Newform orbit 3024.2.k.d

• Label: 3024.2.k.d
• Level: $3024$
• Weight: $2$
• Character orbit: 3024.k
• Analytic conductor: $24.147$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.1467615712\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^5 + (b - 2) * q^7 + 3*b * q^11 - 2*b * q^13 - 6 * q^17 + b * q^19 - 3*b * q^23 + 4 * q^25 + 6*b * q^29 + 3*b * q^31 + (3*b - 6) * q^35 + q^37 - 3 * q^41 - 10 * q^43 + 6 * q^47 + (-4*b + 1) * q^49 + 9*b * q^55 - 6 * q^59 + 8*b * q^61 - 6*b * q^65 - 2 * q^67 - 3*b * q^71 + 2*b * q^73 + (-6*b - 9) * q^77 - 14 * q^79 - 6 * q^83 - 18 * q^85 + 9 * q^89 + (4*b + 6) * q^91 + 3*b * q^95 + 4*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 4 q^{7} - 12 q^{17} + 8 q^{25} - 12 q^{35} + 2 q^{37} - 6 q^{41} - 20 q^{43} + 12 q^{47} + 2 q^{49} - 12 q^{59} - 4 q^{67} - 18 q^{77} - 28 q^{79} - 12 q^{83} - 36 q^{85} + 18 q^{89} + 12 q^{91}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\)	\(757\)	\(785\)	\(1135\)	\(2593\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1889.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

3.00000

0

−2.00000

−

1.73205i

0

0

0

1889.2

0

0

0

3.00000

0

−2.00000

+

1.73205i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3024.2.k.d		2
3.b	odd	2	1		3024.2.k.a		2
4.b	odd	2	1		756.2.f.c	yes	2
7.b	odd	2	1		3024.2.k.a		2
12.b	even	2	1		756.2.f.a	✓	2
21.c	even	2	1	inner	3024.2.k.d		2
28.d	even	2	1		756.2.f.a	✓	2
36.f	odd	6	1		2268.2.x.a		2
36.f	odd	6	1		2268.2.x.b		2
36.h	even	6	1		2268.2.x.g		2
36.h	even	6	1		2268.2.x.h		2
84.h	odd	2	1		756.2.f.c	yes	2
252.s	odd	6	1		2268.2.x.a		2
252.s	odd	6	1		2268.2.x.b		2
252.bi	even	6	1		2268.2.x.g		2
252.bi	even	6	1		2268.2.x.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.f.a	✓	2	12.b	even	2	1
756.2.f.a	✓	2	28.d	even	2	1
756.2.f.c	yes	2	4.b	odd	2	1
756.2.f.c	yes	2	84.h	odd	2	1
2268.2.x.a		2	36.f	odd	6	1
2268.2.x.a		2	252.s	odd	6	1
2268.2.x.b		2	36.f	odd	6	1
2268.2.x.b		2	252.s	odd	6	1
2268.2.x.g		2	36.h	even	6	1
2268.2.x.g		2	252.bi	even	6	1
2268.2.x.h		2	36.h	even	6	1
2268.2.x.h		2	252.bi	even	6	1
3024.2.k.a		2	3.b	odd	2	1
3024.2.k.a		2	7.b	odd	2	1
3024.2.k.d		2	1.a	even	1	1	trivial
3024.2.k.d		2	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5} - 3 \)

\( T_{11}^{2} + 27 \)

\( T_{13}^{2} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( (T - 3)^{2} \)
$7$	\( T^{2} + 4T + 7 \)
$11$	\( T^{2} + 27 \)