Embedded newform 42.6.e.a.25.1

• Label: 42.6.e.a.25.1
• Level: $42$
• Weight: $6$
• Character: 42.25
• Analytic conductor: $6.736$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.25
Dual form			42.6.e.a.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-4.50000 + 7.79423i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(3.00000 + 5.19615i) q^{5} +36.0000 q^{6} +(59.5000 - 115.181i) q^{7} +64.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 9 q^{3} - 16 q^{4} + 6 q^{5} + 72 q^{6} + 119 q^{7} + 128 q^{8} - 81 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−	3.46410i	−0.353553	−	0.612372i
							\(3\)	−4.50000	+	7.79423i	−0.288675	+	0.500000i
\(5\)	3.00000	+	5.19615i	0.0536656	+	0.0929516i								0.891610	−	0.452804i	\(-0.149576\pi\)
							−0.837945	+	0.545755i	\(0.816243\pi\)
\(7\)	59.5000	−	115.181i	0.458957	−	0.888459i
							\(11\)	333.000	−	576.773i	0.829779	−	1.43722i	−0.0684322	−	0.997656i	\(-0.521800\pi\)
0.898211	−	0.439564i	\(-0.144867\pi\)
\(13\)	−559.000			−0.917389			−0.458694	−	0.888594i	\(-0.651683\pi\)
							−0.458694	+	0.888594i	\(0.651683\pi\)
\(17\)	870.000	−	1506.88i	0.730125	−	1.26461i	−0.226705	−	0.973963i	\(-0.572795\pi\)
							0.956830	−	0.290649i	\(-0.0938713\pi\)
\(19\)	−578.500	−	1001.99i	−0.367637	−	0.636766i	0.621558	−	0.783368i	\(-0.286500\pi\)
							−0.989196	+	0.146602i	\(0.953166\pi\)
\(23\)	1734.00	+	3003.38i	0.683486	+	1.18383i	0.973910	+	0.226934i	\(0.0728702\pi\)
							−0.290424	+	0.956898i	\(0.593796\pi\)
\(29\)	3372.00			0.744548			0.372274	−	0.928123i	\(-0.378578\pi\)
							0.372274	+	0.928123i	\(0.378578\pi\)
\(31\)	−3146.50	+	5449.90i	−0.588063	+	1.01855i	0.406423	+	0.913685i	\(0.366776\pi\)
							−0.994486	+	0.104869i	\(0.966558\pi\)
\(37\)	−1565.50	−	2711.53i	−0.187996	−	0.325619i	0.756586	−	0.653894i	\(-0.226866\pi\)
							−0.944582	+	0.328276i	\(0.893533\pi\)
\(41\)	−4866.00			−0.452077			−0.226039	−	0.974118i	\(-0.572578\pi\)
							−0.226039	+	0.974118i	\(0.572578\pi\)
\(43\)	−11407.0			−0.940806			−0.470403	−	0.882452i	\(-0.655892\pi\)
							−0.470403	+	0.882452i	\(0.655892\pi\)
\(47\)	−1155.00	−	2000.52i	−0.0762671	−	0.132099i	0.825369	−	0.564593i	\(-0.190967\pi\)
							−0.901637	+	0.432494i	\(0.857634\pi\)