Properties

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 $2$

Related objects

Downloads

Learn more about

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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{9} + 4q^{11} - 20q^{19} + 16q^{21} - 16q^{29} - 4q^{31} + 20q^{41} - 56q^{49} + 16q^{51} + 4q^{59} + 68q^{61} - 36q^{69} - 12q^{79} + 24q^{81} - 20q^{89} + 40q^{91} - 4q^{99} + O(q^{100}) \)

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
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.24
Significant digits:
Format:

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 \(T_{7}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(7500, [\chi])\).