−0.563912	+	0.976725i	0.433238	+	0.901280i	\(0.357371\pi\)
							−0.997150	+	0.0754449i	\(0.975962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.a.7.5	✓	12
19.11	even	3	inner	38.8.c.a.11.5	yes	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.a.7.5	✓	12	1.1	even	1	trivial
38.8.c.a.11.5	yes	12	19.11	even	3	inner