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Properties

Label 2500.4.a
Level $2500$
Weight $4$
Character orbit 2500.a
Rep. character $\chi_{2500}(1,\cdot)$
Character field $\Q$
Dimension $120$
Newform subspaces $7$
Sturm bound $1500$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2500.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(1500\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2500))\).

Total New Old
Modular forms 1170 120 1050
Cusp forms 1080 120 960
Eisenstein series 90 0 90

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)FrickeDim
\(-\)\(+\)\(-\)\(58\)
\(-\)\(-\)\(+\)\(62\)
Plus space\(+\)\(62\)
Minus space\(-\)\(58\)

Trace form

\( 120 q + 1080 q^{9} + O(q^{10}) \) \( 120 q + 1080 q^{9} + 180 q^{19} + 120 q^{21} + 130 q^{29} - 180 q^{31} - 420 q^{39} - 230 q^{41} + 5160 q^{49} - 1410 q^{51} - 690 q^{59} - 1910 q^{61} - 1360 q^{69} - 1240 q^{71} - 1260 q^{79} + 9080 q^{81} - 1830 q^{89} - 2520 q^{91} + 3350 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2500))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
2500.4.a.a 2500.a 1.a $10$ $147.505$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 2500.4.a.a \(0\) \(-13\) \(0\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-2-2\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)
2500.4.a.b 2500.a 1.a $10$ $147.505$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 2500.4.a.a \(0\) \(13\) \(0\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(2+2\beta _{2}-\beta _{6})q^{7}+(8+\cdots)q^{9}+\cdots\)
2500.4.a.c 2500.a 1.a $14$ $147.505$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 100.4.g.a \(0\) \(-2\) \(0\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{2}+\beta _{3})q^{7}+(9+\beta _{1}+\cdots)q^{9}+\cdots\)
2500.4.a.d 2500.a 1.a $14$ $147.505$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 100.4.g.a \(0\) \(2\) \(0\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1+\beta _{2}-\beta _{3})q^{7}+(9+\beta _{1}+\cdots)q^{9}+\cdots\)
2500.4.a.e 2500.a 1.a $20$ $147.505$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 2500.4.a.e \(0\) \(-1\) \(0\) \(-26\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-2+\beta _{5}-\beta _{6})q^{7}+(9+\beta _{2}+\cdots)q^{9}+\cdots\)
2500.4.a.f 2500.a 1.a $20$ $147.505$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 2500.4.a.e \(0\) \(1\) \(0\) \(26\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(2-\beta _{5}+\beta _{6})q^{7}+(9+\beta _{2}+\cdots)q^{9}+\cdots\)
2500.4.a.g 2500.a 1.a $32$ $147.505$ None 100.4.i.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2500))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2500)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1250))\)\(^{\oplus 2}\)