Embedded newform 38.8.c.a.11.1

• Label: 38.8.c.a.11.1
• Level: $38$
• Weight: $8$
• Character: 38.11
• 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			11.1
Root			\(-46.4369 + 80.4310i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.11
Dual form			38.8.c.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 + 6.92820i) q^{2} +(-45.4369 + 78.6990i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(149.770 - 259.410i) q^{5} +(-363.495 - 629.592i) q^{6} -708.303 q^{7} +512.000 q^{8} +(-3035.52 - 5257.68i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{2} + 12 q^{3} - 384 q^{4} + 124 q^{5} + 96 q^{6} - 1036 q^{7} + 6144 q^{8} - 7232 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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
							\(3\)	−45.4369	+	78.6990i	−0.971593	+	1.68285i	−0.280843	+	0.959754i	\(0.590614\pi\)
−0.690750	+	0.723094i	\(0.742719\pi\)
\(5\)	149.770	−	259.410i	0.535835	−	0.928093i	−0.463288	−	0.886208i	\(-0.653330\pi\)
							0.999122	−	0.0418851i	\(-0.0133363\pi\)
\(7\)	−708.303			−0.780506			−0.390253	−	0.920708i	\(-0.627612\pi\)
							−0.390253	+	0.920708i	\(0.627612\pi\)
\(11\)	952.750			0.215826			0.107913	−	0.994160i	\(-0.465583\pi\)
							0.107913	+	0.994160i	\(0.465583\pi\)
\(13\)	4025.73	+	6972.77i	0.508210	+	0.880245i	0.999955	+	0.00950590i	\(0.00302587\pi\)
							−0.491745	+	0.870739i	\(0.663641\pi\)
\(17\)	9356.62	−	16206.1i	0.461900	−	0.800034i	−0.537156	−	0.843483i	\(-0.680501\pi\)
							0.999056	+	0.0434493i	\(0.0138347\pi\)
\(19\)	−28092.8	−	10230.7i	−0.939630	−	0.342192i
							\(23\)	12015.4	+	20811.3i	0.205917	+	0.356659i	0.950425	−	0.310955i	\(-0.100649\pi\)
−0.744508	+	0.667614i	\(0.767316\pi\)
\(29\)	−105852.	−	183340.i	−0.805943	−	1.39593i	−0.915653	−	0.401970i	\(-0.868326\pi\)
							0.109710	−	0.993964i	\(-0.465008\pi\)
\(31\)	172649.			1.04088			0.520438	−	0.853899i	\(-0.325769\pi\)
							0.520438	+	0.853899i	\(0.325769\pi\)
\(37\)	−273667.			−0.888210			−0.444105	−	0.895975i	\(-0.646478\pi\)
							−0.444105	+	0.895975i	\(0.646478\pi\)
\(41\)	404546.	−	700694.i	0.916694	−	1.58776i	0.112292	−	0.993675i	\(-0.464181\pi\)
							0.804402	−	0.594085i	\(-0.202486\pi\)
\(43\)	384951.	−	666755.i	0.738356	−	1.27887i	−0.214879	−	0.976641i	\(-0.568936\pi\)
							0.953235	−	0.302230i	\(-0.0977309\pi\)
\(47\)	88261.1	+	152873.i	0.124001	+	0.214777i	0.921342	−	0.388753i	\(-0.127094\pi\)
							−0.797341	+	0.603529i	\(0.793761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.a.11.1	yes	12
19.7	even	3	inner	38.8.c.a.7.1	✓	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.a.7.1	✓	12	19.7	even	3	inner
38.8.c.a.11.1	yes	12	1.1	even	1	trivial