Newform orbit 42.4.e.b

• Label: 42.4.e.b
• Level: $42$
• Weight: $4$
• Character orbit: 42.e
• Analytic conductor: $2.478$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47808022024\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
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:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 2) q^{2} + 3 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 15 \zeta_{6} + 15) q^{5} + 6 q^{6} + (7 \zeta_{6} + 14) q^{7} - 8 q^{8} + (9 \zeta_{6} - 9) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} + 15 q^{5} + 12 q^{6} + 35 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(1\)	\(-1 + \zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

25.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.00000

+

1.73205i

1.50000

−

2.59808i

−2.00000

+

3.46410i

7.50000

+

12.9904i

6.00000

17.5000

−

6.06218i

−8.00000

−4.50000

−

7.79423i

−15.0000

+

25.9808i

37.1

1.00000

−

1.73205i

1.50000

+

2.59808i

−2.00000

−

3.46410i

7.50000

−

12.9904i

6.00000

17.5000

+

6.06218i

−8.00000

−4.50000

+

7.79423i

−15.0000

−

25.9808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	42.4.e.b	✓	2
3.b	odd	2	1		126.4.g.a		2
4.b	odd	2	1		336.4.q.d		2
7.b	odd	2	1		294.4.e.e		2
7.c	even	3	1	inner	42.4.e.b	✓	2
7.c	even	3	1		294.4.a.a		1
7.d	odd	6	1		294.4.a.g		1
7.d	odd	6	1		294.4.e.e		2
21.c	even	2	1		882.4.g.l		2
21.g	even	6	1		882.4.a.h		1
21.g	even	6	1		882.4.g.l		2
21.h	odd	6	1		126.4.g.a		2
21.h	odd	6	1		882.4.a.r		1
28.f	even	6	1		2352.4.a.q		1
28.g	odd	6	1		336.4.q.d		2
28.g	odd	6	1		2352.4.a.u		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.e.b	✓	2	1.a	even	1	1	trivial
42.4.e.b	✓	2	7.c	even	3	1	inner
126.4.g.a		2	3.b	odd	2	1
126.4.g.a		2	21.h	odd	6	1
294.4.a.a		1	7.c	even	3	1
294.4.a.g		1	7.d	odd	6	1
294.4.e.e		2	7.b	odd	2	1
294.4.e.e		2	7.d	odd	6	1
336.4.q.d		2	4.b	odd	2	1
336.4.q.d		2	28.g	odd	6	1
882.4.a.h		1	21.g	even	6	1
882.4.a.r		1	21.h	odd	6	1
882.4.g.l		2	21.c	even	2	1
882.4.g.l		2	21.g	even	6	1
2352.4.a.q		1	28.f	even	6	1
2352.4.a.u		1	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 15T_{5} + 225 \) acting on \(S_{4}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 4 \)
$3$	\( T^{2} - 3T + 9 \)
$5$	\( T^{2} - 15T + 225 \)
$7$	\( T^{2} - 35T + 343 \)
$11$	\( T^{2} - 9T + 81 \)