Properties

Label 150.2.a
Level 150
Weight 2
Character orbit a
Rep. character \(\chi_{150}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 3
Sturm bound 60
Trace bound 3

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Defining parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(150))\).

Total New Old
Modular forms 42 3 39
Cusp forms 19 3 16
Eisenstein series 23 0 23

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

\( 3q + q^{2} - q^{3} + 3q^{4} + q^{6} + 4q^{7} + q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} - q^{3} + 3q^{4} + q^{6} + 4q^{7} + q^{8} + 3q^{9} + 4q^{11} - q^{12} - 2q^{13} + 3q^{16} - 6q^{17} + q^{18} - 4q^{19} - 8q^{21} + q^{24} - 14q^{26} - q^{27} + 4q^{28} - 6q^{29} - 8q^{31} + q^{32} - 10q^{34} + 3q^{36} - 2q^{37} - 4q^{38} - 10q^{39} - 2q^{41} - 4q^{42} + 4q^{43} + 4q^{44} + 8q^{46} - q^{48} + 3q^{49} + 2q^{51} - 2q^{52} + 6q^{53} + q^{54} + 4q^{57} - 6q^{58} + 20q^{59} - 6q^{61} + 8q^{62} + 4q^{63} + 3q^{64} + 4q^{66} + 4q^{67} - 6q^{68} + 8q^{69} + 24q^{71} + q^{72} - 2q^{73} - 6q^{74} - 4q^{76} + 2q^{78} + 8q^{79} + 3q^{81} - 6q^{82} - 12q^{83} - 8q^{84} + 12q^{86} + 6q^{87} - 2q^{89} + 16q^{91} - 8q^{93} + 16q^{94} + q^{96} - 2q^{97} + 9q^{98} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.2.a.a \(1\) \(1.198\) \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
150.2.a.b \(1\) \(1.198\) \(\Q\) None \(1\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots\)
150.2.a.c \(1\) \(1.198\) \(\Q\) None \(1\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(150)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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