Embedded newform 38.8.c.a.11.2

• Label: 38.8.c.a.11.2
• 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			11.2
Root			\(-21.5062 + 37.2499i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.11
Dual form			38.8.c.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 + 6.92820i) q^{2} +(-20.5062 + 35.5178i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(-273.130 + 473.076i) q^{5} +(-164.050 - 284.143i) q^{6} -154.247 q^{7} +512.000 q^{8} +(252.489 + 437.324i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{2} + 12 q^{3} - 384 q^{4} + 124 q^{5} + 96 q^{6} - 1036 q^{7} + 6144 q^{8} - 7232 q^{9}+O(q^{10}) \)

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
							\(3\)	−20.5062	+	35.5178i	−0.438492	+	0.759490i	−0.997573	−	0.0696227i	\(-0.977820\pi\)
0.559082	+	0.829113i	\(0.311154\pi\)
\(5\)	−273.130	+	473.076i	−0.977181	+	1.69253i	−0.304638	+	0.952468i	\(0.598536\pi\)
							−0.672543	+	0.740058i	\(0.734798\pi\)
\(7\)	−154.247			−0.169971			−0.0849855	−	0.996382i	\(-0.527084\pi\)
							−0.0849855	+	0.996382i	\(0.527084\pi\)
\(11\)	−4153.91			−0.940984			−0.470492	−	0.882404i	\(-0.655924\pi\)
							−0.470492	+	0.882404i	\(0.655924\pi\)
\(13\)	4201.88	+	7277.88i	0.530447	+	0.918762i	0.999369	+	0.0355220i	\(0.0113094\pi\)
							−0.468921	+	0.883240i	\(0.655357\pi\)
\(17\)	16175.7	−	28017.1i	0.798531	−	1.38310i	−0.122041	−	0.992525i	\(-0.538944\pi\)
							0.920573	−	0.390572i	\(-0.127723\pi\)
\(19\)	6628.11	−	29153.7i	0.221693	−	0.975117i
							\(23\)	32161.4	+	55705.2i	0.551173	+	0.954659i	0.998190	+	0.0601338i	\(0.0191527\pi\)
−0.447018	+	0.894525i	\(0.647514\pi\)
\(29\)	101353.	+	175549.i	0.771693	+	1.33661i	0.936635	+	0.350307i	\(0.113923\pi\)
							−0.164942	+	0.986303i	\(0.552744\pi\)
\(31\)	−132507.			−0.798861			−0.399431	−	0.916763i	\(-0.630792\pi\)
							−0.399431	+	0.916763i	\(0.630792\pi\)
\(37\)	93680.1			0.304047			0.152024	−	0.988377i	\(-0.451421\pi\)
							0.152024	+	0.988377i	\(0.451421\pi\)
\(41\)	−248414.	+	430265.i	−0.562901	+	0.974973i	0.434340	+	0.900749i	\(0.356982\pi\)
							−0.997242	+	0.0742246i	\(0.976352\pi\)
\(43\)	−172620.	+	298987.i	−0.331095	+	0.573473i	−0.982727	−	0.185062i	\(-0.940751\pi\)
							0.651632	+	0.758535i	\(0.274085\pi\)
\(47\)	−20822.9	−	36066.4i	−0.0292549	−	0.0506711i	0.851027	−	0.525122i	\(-0.175980\pi\)
							−0.880282	+	0.474450i	\(0.842647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.a.11.2	yes	12
19.7	even	3	inner	38.8.c.a.7.2	✓	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.a.7.2	✓	12	19.7	even	3	inner
38.8.c.a.11.2	yes	12	1.1	even	1	trivial