Properties

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

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Defining parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 225.a (trivial)
Character field: \(\Q\)
Newforms: \( 6 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(225))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
225.2.a.a \(1\) \(1.797\) \(\Q\) None \(-2\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(q-2q^{2}+2q^{4}-3q^{7}-2q^{11}+q^{13}+\cdots\)
225.2.a.b \(1\) \(1.797\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-q^{2}-q^{4}+3q^{8}+4q^{11}+2q^{13}+\cdots\)
225.2.a.c \(1\) \(1.797\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(+\) \(+\) \(q-2q^{4}-5q^{7}-5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.d \(1\) \(1.797\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(5\) \(+\) \(-\) \(q-2q^{4}+5q^{7}+5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.e \(1\) \(1.797\) \(\Q\) None \(2\) \(0\) \(0\) \(3\) \(-\) \(+\) \(q+2q^{2}+2q^{4}+3q^{7}-2q^{11}-q^{13}+\cdots\)
225.2.a.f \(2\) \(1.797\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta q^{2}+3q^{4}-\beta q^{8}-q^{16}+2\beta q^{17}+\cdots\)

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

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