Properties

Label 136.2.a
Level $136$
Weight $2$
Character orbit 136.a
Rep. character $\chi_{136}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
Sturm bound $36$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(136))\).

Total New Old
Modular forms 22 4 18
Cusp forms 15 4 11
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.

\(2\)\(17\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q - 2 q^{3} + 2 q^{5} + 8 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{3} + 2 q^{5} + 8 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{17} - 8 q^{21} + 12 q^{23} - 8 q^{25} - 20 q^{27} - 6 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{35} - 2 q^{37} + 4 q^{39} + 4 q^{41} - 12 q^{43} + 10 q^{45} - 12 q^{49} - 6 q^{51} - 4 q^{53} + 16 q^{55} - 8 q^{57} + 12 q^{59} + 22 q^{61} + 24 q^{63} - 4 q^{65} - 12 q^{67} - 16 q^{69} + 28 q^{71} - 6 q^{75} + 24 q^{77} - 4 q^{79} + 36 q^{83} + 2 q^{85} + 16 q^{87} - 24 q^{89} - 24 q^{91} - 8 q^{93} - 8 q^{95} - 12 q^{97} + 22 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(136))\) 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 17
136.2.a.a 136.a 1.a $1$ $1.086$ \(\Q\) None \(0\) \(-2\) \(-2\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{5}-2q^{7}+q^{9}-6q^{11}+\cdots\)
136.2.a.b 136.a 1.a $1$ $1.086$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}+2q^{11}-6q^{13}-q^{17}+\cdots\)
136.2.a.c 136.a 1.a $2$ $1.086$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(4\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+2q^{5}+(1+\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(136))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(136)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 2}\)