Properties

Label 30.2.a
Level $30$
Weight $2$
Character orbit 30.a
Rep. character $\chi_{30}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $12$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(30))\).

Total New Old
Modular forms 10 1 9
Cusp forms 3 1 2
Eisenstein series 7 0 7

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

Trace form

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2q^{13} + 4q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} - 4q^{19} - q^{20} - 4q^{21} - q^{24} + q^{25} - 2q^{26} + q^{27} - 4q^{28} - 6q^{29} + q^{30} + 8q^{31} - q^{32} - 6q^{34} + 4q^{35} + q^{36} + 2q^{37} + 4q^{38} + 2q^{39} + q^{40} - 6q^{41} + 4q^{42} - 4q^{43} - q^{45} + q^{48} + 9q^{49} - q^{50} + 6q^{51} + 2q^{52} - 6q^{53} - q^{54} + 4q^{56} - 4q^{57} + 6q^{58} - q^{60} - 10q^{61} - 8q^{62} - 4q^{63} + q^{64} - 2q^{65} - 4q^{67} + 6q^{68} - 4q^{70} - q^{72} + 2q^{73} - 2q^{74} + q^{75} - 4q^{76} - 2q^{78} + 8q^{79} - q^{80} + q^{81} + 6q^{82} + 12q^{83} - 4q^{84} - 6q^{85} + 4q^{86} - 6q^{87} + 18q^{89} + q^{90} - 8q^{91} + 8q^{93} + 4q^{95} - q^{96} + 2q^{97} - 9q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(30))\) 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
30.2.a.a \(1\) \(0.240\) \(\Q\) None \(-1\) \(1\) \(-1\) \(-4\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

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

Hecke characteristic polynomials

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