Embedded newform 42.6.e.a.37.1

• Label: 42.6.e.a.37.1
• Level: $42$
• Weight: $6$
• Character: 42.37
• Analytic conductor: $6.736$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• 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			37.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.37
Dual form			42.6.e.a.25.1

$q$-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 \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

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

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
\(5\)	3.00000	−	5.19615i	0.0536656	−	0.0929516i								−0.837945	−	0.545755i	\(-0.816243\pi\)
							0.891610	+	0.452804i	\(0.149576\pi\)
\(7\)	59.5000	+	115.181i	0.458957	+	0.888459i
							\(11\)	333.000	+	576.773i	0.829779	+	1.43722i	0.898211	+	0.439564i	\(0.144867\pi\)
−0.0684322	+	0.997656i	\(0.521800\pi\)
\(13\)	−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	0.956830	+	0.290649i	\(0.0938713\pi\)
							−0.226705	+	0.973963i	\(0.572795\pi\)
\(19\)	−578.500	+	1001.99i	−0.367637	+	0.636766i	−0.989196	−	0.146602i	\(-0.953166\pi\)
							0.621558	+	0.783368i	\(0.286500\pi\)
\(23\)	1734.00	−	3003.38i	0.683486	−	1.18383i	−0.290424	−	0.956898i	\(-0.593796\pi\)
							0.973910	−	0.226934i	\(-0.0728702\pi\)
\(29\)	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	−0.994486	−	0.104869i	\(-0.966558\pi\)
							0.406423	−	0.913685i	\(-0.366776\pi\)
\(37\)	−1565.50	+	2711.53i	−0.187996	+	0.325619i	−0.944582	−	0.328276i	\(-0.893533\pi\)
							0.756586	+	0.653894i	\(0.226866\pi\)
\(41\)	−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	−0.901637	−	0.432494i	\(-0.857634\pi\)
							0.825369	+	0.564593i	\(0.190967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.6.e.a.37.1	yes	2
3.2	odd	2		126.6.g.c.37.1		2
4.3	odd	2		336.6.q.c.289.1		2
7.2	even	3		294.6.a.l.1.1		1
7.3	odd	6		294.6.e.e.67.1		2
7.4	even	3	inner	42.6.e.a.25.1	✓	2
7.5	odd	6		294.6.a.j.1.1		1
7.6	odd	2		294.6.e.e.79.1		2
21.2	odd	6		882.6.a.f.1.1		1
21.5	even	6		882.6.a.e.1.1		1
21.11	odd	6		126.6.g.c.109.1		2
28.11	odd	6		336.6.q.c.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.a.25.1	✓	2	7.4	even	3	inner
42.6.e.a.37.1	yes	2	1.1	even	1	trivial
126.6.g.c.37.1		2	3.2	odd	2
126.6.g.c.109.1		2	21.11	odd	6
294.6.a.j.1.1		1	7.5	odd	6
294.6.a.l.1.1		1	7.2	even	3
294.6.e.e.67.1		2	7.3	odd	6
294.6.e.e.79.1		2	7.6	odd	2
336.6.q.c.193.1		2	28.11	odd	6
336.6.q.c.289.1		2	4.3	odd	2
882.6.a.e.1.1		1	21.5	even	6
882.6.a.f.1.1		1	21.2	odd	6