Embedded newform 42.6.e.c.37.2

• Label: 42.6.e.c.37.2
• Level: $42$
• Weight: $6$
• Character: 42.37
• Analytic conductor: $6.736$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.73612043215\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{9601})\)
Defining polynomial:	\( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.2
Root			\(-24.2462 + 41.9956i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.37
Dual form			42.6.e.c.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{2} +(4.50000 + 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(37.7462 - 65.3783i) q^{5} -36.0000 q^{6} +(99.4847 - 83.1252i) q^{7} +64.0000 q^{8} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 18 q^{3} - 32 q^{4} + 53 q^{5} - 144 q^{6} + 6 q^{7} + 256 q^{8} - 162 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	+	3.46410i	−0.353553	+	0.612372i
							\(3\)	4.50000	+	7.79423i	0.288675	+	0.500000i
\(5\)	37.7462	−	65.3783i	0.675224	−	1.16952i								−0.301179	−	0.953568i	\(-0.597380\pi\)
							0.976403	−	0.215955i	\(-0.0692864\pi\)
\(7\)	99.4847	−	83.1252i	0.767381	−	0.641191i
							\(11\)	74.7309	+	129.438i	0.186217	+	0.322537i	0.943986	−	0.329986i	\(-0.107044\pi\)
−0.757769	+	0.652523i	\(0.773711\pi\)
\(13\)	349.416			0.573435			0.286717	−	0.958015i	\(-0.407436\pi\)
							0.286717	+	0.958015i	\(0.407436\pi\)
\(17\)	574.923	+	995.797i	0.482489	+	0.835696i	0.999798	−	0.0201029i	\(-0.00639938\pi\)
							−0.517309	+	0.855799i	\(0.673066\pi\)
\(19\)	1397.60	−	2420.72i	0.888176	−	1.53837i	0.0461468	−	0.998935i	\(-0.485306\pi\)
							0.842029	−	0.539432i	\(-0.181361\pi\)
\(23\)	−906.985	+	1570.94i	−0.357504	+	0.619214i	−0.987543	−	0.157349i	\(-0.949705\pi\)
							0.630040	+	0.776563i	\(0.283039\pi\)
\(29\)	−759.033			−0.167597			−0.0837984	−	0.996483i	\(-0.526705\pi\)
							−0.0837984	+	0.996483i	\(0.526705\pi\)
\(31\)	−4515.87	−	7821.72i	−0.843990	−	1.46183i	−0.886495	−	0.462738i	\(-0.846867\pi\)
							0.0425050	−	0.999096i	\(-0.486466\pi\)
\(37\)	−3897.45	+	6750.57i	−0.468032	+	0.810655i	−0.999333	−	0.0365280i	\(-0.988370\pi\)
							0.531300	+	0.847183i	\(0.321704\pi\)
\(41\)	7640.49			0.709842			0.354921	−	0.934896i	\(-0.384508\pi\)
							0.354921	+	0.934896i	\(0.384508\pi\)
\(43\)	12188.8			1.00529			0.502645	−	0.864493i	\(-0.332360\pi\)
							0.502645	+	0.864493i	\(0.332360\pi\)
\(47\)	−12299.4	+	21303.2i	−0.812156	+	1.40670i	0.0991964	+	0.995068i	\(0.468373\pi\)
							−0.911352	+	0.411627i	\(0.864961\pi\)