Properties

Label 225.2.a
Level $225$
Weight $2$
Character orbit 225.a
Rep. character $\chi_{225}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $6$
Sturm bound $60$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(60\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(225))\).

Total New Old
Modular forms 42 10 32
Cusp forms 19 7 12
Eisenstein series 23 3 20

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

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7 q - q^{2} + 5 q^{4} + 3 q^{8} + O(q^{10}) \) \( 7 q - q^{2} + 5 q^{4} + 3 q^{8} + 2 q^{13} + 12 q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{22} - 6 q^{26} - 18 q^{29} - 4 q^{31} - 5 q^{32} - 14 q^{34} + 10 q^{37} - 4 q^{38} + 6 q^{41} - 4 q^{43} - 12 q^{44} - 16 q^{46} + 8 q^{47} + 19 q^{49} - 2 q^{52} - 10 q^{53} - 2 q^{58} + 24 q^{59} - 10 q^{61} - 51 q^{64} - 12 q^{67} - 2 q^{68} + 24 q^{71} - 10 q^{73} - 18 q^{74} + 4 q^{76} + 24 q^{79} + 10 q^{82} + 12 q^{83} + 12 q^{88} + 6 q^{89} + 44 q^{91} + 40 q^{94} - 2 q^{97} + 7 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(225))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
225.2.a.a 225.a 1.a $1$ $1.797$ \(\Q\) None \(-2\) \(0\) \(0\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-3q^{7}-2q^{11}+q^{13}+\cdots\)
225.2.a.b 225.a 1.a $1$ $1.797$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{8}+4q^{11}+2q^{13}+\cdots\)
225.2.a.c 225.a 1.a $1$ $1.797$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}-5q^{7}-5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.d 225.a 1.a $1$ $1.797$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+5q^{7}+5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.e 225.a 1.a $1$ $1.797$ \(\Q\) None \(2\) \(0\) \(0\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+3q^{7}-2q^{11}-q^{13}+\cdots\)
225.2.a.f 225.a 1.a $2$ $1.797$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-\beta q^{2}+3q^{4}-\beta q^{8}-q^{16}+2\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(225)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)