Embedded newform 38.8.c.b.11.1

• Label: 38.8.c.b.11.1
• Level: $38$
• Weight: $8$
• Character: 38.11
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $14$
• 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:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	x^14 - x^13 + 9740*x^12 + 37173*x^11 + 71393485*x^10 + 196352446*x^9 + 218671355941*x^8 - 33116965187*x^7 + 501909199155445*x^6 + 488917878843726*x^5 + 115037439673768301*x^4 + 1132246613431747493*x^3 + 24536677825193566084*x^2 + 126800218755690130647*x + 666410398757822143881
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.1
Root			\(40.1624 + 69.5634i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.11
Dual form			38.8.c.b.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.00000 - 6.92820i) q^{2} +(-36.1624 + 62.6352i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(-36.8530 + 63.8313i) q^{5} +(289.300 + 501.082i) q^{6} +866.283 q^{7} -512.000 q^{8} +(-1521.94 - 2636.09i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 56 q^{2} + 55 q^{3} - 448 q^{4} - 126 q^{5} - 440 q^{6} + 1928 q^{7} - 7168 q^{8} - 4602 q^{9}+O(q^{10}) \)

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
							\(3\)	−36.1624	+	62.6352i	−0.773274	+	1.33935i	0.162486	+	0.986711i	\(0.448049\pi\)
−0.935760	+	0.352639i	\(0.885284\pi\)
\(5\)	−36.8530	+	63.8313i	−0.131849	+	0.228370i	−0.924389	−	0.381450i	\(-0.875425\pi\)
							0.792540	+	0.609820i	\(0.208758\pi\)
\(7\)	866.283			0.954590			0.477295	−	0.878743i	\(-0.341617\pi\)
							0.477295	+	0.878743i	\(0.341617\pi\)
\(11\)	−8185.82			−1.85433			−0.927167	−	0.374649i	\(-0.877763\pi\)
							−0.927167	+	0.374649i	\(0.877763\pi\)
\(13\)	−5184.73	−	8980.21i	−0.654522	−	1.13367i	−0.982013	−	0.188811i	\(-0.939537\pi\)
							0.327491	−	0.944854i	\(-0.393797\pi\)
\(17\)	17926.8	−	31050.1i	0.884976	−	1.53282i	0.0392357	−	0.999230i	\(-0.487508\pi\)
							0.845741	−	0.533594i	\(-0.179159\pi\)
\(19\)	−28285.1	+	9686.40i	−0.946062	+	0.323985i
							\(23\)	−24270.0	−	42036.8i	−0.415932	−	0.720415i	0.579594	−	0.814905i	\(-0.303211\pi\)
−0.995526	+	0.0944907i	\(0.969878\pi\)
\(29\)	−2614.49	−	4528.43i	−0.0199065	−	0.0344790i	0.855901	−	0.517140i	\(-0.173004\pi\)
							−0.875807	+	0.482661i	\(0.839670\pi\)
\(31\)	2036.53			0.0122779			0.00613895	−	0.999981i	\(-0.498046\pi\)
							0.00613895	+	0.999981i	\(0.498046\pi\)
\(37\)	40019.7			0.129888			0.0649438	−	0.997889i	\(-0.479313\pi\)
							0.0649438	+	0.997889i	\(0.479313\pi\)
\(41\)	−180706.	+	312992.i	−0.409477	+	0.709235i	−0.994831	−	0.101543i	\(-0.967622\pi\)
							0.585354	+	0.810778i	\(0.300955\pi\)
\(43\)	−182088.	+	315386.i	−0.349255	+	0.604927i	−0.986117	−	0.166050i	\(-0.946899\pi\)
							0.636862	+	0.770978i	\(0.280232\pi\)
\(47\)	−84667.2	−	146648.i	−0.118952	−	0.206031i	0.800400	−	0.599466i	\(-0.204620\pi\)
							−0.919353	+	0.393434i	\(0.871287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.b.11.1	yes	14
19.7	even	3	inner	38.8.c.b.7.1	✓	14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.b.7.1	✓	14	19.7	even	3	inner
38.8.c.b.11.1	yes	14	1.1	even	1	trivial