Properties

Label 78.2.a
Level $78$
Weight $2$
Character orbit 78.a
Rep. character $\chi_{78}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $28$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(28\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(78))\).

Total New Old
Modular forms 18 1 17
Cusp forms 11 1 10
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.

\(2\)\(3\)\(13\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(-\)\(-\)\(4\)\(1\)\(3\)\(3\)\(1\)\(2\)\(1\)\(0\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)\(0\)\(3\)\(2\)\(0\)\(2\)\(1\)\(0\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(2\)\(0\)\(2\)\(1\)\(0\)\(1\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)\(0\)\(2\)\(1\)\(0\)\(1\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(2\)\(0\)\(2\)\(1\)\(0\)\(1\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(4\)\(0\)\(4\)\(3\)\(0\)\(3\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)\(0\)\(1\)\(0\)\(0\)\(0\)\(1\)\(0\)\(1\)
Plus space\(+\)\(8\)\(0\)\(8\)\(5\)\(0\)\(5\)\(3\)\(0\)\(3\)
Minus space\(-\)\(10\)\(1\)\(9\)\(6\)\(1\)\(5\)\(4\)\(0\)\(4\)

Trace form

\( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{21}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(78))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 13
78.2.a.a 78.a 1.a $1$ $0.623$ \(\Q\) None 78.2.a.a \(-1\) \(-1\) \(2\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(78))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(78)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)