Properties

Label 1200.2.a
Level $1200$
Weight $2$
Character orbit 1200.a
Rep. character $\chi_{1200}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $19$
Sturm bound $480$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1200))\).

Total New Old
Modular forms 276 19 257
Cusp forms 205 19 186
Eisenstein series 71 0 71

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.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(12\)

Trace form

\( 19q + q^{3} + 19q^{9} + O(q^{10}) \) \( 19q + q^{3} + 19q^{9} - 4q^{11} + 2q^{13} - 2q^{17} - 16q^{23} + q^{27} + 10q^{29} + 12q^{31} + 4q^{33} + 10q^{37} + 6q^{39} + 6q^{41} + 12q^{43} + 24q^{47} + 35q^{49} - 2q^{51} + 2q^{53} + 4q^{57} + 28q^{59} - 14q^{61} + 4q^{67} - 8q^{69} + 24q^{71} - 2q^{73} + 48q^{79} + 19q^{81} + 20q^{83} - 18q^{87} - 10q^{89} + 20q^{91} + 8q^{93} - 10q^{97} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\) 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
1200.2.a.a \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{17}+\cdots\)
1200.2.a.b \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(q-q^{3}-3q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
1200.2.a.c \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(q-q^{3}-3q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
1200.2.a.d \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.e \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.f \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-6q^{11}+5q^{13}+\cdots\)
1200.2.a.g \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q-q^{3}+2q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
1200.2.a.h \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-q^{3}+2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1200.2.a.i \(1\) \(9.582\) \(\Q\) None \(0\) \(-1\) \(0\) \(5\) \(+\) \(+\) \(-\) \(q-q^{3}+5q^{7}+q^{9}+6q^{11}-3q^{13}+\cdots\)
1200.2.a.j \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(-5\) \(+\) \(-\) \(+\) \(q+q^{3}-5q^{7}+q^{9}+6q^{11}+3q^{13}+\cdots\)
1200.2.a.k \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{3}-4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
1200.2.a.l \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q+q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
1200.2.a.m \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
1200.2.a.n \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-6q^{11}-5q^{13}+\cdots\)
1200.2.a.o \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+q^{9}+4q^{11}-6q^{13}+6q^{17}+\cdots\)
1200.2.a.p \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+q^{3}+3q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
1200.2.a.q \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(3\) \(+\) \(-\) \(+\) \(q+q^{3}+3q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
1200.2.a.r \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+q^{3}+4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
1200.2.a.s \(1\) \(9.582\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1200))\) into lower level spaces

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