Properties

Label 63.2.a
Level $63$
Weight $2$
Character orbit 63.a
Rep. character $\chi_{63}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $16$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(63))\).

Total New Old
Modular forms 12 3 9
Cusp forms 5 3 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.

\(3\)\(7\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} + q^{4} + 2q^{5} + q^{7} - 3q^{8} + O(q^{10}) \) \( 3q + q^{2} + q^{4} + 2q^{5} + q^{7} - 3q^{8} - 10q^{10} - 4q^{11} + 2q^{13} - q^{14} - 11q^{16} + 6q^{17} - 4q^{19} - 2q^{20} + 8q^{22} + 13q^{25} - 2q^{26} + 3q^{28} + 2q^{29} - 8q^{31} + 5q^{32} + 18q^{34} - 2q^{35} + 10q^{37} + 4q^{38} + 6q^{40} - 2q^{41} - 12q^{43} + 4q^{44} - 12q^{46} + 3q^{49} - q^{50} + 6q^{52} - 6q^{53} - 32q^{55} + 3q^{56} + 2q^{58} - 12q^{59} - 22q^{61} + 9q^{64} - 4q^{65} - 4q^{67} - 6q^{68} - 14q^{70} + 22q^{73} + 6q^{74} - 12q^{76} + 4q^{77} - 2q^{80} + 34q^{82} + 12q^{83} - 12q^{85} - 4q^{86} + 14q^{89} + 6q^{91} + 24q^{94} + 8q^{95} + 46q^{97} + q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(63))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
63.2.a.a \(1\) \(0.503\) \(\Q\) None \(1\) \(0\) \(2\) \(-1\) \(-\) \(+\) \(q+q^{2}-q^{4}+2q^{5}-q^{7}-3q^{8}+2q^{10}+\cdots\)
63.2.a.b \(2\) \(0.503\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(q+\beta q^{2}+q^{4}-2\beta q^{5}+q^{7}-\beta q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(63)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 + T^{2} + 4 T^{4} \))
$3$ 1
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))
$7$ (\( 1 + T \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 10 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 34 T^{2} + 529 T^{4} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 26 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 58 T^{2} + 2809 T^{4} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 70 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 34 T^{2} + 5041 T^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))
$89$ (\( 1 - 14 T + 89 T^{2} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 18 T + 97 T^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))
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