Properties

Label 90.2.a
Level 90
Weight 2
Character orbit a
Rep. character \(\chi_{90}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 36
Trace bound 5

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 90.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(90))\).

Total New Old
Modular forms 26 3 23
Cusp forms 11 3 8
Eisenstein series 15 0 15

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.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} + 3q^{4} + q^{5} + q^{8} + O(q^{10}) \) \( 3q + q^{2} + 3q^{4} + q^{5} + q^{8} - q^{10} - 6q^{13} - 4q^{14} + 3q^{16} - 6q^{17} - 12q^{19} + q^{20} - 12q^{22} + 3q^{25} + 2q^{26} + 6q^{29} + q^{32} + 6q^{34} - 4q^{35} + 18q^{37} - 4q^{38} - q^{40} + 6q^{41} + 12q^{43} + 3q^{49} + q^{50} - 6q^{52} + 6q^{53} + 12q^{55} - 4q^{56} + 18q^{58} - 6q^{61} + 8q^{62} + 3q^{64} + 2q^{65} - 12q^{67} - 6q^{68} - 8q^{70} - 18q^{73} + 2q^{74} - 12q^{76} + q^{80} + 6q^{82} - 12q^{83} - 18q^{85} - 4q^{86} - 12q^{88} - 18q^{89} - 24q^{91} - 4q^{95} + 6q^{97} + 9q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(90))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
90.2.a.a \(1\) \(0.719\) \(\Q\) None \(-1\) \(0\) \(1\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
90.2.a.b \(1\) \(0.719\) \(\Q\) None \(1\) \(0\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
90.2.a.c \(1\) \(0.719\) \(\Q\) None \(1\) \(0\) \(1\) \(-4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))
$3$ 1
$5$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))
$11$ (\( 1 - 6 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))
$37$ (\( 1 - 8 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))
$73$ (\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))
$89$ (\( 1 - 12 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))
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