Properties

Label 2400.2.a
Level $2400$
Weight $2$
Character orbit 2400.a
Rep. character $\chi_{2400}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $36$
Sturm bound $960$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(960\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2400))\).

Total New Old
Modular forms 528 38 490
Cusp forms 433 38 395
Eisenstein series 95 0 95

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(16\)
Minus space\(-\)\(22\)

Trace form

\( 38 q + 38 q^{9} + O(q^{10}) \) \( 38 q + 38 q^{9} - 12 q^{13} + 12 q^{17} - 8 q^{21} - 4 q^{29} - 8 q^{33} - 12 q^{37} - 4 q^{41} + 6 q^{49} + 12 q^{53} - 8 q^{57} + 4 q^{61} - 4 q^{73} + 64 q^{77} + 38 q^{81} + 12 q^{89} - 24 q^{93} + 44 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2400))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
2400.2.a.a 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
2400.2.a.b 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
2400.2.a.c 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}-6q^{11}+2q^{13}+\cdots\)
2400.2.a.d 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
2400.2.a.e 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-4q^{11}+3q^{13}+\cdots\)
2400.2.a.f 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-q^{13}+3q^{19}+q^{21}+\cdots\)
2400.2.a.g 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}+q^{13}-3q^{19}+q^{21}+\cdots\)
2400.2.a.h 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}+4q^{11}-3q^{13}+\cdots\)
2400.2.a.i 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
2400.2.a.j 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}+4q^{11}-2q^{13}+2q^{17}+\cdots\)
2400.2.a.k 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}-4q^{11}-7q^{13}+\cdots\)
2400.2.a.l 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}-5q^{13}-5q^{19}+\cdots\)
2400.2.a.m 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}+5q^{13}+5q^{19}+\cdots\)
2400.2.a.n 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}+4q^{11}+7q^{13}+\cdots\)
2400.2.a.o 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}-4q^{13}-8q^{19}+\cdots\)
2400.2.a.p 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}+4q^{13}+8q^{19}+\cdots\)
2400.2.a.q 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
2400.2.a.r 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
2400.2.a.s 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{7}+q^{9}-4q^{13}+8q^{19}+\cdots\)
2400.2.a.t 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{7}+q^{9}+4q^{13}-8q^{19}+\cdots\)
2400.2.a.u 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}+q^{9}-4q^{11}+7q^{13}+\cdots\)
2400.2.a.v 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}+q^{9}-5q^{13}+5q^{19}+\cdots\)
2400.2.a.w 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}+q^{9}+5q^{13}-5q^{19}+\cdots\)
2400.2.a.x 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}+q^{9}+4q^{11}-7q^{13}+\cdots\)
2400.2.a.y 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
2400.2.a.z 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
2400.2.a.ba 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}-4q^{11}-3q^{13}+\cdots\)
2400.2.a.bb 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}-q^{13}-3q^{19}+q^{21}+\cdots\)
2400.2.a.bc 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}+q^{13}+3q^{19}+q^{21}+\cdots\)
2400.2.a.bd 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}+4q^{11}+3q^{13}+\cdots\)
2400.2.a.be 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
2400.2.a.bf 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}+6q^{11}+2q^{13}+\cdots\)
2400.2.a.bg 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
2400.2.a.bh 2400.a 1.a $1$ $19.164$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
2400.2.a.bi 2400.a 1.a $2$ $19.164$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}-\beta q^{11}+\beta q^{13}+\cdots\)
2400.2.a.bj 2400.a 1.a $2$ $19.164$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}-\beta q^{11}-\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2400)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\)\(^{\oplus 2}\)