Embedded newform 38.10.c.b.7.4

• Label: 38.10.c.b.7.4
• Level: $38$
• Weight: $10$
• Character: 38.7
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5713617742\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 104960*x^14 - 3899480*x^13 + 8040649724*x^12 - 270026959304*x^11 + 275078771318560*x^10 - 7559006944968384*x^9 + 6786319237849450512*x^8 - 132172428275143142112*x^7 + 58413131602192832245632*x^6 + 306187936352529387788160*x^5 + 338231098965100183334201088*x^4 + 833605976716998105836915712*x^3 + 801869139824974932731497058304*x^2 + 7085244797393532409215050711040*x + 1344731733940407880332419688038400
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{5}\cdot 19^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.4
Root			\(26.7983 + 46.4161i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.10.c.b.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.00000 - 13.8564i) q^{2} +(-22.2983 - 38.6218i) q^{3} +(-128.000 + 221.703i) q^{4} +(397.433 + 688.374i) q^{5} +(-356.773 + 617.949i) q^{6} +2271.77 q^{7} +4096.00 q^{8} +(8847.07 - 15323.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 128 q^{2} + 70 q^{3} - 2048 q^{4} - 341 q^{5} + 1120 q^{6} + 3704 q^{7} + 65536 q^{8} - 53064 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.00000	−	13.8564i	−0.353553	−	0.612372i
							\(3\)	−22.2983	−	38.6218i	−0.158938	−	0.275288i	0.775548	−	0.631288i	\(-0.217474\pi\)
−0.934486	+	0.356000i	\(0.884140\pi\)
\(5\)	397.433	+	688.374i	0.284380	+	0.492560i	0.972459	−	0.233076i	\(-0.0748790\pi\)
							−0.688079	+	0.725636i	\(0.741546\pi\)
\(7\)	2271.77			0.357621			0.178810	−	0.983884i	\(-0.442775\pi\)
							0.178810	+	0.983884i	\(0.442775\pi\)
\(11\)	−84129.7			−1.73254			−0.866268	−	0.499580i	\(-0.833488\pi\)
							−0.866268	+	0.499580i	\(0.833488\pi\)
\(13\)	46416.7	−	80396.1i	0.450743	−	0.780711i	−0.547689	−	0.836682i	\(-0.684492\pi\)
							0.998432	+	0.0559715i	\(0.0178256\pi\)
\(17\)	109662.	+	189940.i	0.318445	+	0.551564i	0.980164	−	0.198189i	\(-0.0635059\pi\)
							−0.661718	+	0.749752i	\(0.730173\pi\)
\(19\)	−547155.	−	152672.i	−0.963207	−	0.268762i
							\(23\)	−664676.	+	1.15125e6i	−0.495261	+	0.857818i	−0.999985	−	0.00546299i	\(-0.998261\pi\)
0.504724	+	0.863281i	\(0.331594\pi\)
\(29\)	−3.09534e6	+	5.36129e6i	−0.812677	+	1.40760i	0.0983071	+	0.995156i	\(0.468657\pi\)
							−0.910984	+	0.412442i	\(0.864676\pi\)
\(31\)	−6.00995e6			−1.16881			−0.584404	−	0.811463i	\(-0.698672\pi\)
							−0.584404	+	0.811463i	\(0.698672\pi\)
\(37\)	−1.63924e7			−1.43792			−0.718962	−	0.695050i	\(-0.755382\pi\)
							−0.718962	+	0.695050i	\(0.755382\pi\)
\(41\)	−1.66162e7	−	2.87801e7i	−0.918343	−	1.59062i	−0.801932	−	0.597415i	\(-0.796194\pi\)
							−0.116411	−	0.993201i	\(-0.537139\pi\)
\(43\)	1.35866e7	+	2.35328e7i	0.606044	+	1.04970i	0.991886	+	0.127134i	\(0.0405778\pi\)
							−0.385841	+	0.922565i	\(0.626089\pi\)
\(47\)	1.42582e7	−	2.46960e7i	0.426212	−	0.738221i	−0.570321	−	0.821422i	\(-0.693181\pi\)
							0.996533	+	0.0832014i	\(0.0265145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.c.b.7.4	✓	16
19.11	even	3	inner	38.10.c.b.11.4	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.c.b.7.4	✓	16	1.1	even	1	trivial
38.10.c.b.11.4	yes	16	19.11	even	3	inner