Embedded newform 38.8.c.a.7.4

• Label: 38.8.c.a.7.4
• 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			7.4
Root			\(19.2647 + 33.3674i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.8.c.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(20.2647 + 35.0995i) q^{3} +(-32.0000 + 55.4256i) q^{4} +(-75.0460 - 129.983i) q^{5} +(162.117 - 280.796i) q^{6} -892.194 q^{7} +512.000 q^{8} +(272.185 - 471.439i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{2} + 12 q^{3} - 384 q^{4} + 124 q^{5} + 96 q^{6} - 1036 q^{7} + 6144 q^{8} - 7232 q^{9}+O(q^{10}) \)

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
							\(3\)	20.2647	+	35.0995i	0.433327	+	0.750544i	0.997157	−	0.0753465i	\(-0.0240063\pi\)
−0.563831	+	0.825890i	\(0.690673\pi\)
\(5\)	−75.0460	−	129.983i	−0.268493	−	0.465043i	0.699980	−	0.714162i	\(-0.253192\pi\)
							−0.968473	+	0.249119i	\(0.919859\pi\)
\(7\)	−892.194			−0.983141			−0.491571	−	0.870838i	\(-0.663577\pi\)
							−0.491571	+	0.870838i	\(0.663577\pi\)
\(11\)	7582.57			1.71768			0.858839	−	0.512246i	\(-0.171186\pi\)
							0.858839	+	0.512246i	\(0.171186\pi\)
\(13\)	3996.97	−	6922.95i	0.504579	−	0.873956i	−0.495407	−	0.868661i	\(-0.664981\pi\)
							0.999986	−	0.00529532i	\(-0.00168556\pi\)
\(17\)	−5167.65	−	8950.63i	−0.255107	−	0.441857i	0.709818	−	0.704385i	\(-0.248777\pi\)
							−0.964924	+	0.262528i	\(0.915444\pi\)
\(19\)	−12223.5	−	27284.8i	−0.408844	−	0.912604i
							\(23\)	22770.5	−	39439.6i	0.390233	−	0.675904i	−0.602247	−	0.798310i	\(-0.705728\pi\)
0.992480	+	0.122406i	\(0.0390611\pi\)
\(29\)	−45965.8	+	79615.1i	−0.349979	+	0.606181i	−0.986245	−	0.165288i	\(-0.947145\pi\)
							0.636266	+	0.771469i	\(0.280478\pi\)
\(31\)	−22164.3			−0.133625			−0.0668125	−	0.997766i	\(-0.521283\pi\)
							−0.0668125	+	0.997766i	\(0.521283\pi\)
\(37\)	98723.9			0.320418			0.160209	−	0.987083i	\(-0.448783\pi\)
							0.160209	+	0.987083i	\(0.448783\pi\)
\(41\)	−421851.	−	730668.i	−0.955907	−	1.65568i	−0.732278	−	0.681005i	\(-0.761543\pi\)
							−0.223629	−	0.974674i	\(-0.571790\pi\)
\(43\)	353616.	+	612480.i	0.678253	+	1.17477i	0.975507	+	0.219971i	\(0.0705961\pi\)
							−0.297253	+	0.954799i	\(0.596071\pi\)
\(47\)	−275565.	+	477293.i	−0.387152	+	0.670567i	−0.992065	−	0.125724i	\(-0.959875\pi\)
							0.604913	+	0.796291i	\(0.293208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.a.7.4	✓	12
19.11	even	3	inner	38.8.c.a.11.4	yes	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.a.7.4	✓	12	1.1	even	1	trivial
38.8.c.a.11.4	yes	12	19.11	even	3	inner