Embedded newform 38.8.c.a.7.3

• Label: 38.8.c.a.7.3
• 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			7.3
Root			\(-12.7080 - 22.0110i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.8.c.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(-11.7080 - 20.2789i) q^{3} +(-32.0000 + 55.4256i) q^{4} +(68.7885 + 119.145i) q^{5} +(-93.6643 + 162.231i) q^{6} +1024.63 q^{7} +512.000 q^{8} +(819.344 - 1419.14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{2} + 12 q^{3} - 384 q^{4} + 124 q^{5} + 96 q^{6} - 1036 q^{7} + 6144 q^{8} - 7232 q^{9}+O(q^{10}) \)

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
							\(3\)	−11.7080	−	20.2789i	−0.250357	−	0.433631i	0.713267	−	0.700892i	\(-0.247215\pi\)
−0.963624	+	0.267261i	\(0.913881\pi\)
\(5\)	68.7885	+	119.145i	0.246105	+	0.426267i	0.962442	−	0.271488i	\(-0.0875158\pi\)
							−0.716336	+	0.697755i	\(0.754182\pi\)
\(7\)	1024.63			1.12907			0.564537	−	0.825408i	\(-0.309055\pi\)
							0.564537	+	0.825408i	\(0.309055\pi\)
\(11\)	−1607.88			−0.364234			−0.182117	−	0.983277i	\(-0.558295\pi\)
							−0.182117	+	0.983277i	\(0.558295\pi\)
\(13\)	−262.324	+	454.359i	−0.0331159	+	0.0573584i	−0.882108	−	0.471047i	\(-0.843876\pi\)
							0.848992	+	0.528405i	\(0.177210\pi\)
\(17\)	−7745.95	−	13416.4i	−0.382388	−	0.662315i	0.609015	−	0.793158i	\(-0.291565\pi\)
							−0.991403	+	0.130844i	\(0.958231\pi\)
\(19\)	128.623	−	29897.4i	0.00430211	−	0.999991i
							\(23\)	13699.7	−	23728.6i	0.234782	−	0.406654i	−0.724427	−	0.689351i	\(-0.757896\pi\)
0.959209	+	0.282697i	\(0.0912291\pi\)
\(29\)	67109.5	−	116237.i	0.510965	−	0.885017i	−0.488955	−	0.872309i	\(-0.662622\pi\)
							0.999919	−	0.0127076i	\(-0.00404505\pi\)
\(31\)	79377.0			0.478552			0.239276	−	0.970952i	\(-0.423090\pi\)
							0.239276	+	0.970952i	\(0.423090\pi\)
\(37\)	239193.			0.776322			0.388161	−	0.921592i	\(-0.373111\pi\)
							0.388161	+	0.921592i	\(0.373111\pi\)
\(41\)	353896.	+	612966.i	0.801922	+	1.38897i	0.918350	+	0.395770i	\(0.129522\pi\)
							−0.116428	+	0.993199i	\(0.537144\pi\)
\(43\)	−165133.	−	286019.i	−0.316734	−	0.548600i	0.663070	−	0.748557i	\(-0.269253\pi\)
							−0.979805	+	0.199957i	\(0.935920\pi\)
\(47\)	78865.8	−	136600.i	0.110802	−	0.191914i	−0.805292	−	0.592878i	\(-0.797992\pi\)
							0.916094	+	0.400964i	\(0.131325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.a.7.3	✓	12
19.11	even	3	inner	38.8.c.a.11.3	yes	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.a.7.3	✓	12	1.1	even	1	trivial
38.8.c.a.11.3	yes	12	19.11	even	3	inner