Newform orbit 7500.2.d.g

• Label: 7500.2.d.g
• Level: $7500$
• Weight: $2$
• Character orbit: 7500.d
• Analytic conductor: $59.888$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(59.8878015160\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
1249.1		0			−	1.00000i		0			0			0			−	4.13266i		0		−1.00000				0
1249.2		0			−	1.00000i		0			0			0			−	3.54704i		0		−1.00000				0
1249.3		0			−	1.00000i		0			0			0			−	1.57893i		0		−1.00000				0
1249.4		0			−	1.00000i		0			0			0			−	1.31873i		0		−1.00000				0
1249.5		0			−	1.00000i		0			0			0			−	0.957526i		0		−1.00000				0
1249.6		0			−	1.00000i		0			0			0			−	0.595901i		0		−1.00000				0
1249.7		0			−	1.00000i		0			0			0				1.04684i		0		−1.00000				0
1249.8		0			−	1.00000i		0			0			0				2.44380i		0		−1.00000				0
1249.9		0			−	1.00000i		0			0			0				3.78808i		0		−1.00000				0
1249.10		0			−	1.00000i		0			0			0				3.80992i		0		−1.00000				0
1249.11		0			−	1.00000i		0			0			0				4.41540i		0		−1.00000				0
1249.12		0			−	1.00000i		0			0			0				4.62675i		0		−1.00000				0
1249.13		0				1.00000i		0			0			0			−	4.62675i		0		−1.00000				0
1249.14		0				1.00000i		0			0			0			−	4.41540i		0		−1.00000				0
1249.15		0				1.00000i		0			0			0			−	3.80992i		0		−1.00000				0
1249.16		0				1.00000i		0			0			0			−	3.78808i		0		−1.00000				0
1249.17		0				1.00000i		0			0			0			−	2.44380i		0		−1.00000				0
1249.18		0				1.00000i		0			0			0			−	1.04684i		0		−1.00000				0
1249.19		0				1.00000i		0			0			0				0.595901i		0		−1.00000				0
1249.20		0				1.00000i		0			0			0				0.957526i		0		−1.00000				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7500.2.d.g		24
5.b	even	2	1	inner	7500.2.d.g		24
5.c	odd	4	1		7500.2.a.m		12
5.c	odd	4	1		7500.2.a.n		12
25.d	even	5	1		300.2.o.a	✓	24
25.d	even	5	1		1500.2.o.c		24
25.e	even	10	1		300.2.o.a	✓	24
25.e	even	10	1		1500.2.o.c		24
25.f	odd	20	2		1500.2.m.c		24
25.f	odd	20	2		1500.2.m.d		24
75.h	odd	10	1		900.2.w.c		24
75.j	odd	10	1		900.2.w.c		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.2.o.a	✓	24	25.d	even	5	1
300.2.o.a	✓	24	25.e	even	10	1
900.2.w.c		24	75.h	odd	10	1
900.2.w.c		24	75.j	odd	10	1
1500.2.m.c		24	25.f	odd	20	2
1500.2.m.d		24	25.f	odd	20	2
1500.2.o.c		24	25.d	even	5	1
1500.2.o.c		24	25.e	even	10	1
7500.2.a.m		12	5.c	odd	4	1
7500.2.a.n		12	5.c	odd	4	1
7500.2.d.g		24	1.a	even	1	1	trivial
7500.2.d.g		24	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^24 + 112*T7^22 + 5396*T7^20 + 146190*T7^18 + 2444070*T7^16 + 26062572*T7^14 + 177495989*T7^12 + 757283742*T7^10 + 1964246465*T7^8 + 2998987920*T7^6 + 2559664096*T7^4 + 1093572352*T7^2 + 172554496