Properties

Label 450.2.a
Level 450
Weight 2
Character orbit a
Rep. character \(\chi_{450}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 7
Sturm bound 180
Trace bound 11

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Defining parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.a (trivial)
Character field: \(\Q\)
Newforms: \( 7 \)
Sturm bound: \(180\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(450))\).

Total New Old
Modular forms 114 7 107
Cusp forms 67 7 60
Eisenstein series 47 0 47

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

Trace form

\(7q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 12q^{22} \) \(\mathstrut +\mathstrut 6q^{26} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut +\mathstrut 8q^{34} \) \(\mathstrut -\mathstrut 18q^{37} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut -\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 6q^{53} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 18q^{58} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 7q^{64} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut 48q^{71} \) \(\mathstrut +\mathstrut 18q^{73} \) \(\mathstrut +\mathstrut 10q^{74} \) \(\mathstrut -\mathstrut 2q^{76} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut -\mathstrut 6q^{82} \) \(\mathstrut +\mathstrut 12q^{83} \) \(\mathstrut -\mathstrut 20q^{86} \) \(\mathstrut +\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 28q^{89} \) \(\mathstrut -\mathstrut 16q^{91} \) \(\mathstrut -\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 9q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(450))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
450.2.a.a \(1\) \(3.593\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-6q^{11}+4q^{13}+\cdots\)
450.2.a.b \(1\) \(3.593\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{11}-6q^{13}+\cdots\)
450.2.a.c \(1\) \(3.593\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}+4q^{13}+\cdots\)
450.2.a.d \(1\) \(3.593\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-2q^{13}-4q^{14}+\cdots\)
450.2.a.e \(1\) \(3.593\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}+6q^{11}+4q^{13}+\cdots\)
450.2.a.f \(1\) \(3.593\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}+6q^{13}+\cdots\)
450.2.a.g \(1\) \(3.593\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}-4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(450))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(450)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)