Properties

Label 75.2.a
Level 75
Weight 2
Character orbit a
Rep. character \(\chi_{75}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 20
Trace bound 2

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 75.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(75))\).

Total New Old
Modular forms 16 3 13
Cusp forms 5 3 2
Eisenstein series 11 0 11

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.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} + q^{3} + 3q^{4} - 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + q^{3} + 3q^{4} - 3q^{6} - 3q^{8} + 3q^{9} - q^{12} + 2q^{13} - 12q^{14} - 9q^{16} - 2q^{17} + q^{18} - 6q^{19} + 6q^{21} - 4q^{22} - 3q^{24} + 6q^{26} + q^{27} + 18q^{29} - 6q^{31} + 5q^{32} - 4q^{33} + 6q^{34} + 3q^{36} + 10q^{37} + 4q^{38} - 6q^{41} - 4q^{43} + 12q^{44} + 24q^{46} - 8q^{47} - q^{48} - 3q^{49} - 6q^{51} - 2q^{52} + 10q^{53} - 3q^{54} + 4q^{57} - 2q^{58} - 24q^{59} + 12q^{61} - 9q^{64} - 12q^{66} - 12q^{67} + 2q^{68} - 12q^{69} - 24q^{71} - 3q^{72} - 10q^{73} + 18q^{74} - 24q^{76} + 2q^{78} + 3q^{81} + 10q^{82} - 12q^{83} + 12q^{84} - 2q^{87} + 12q^{88} - 6q^{89} - 6q^{91} + 21q^{96} - 2q^{97} - 7q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
75.2.a.a \(1\) \(0.599\) \(\Q\) None \(-2\) \(1\) \(0\) \(3\) \(-\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+3q^{7}+\cdots\)
75.2.a.b \(1\) \(0.599\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
75.2.a.c \(1\) \(0.599\) \(\Q\) None \(2\) \(-1\) \(0\) \(-3\) \(+\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-3q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

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

Hecke Characteristic Polynomials

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