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Properties

Label 1250.4.a
Level $1250$
Weight $4$
Character orbit 1250.a
Rep. character $\chi_{1250}(1,\cdot)$
Character field $\Q$
Dimension $120$
Newform subspaces $14$
Sturm bound $750$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1250.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(750\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1250))\).

Total New Old
Modular forms 593 120 473
Cusp forms 533 120 413
Eisenstein series 60 0 60

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
\(+\)\(+\)\(+\)\(32\)
\(+\)\(-\)\(-\)\(28\)
\(-\)\(+\)\(-\)\(26\)
\(-\)\(-\)\(+\)\(34\)
Plus space\(+\)\(66\)
Minus space\(-\)\(54\)

Trace form

\( 120 q + 480 q^{4} + 1080 q^{9} + O(q^{10}) \) \( 120 q + 480 q^{4} + 1080 q^{9} + 1920 q^{16} - 360 q^{19} - 240 q^{21} + 220 q^{26} + 130 q^{29} + 360 q^{31} - 20 q^{34} + 4320 q^{36} + 840 q^{39} - 230 q^{41} + 7320 q^{49} - 1410 q^{51} - 1140 q^{54} - 690 q^{59} + 1330 q^{61} + 7680 q^{64} - 60 q^{66} + 2720 q^{69} + 2480 q^{71} - 1220 q^{74} - 1440 q^{76} - 1260 q^{79} + 11000 q^{81} - 960 q^{84} + 1740 q^{86} - 1350 q^{89} + 3420 q^{91} + 2030 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1250))\) 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
1250.4.a.a 1250.a 1.a $4$ $73.752$ 4.4.33625.1 None 1250.4.a.a \(-8\) \(-3\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2\beta _{2}+\beta _{3})q^{3}+4q^{4}+(-4\beta _{2}+\cdots)q^{6}+\cdots\)
1250.4.a.b 1250.a 1.a $4$ $73.752$ 4.4.33625.1 None 1250.4.a.a \(8\) \(3\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2\beta _{2}-\beta _{3})q^{3}+4q^{4}+(-4\beta _{2}+\cdots)q^{6}+\cdots\)
1250.4.a.c 1250.a 1.a $6$ $73.752$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 1250.4.a.c \(-12\) \(8\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
1250.4.a.d 1250.a 1.a $6$ $73.752$ 6.6.2140313125.1 None 50.4.d.a \(-12\) \(8\) \(0\) \(29\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
1250.4.a.e 1250.a 1.a $6$ $73.752$ 6.6.2140313125.1 None 50.4.d.a \(12\) \(-8\) \(0\) \(-29\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+4q^{4}+\cdots\)
1250.4.a.f 1250.a 1.a $6$ $73.752$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 1250.4.a.c \(12\) \(-8\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+4q^{4}+\cdots\)
1250.4.a.g 1250.a 1.a $8$ $73.752$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 50.4.d.b \(-16\) \(4\) \(0\) \(27\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1}+\beta _{3})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
1250.4.a.h 1250.a 1.a $8$ $73.752$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 1250.4.a.h \(-16\) \(9\) \(0\) \(22\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(-2+2\beta _{1}+\cdots)q^{6}+\cdots\)
1250.4.a.i 1250.a 1.a $8$ $73.752$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 1250.4.a.h \(16\) \(-9\) \(0\) \(-22\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
1250.4.a.j 1250.a 1.a $8$ $73.752$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 50.4.d.b \(16\) \(-4\) \(0\) \(-27\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta _{1}-\beta _{3})q^{3}+4q^{4}+\cdots\)
1250.4.a.k 1250.a 1.a $12$ $73.752$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1250.4.a.k \(-24\) \(-14\) \(0\) \(-12\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta _{5})q^{3}+4q^{4}+(2-2\beta _{5}+\cdots)q^{6}+\cdots\)
1250.4.a.l 1250.a 1.a $12$ $73.752$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1250.4.a.k \(24\) \(14\) \(0\) \(12\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{5})q^{3}+4q^{4}+(2-2\beta _{5}+\cdots)q^{6}+\cdots\)
1250.4.a.m 1250.a 1.a $16$ $73.752$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 50.4.e.a \(-32\) \(-12\) \(0\) \(-56\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}+(2-2\beta _{3}+\cdots)q^{6}+\cdots\)
1250.4.a.n 1250.a 1.a $16$ $73.752$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 50.4.e.a \(32\) \(12\) \(0\) \(56\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{3})q^{3}+4q^{4}+(2-2\beta _{3}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1250))\) into lower level spaces

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