Properties

Label 300.2.a
Level $300$
Weight $2$
Character orbit 300.a
Rep. character $\chi_{300}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $120$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(300))\).

Total New Old
Modular forms 78 4 74
Cusp forms 43 4 39
Eisenstein series 35 0 35

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

Trace form

\( 4 q + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{9} + 4 q^{11} + 10 q^{19} + 6 q^{21} - 24 q^{29} + 6 q^{31} + 10 q^{39} - 20 q^{41} + 6 q^{49} - 4 q^{51} - 4 q^{59} - 22 q^{61} - 4 q^{69} - 8 q^{79} + 4 q^{81} - 20 q^{89} - 10 q^{91} + 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(300))\) 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
300.2.a.a 300.a 1.a $1$ $2.396$ \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}-4q^{11}-4q^{17}+\cdots\)
300.2.a.b 300.a 1.a $1$ $2.396$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}+6q^{11}-5q^{13}+\cdots\)
300.2.a.c 300.a 1.a $1$ $2.396$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}+6q^{11}+5q^{13}+\cdots\)
300.2.a.d 300.a 1.a $1$ $2.396$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)