Properties

Label 169.2.a
Level $169$
Weight $2$
Character orbit 169.a
Rep. character $\chi_{169}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $3$
Sturm bound $30$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(30\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(169))\).

Total New Old
Modular forms 21 19 2
Cusp forms 8 8 0
Eisenstein series 13 11 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(13\)Dim.
\(+\)\(3\)
\(-\)\(5\)

Trace form

\( 8 q + 2 q^{4} - 4 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{4} - 4 q^{9} + 4 q^{10} + 4 q^{12} - 10 q^{14} - 6 q^{16} + 2 q^{17} + 6 q^{22} + 2 q^{23} - 14 q^{25} - 6 q^{27} + 4 q^{29} - 14 q^{30} + 8 q^{35} - 12 q^{36} + 12 q^{38} + 16 q^{42} + 10 q^{43} - 18 q^{48} - 22 q^{49} + 10 q^{51} - 4 q^{53} + 12 q^{55} - 8 q^{56} + 10 q^{61} + 14 q^{62} - 20 q^{64} + 10 q^{66} - 36 q^{68} + 12 q^{69} + 14 q^{74} + 22 q^{75} + 16 q^{77} - 2 q^{79} - 24 q^{81} - 10 q^{82} - 24 q^{87} + 30 q^{88} - 30 q^{90} + 12 q^{92} + 22 q^{94} + 18 q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 13
169.2.a.a 169.a 1.a $2$ $1.349$ \(\Q(\sqrt{3}) \) None \(0\) \(4\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+2q^{3}+q^{4}-\beta q^{5}+2\beta q^{6}+\cdots\)
169.2.a.b 169.a 1.a $3$ $1.349$ \(\Q(\zeta_{14})^+\) None \(-2\) \(-2\) \(-4\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
169.2.a.c 169.a 1.a $3$ $1.349$ \(\Q(\zeta_{14})^+\) None \(2\) \(-2\) \(4\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)