Properties

Label 225.2.a
Level 225
Weight 2
Character orbit a
Rep. character \(\chi_{225}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 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: \(\Q\)
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

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

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

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - T^{2} + 4 T^{4} \))
$3$ 1
$5$ 1
$7$ (\( 1 + 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 14 T^{2} + 289 T^{4} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 34 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 10 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))
$31$ (\( 1 + 3 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))
$41$ (\( 1 - 8 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))
$43$ (\( 1 - T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))
$47$ (\( 1 + 2 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 14 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 4 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 86 T^{2} + 2809 T^{4} \))
$59$ (\( 1 - 10 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))
$61$ (\( 1 - 7 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 3 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( ( 1 + 73 T^{2} )^{2} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 154 T^{2} + 6889 T^{4} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))
$97$ (\( 1 - 17 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 5 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))
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