Embedded newform 38.8.c.b.11.4

• Label: 38.8.c.b.11.4
• 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			11.4
Root			\(-3.15555 - 5.46558i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.11
Dual form			38.8.c.b.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.00000 - 6.92820i) q^{2} +(7.15555 - 12.3938i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(-222.650 + 385.641i) q^{5} +(-57.2444 - 99.1503i) q^{6} +1436.09 q^{7} -512.000 q^{8} +(991.096 + 1716.63i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 56 q^{2} + 55 q^{3} - 448 q^{4} - 126 q^{5} - 440 q^{6} + 1928 q^{7} - 7168 q^{8} - 4602 q^{9}+O(q^{10}) \)

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
							\(3\)	7.15555	−	12.3938i	0.153010	−	0.265020i	−0.779323	−	0.626623i	\(-0.784437\pi\)
0.932333	+	0.361602i	\(0.117770\pi\)
\(5\)	−222.650	+	385.641i	−0.796576	+	1.37971i	0.125257	+	0.992124i	\(0.460025\pi\)
							−0.921833	+	0.387587i	\(0.873309\pi\)
\(7\)	1436.09			1.58248			0.791242	−	0.611503i	\(-0.209435\pi\)
							0.791242	+	0.611503i	\(0.209435\pi\)
\(11\)	5160.07			1.16891			0.584455	−	0.811426i	\(-0.301308\pi\)
							0.584455	+	0.811426i	\(0.301308\pi\)
\(13\)	−4986.52	−	8636.91i	−0.629500	−	1.09033i	−0.987652	−	0.156663i	\(-0.949926\pi\)
							0.358152	−	0.933663i	\(-0.383407\pi\)
\(17\)	−5591.05	+	9683.97i	−0.276008	+	0.478060i	−0.970389	−	0.241548i	\(-0.922345\pi\)
							0.694381	+	0.719608i	\(0.255678\pi\)
\(19\)	28898.8	+	7663.77i	0.966588	+	0.256333i
							\(23\)	33402.2	+	57854.3i	0.572437	+	0.991490i	0.996315	+	0.0857710i	\(0.0273353\pi\)
−0.423878	+	0.905719i	\(0.639331\pi\)
\(29\)	58975.7	+	102149.i	0.449035	+	0.777752i	0.998323	−	0.0578813i	\(-0.0184345\pi\)
							−0.549288	+	0.835633i	\(0.685101\pi\)
\(31\)	154801.			0.933271			0.466635	−	0.884450i	\(-0.345466\pi\)
							0.466635	+	0.884450i	\(0.345466\pi\)
\(37\)	−478762.			−1.55387			−0.776933	−	0.629584i	\(-0.783225\pi\)
							−0.776933	+	0.629584i	\(0.783225\pi\)
\(41\)	−178242.	+	308724.i	−0.403893	+	0.699563i	−0.994192	−	0.107621i	\(-0.965677\pi\)
							0.590299	+	0.807185i	\(0.299010\pi\)
\(43\)	−17516.3	+	30339.0i	−0.0335971	+	0.0581919i	−0.882335	−	0.470622i	\(-0.844030\pi\)
							0.848738	+	0.528814i	\(0.177363\pi\)
\(47\)	−574302.	−	994720.i	−0.806859	−	1.39752i	−0.915029	−	0.403388i	\(-0.867832\pi\)
							0.108170	−	0.994132i	\(-0.465501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.c.b.11.4	yes	14
19.7	even	3	inner	38.8.c.b.7.4	✓	14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.c.b.7.4	✓	14	19.7	even	3	inner
38.8.c.b.11.4	yes	14	1.1	even	1	trivial