Properties

Label 156.2.a
Level $156$
Weight $2$
Character orbit 156.a
Rep. character $\chi_{156}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $56$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(156))\).

Total New Old
Modular forms 34 2 32
Cusp forms 23 2 21
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.

\(2\)\(3\)\(13\)FrickeDim.
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(1\)

Trace form

\( 2 q - 4 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{15} - 4 q^{17} + 4 q^{21} + 6 q^{25} - 12 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{35} + 12 q^{37} - 4 q^{41} - 4 q^{45} - 4 q^{47} - 6 q^{49} - 8 q^{51} - 4 q^{53} + 16 q^{55} + 4 q^{57} + 4 q^{59} - 12 q^{61} - 4 q^{65} - 8 q^{67} + 28 q^{71} + 4 q^{73} - 16 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} + 12 q^{83} - 8 q^{85} - 4 q^{89} + 12 q^{93} + 8 q^{95} - 12 q^{97} - 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(156))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 13
156.2.a.a 156.a 1.a $1$ $1.246$ \(\Q\) None \(0\) \(-1\) \(-4\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
156.2.a.b 156.a 1.a $1$ $1.246$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}+q^{13}-6q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(156))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(156)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)