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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
2400.2.a.a \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(q-q^{3}-4q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
2400.2.a.b \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(q-q^{3}-4q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
2400.2.a.c \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(q-q^{3}-2q^{7}+q^{9}-6q^{11}+2q^{13}+\cdots\)
2400.2.a.d \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
2400.2.a.e \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-4q^{11}+3q^{13}+\cdots\)
2400.2.a.f \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-q^{13}+3q^{19}+q^{21}+\cdots\)
2400.2.a.g \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q-q^{3}-q^{7}+q^{9}+q^{13}-3q^{19}+q^{21}+\cdots\)
2400.2.a.h \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q-q^{3}-q^{7}+q^{9}+4q^{11}-3q^{13}+\cdots\)
2400.2.a.i \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
2400.2.a.j \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+q^{9}+4q^{11}-2q^{13}+2q^{17}+\cdots\)
2400.2.a.k \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(+\) \(+\) \(-\) \(q-q^{3}+3q^{7}+q^{9}-4q^{11}-7q^{13}+\cdots\)
2400.2.a.l \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(+\) \(+\) \(+\) \(q-q^{3}+3q^{7}+q^{9}-5q^{13}-5q^{19}+\cdots\)
2400.2.a.m \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(+\) \(+\) \(-\) \(q-q^{3}+3q^{7}+q^{9}+5q^{13}+5q^{19}+\cdots\)
2400.2.a.n \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(-\) \(+\) \(+\) \(q-q^{3}+3q^{7}+q^{9}+4q^{11}+7q^{13}+\cdots\)
2400.2.a.o \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(-\) \(q-q^{3}+4q^{7}+q^{9}-4q^{13}-8q^{19}+\cdots\)
2400.2.a.p \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q-q^{3}+4q^{7}+q^{9}+4q^{13}+8q^{19}+\cdots\)
2400.2.a.q \(1\) \(19.164\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q-q^{3}+4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
2400.2.a.r \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(q+q^{3}-4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
2400.2.a.s \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q+q^{3}-4q^{7}+q^{9}-4q^{13}+8q^{19}+\cdots\)
2400.2.a.t \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q+q^{3}-4q^{7}+q^{9}+4q^{13}-8q^{19}+\cdots\)
2400.2.a.u \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(q+q^{3}-3q^{7}+q^{9}-4q^{11}+7q^{13}+\cdots\)
2400.2.a.v \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(q+q^{3}-3q^{7}+q^{9}-5q^{13}+5q^{19}+\cdots\)
2400.2.a.w \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(-\) \(q+q^{3}-3q^{7}+q^{9}+5q^{13}-5q^{19}+\cdots\)
2400.2.a.x \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(+\) \(-\) \(-\) \(q+q^{3}-3q^{7}+q^{9}+4q^{11}-7q^{13}+\cdots\)
2400.2.a.y \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{9}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
2400.2.a.z \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{9}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
2400.2.a.ba \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}-4q^{11}-3q^{13}+\cdots\)
2400.2.a.bb \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(+\) \(q+q^{3}+q^{7}+q^{9}-q^{13}-3q^{19}+q^{21}+\cdots\)
2400.2.a.bc \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}+q^{13}+3q^{19}+q^{21}+\cdots\)
2400.2.a.bd \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(+\) \(q+q^{3}+q^{7}+q^{9}+4q^{11}+3q^{13}+\cdots\)
2400.2.a.be \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+q^{3}+2q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
2400.2.a.bf \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{7}+q^{9}+6q^{11}+2q^{13}+\cdots\)
2400.2.a.bg \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+q^{3}+4q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
2400.2.a.bh \(1\) \(19.164\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+q^{3}+4q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
2400.2.a.bi \(2\) \(19.164\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{7}+q^{9}-\beta q^{11}+\beta q^{13}+\cdots\)
2400.2.a.bj \(2\) \(19.164\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{7}+q^{9}-\beta q^{11}-\beta q^{13}+\cdots\)

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

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