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Properties

Label 177.2.a
Level $177$
Weight $2$
Character orbit 177.a
Rep. character $\chi_{177}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $4$
Sturm bound $40$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(177))\).

Total New Old
Modular forms 22 9 13
Cusp forms 19 9 10
Eisenstein series 3 0 3

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

\(3\)\(59\)FrickeDim
\(+\)\(+\)$+$\(2\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(2\)
\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(5\)

Trace form

\( 9 q - 3 q^{2} - q^{3} + 3 q^{4} - 6 q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + 9 q^{9} + O(q^{10}) \) \( 9 q - 3 q^{2} - q^{3} + 3 q^{4} - 6 q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + 9 q^{9} + 10 q^{10} + q^{12} - 6 q^{13} + 20 q^{14} - 2 q^{15} + 3 q^{16} - 2 q^{17} - 3 q^{18} - 8 q^{19} - 14 q^{20} - 8 q^{21} - 16 q^{22} - 8 q^{23} - 9 q^{24} - q^{25} + 2 q^{26} - q^{27} + 4 q^{28} - 2 q^{29} + 10 q^{30} + 4 q^{31} - 19 q^{32} + 4 q^{33} + 2 q^{34} + 16 q^{35} + 3 q^{36} - 22 q^{37} + 32 q^{38} + 2 q^{39} + 26 q^{40} + 2 q^{41} - 8 q^{42} + 12 q^{43} - 6 q^{45} - 4 q^{46} + 4 q^{47} + 17 q^{48} + 29 q^{49} - 37 q^{50} - 2 q^{51} - 46 q^{52} + 10 q^{53} - q^{54} + 4 q^{55} + 20 q^{56} - 12 q^{57} + 18 q^{58} + q^{59} - 6 q^{60} - 2 q^{61} - 24 q^{62} - 4 q^{63} - 9 q^{64} + 8 q^{65} - 4 q^{66} + 20 q^{67} + 2 q^{68} + 4 q^{69} - 4 q^{70} + 36 q^{71} - 3 q^{72} + 10 q^{73} - 26 q^{74} + q^{75} - 36 q^{76} + 12 q^{77} + 26 q^{78} - 34 q^{80} + 9 q^{81} + 54 q^{82} - 16 q^{84} - 40 q^{85} + 4 q^{86} - 10 q^{87} - 36 q^{88} - 30 q^{89} + 10 q^{90} + 8 q^{91} - 4 q^{92} - 20 q^{93} + 24 q^{94} - 12 q^{95} - 29 q^{96} + 6 q^{97} + 17 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(177))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 59
177.2.a.a 177.a 1.a $2$ $1.413$ \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(-6\) \(-7\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}-3q^{5}+\cdots\)
177.2.a.b 177.a 1.a $2$ $1.413$ \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(-7\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
177.2.a.c 177.a 1.a $2$ $1.413$ \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(2\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
177.2.a.d 177.a 1.a $3$ $1.413$ 3.3.229.1 None \(0\) \(-3\) \(-2\) \(9\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(177))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(177)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 2}\)