Embedded newform 38.8.c.a.7.6

• Label: 38.8.c.a.7.6
• 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			7.6
Root			\(35.5320 + 61.5431i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.8.c.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(36.5320 + 63.2752i) q^{3} +(-32.0000 + 55.4256i) q^{4} +(-61.9899 - 107.370i) q^{5} +(292.256 - 506.202i) q^{6} +1245.26 q^{7} +512.000 q^{8} +(-1575.67 + 2729.14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{2} + 12 q^{3} - 384 q^{4} + 124 q^{5} + 96 q^{6} - 1036 q^{7} + 6144 q^{8} - 7232 q^{9}+O(q^{10}) \)

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
							\(3\)	36.5320	+	63.2752i	0.781175	+	1.35304i	0.931258	+	0.364362i	\(0.118713\pi\)
−0.150082	+	0.988674i	\(0.547954\pi\)
\(5\)	−61.9899	−	107.370i	−0.221782	−	0.384137i	0.733567	−	0.679617i	\(-0.237854\pi\)
							−0.955349	+	0.295479i	\(0.904521\pi\)
\(7\)	1245.26			1.37220			0.686101	−	0.727507i	\(-0.259321\pi\)
							0.686101	+	0.727507i	\(0.259321\pi\)
\(11\)	−746.907			−0.169197			−0.0845984	−	0.996415i	\(-0.526961\pi\)
							−0.0845984	+	0.996415i	\(0.526961\pi\)
\(13\)	−6386.14	+	11061.1i	−0.806189	+	1.39636i	0.109297	+	0.994009i	\(0.465140\pi\)
							−0.915485	+	0.402351i	\(0.868193\pi\)
\(17\)	12612.0	+	21844.5i	0.622603	+	1.07838i	0.988999	+	0.147921i	\(0.0472581\pi\)
							−0.366396	+	0.930459i	\(0.619409\pi\)
\(19\)	−3524.87	+	29689.2i	−0.117898	+	0.993026i
							\(23\)	6118.65	−	10597.8i	0.104860	−	0.181622i	−0.808821	−	0.588055i	\(-0.799894\pi\)
0.913681	+	0.406433i	\(0.133227\pi\)
\(29\)	−13897.8	+	24071.6i	−0.105816	+	0.183279i	−0.914071	−	0.405553i	\(-0.867079\pi\)
							0.808255	+	0.588832i	\(0.200412\pi\)
\(31\)	−172184.			−1.03807			−0.519036	−	0.854752i	\(-0.673709\pi\)
							−0.519036	+	0.854752i	\(0.673709\pi\)
\(37\)	−21713.3			−0.0704724			−0.0352362	−	0.999379i	\(-0.511218\pi\)
							−0.0352362	+	0.999379i	\(0.511218\pi\)
\(41\)	52665.7	+	91219.6i	0.119339	+	0.206702i	0.919506	−	0.393076i	\(-0.128589\pi\)
							−0.800167	+	0.599778i	\(0.795256\pi\)
\(43\)	422519.	+	731825.i	0.810414	+	1.40368i	0.912575	+	0.408910i	\(0.134091\pi\)
							−0.102161	+	0.994768i	\(0.532576\pi\)
\(47\)	620114.	−	1.07407e6i	0.871223	−	1.50900i	0.0104906	−	0.999945i	\(-0.496661\pi\)
							0.860732	−	0.509058i	\(-0.170006\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.a.7.6	✓	12
19.11	even	3	inner	38.8.c.a.11.6	yes	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.a.7.6	✓	12	1.1	even	1	trivial
38.8.c.a.11.6	yes	12	19.11	even	3	inner