Properties

Label 171.2.a
Level $171$
Weight $2$
Character orbit 171.a
Rep. character $\chi_{171}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $5$
Sturm bound $40$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(40\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(171))\).

Total New Old
Modular forms 24 8 16
Cusp forms 17 8 9
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.

\(3\)\(19\)FrickeDim
\(+\)\(-\)$-$\(4\)
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(7\)

Trace form

\( 8 q + 3 q^{2} + 11 q^{4} + q^{5} - q^{7} + 3 q^{8} + O(q^{10}) \) \( 8 q + 3 q^{2} + 11 q^{4} + q^{5} - q^{7} + 3 q^{8} - 10 q^{10} - q^{11} + 6 q^{13} - 4 q^{14} + 17 q^{16} + 7 q^{17} + 2 q^{19} + 8 q^{20} - 20 q^{22} - 4 q^{23} + 13 q^{25} - 14 q^{26} - 30 q^{28} + 4 q^{29} - 4 q^{31} - 21 q^{32} - 22 q^{34} - 15 q^{35} + 20 q^{37} - 3 q^{38} - 66 q^{40} + 16 q^{41} - 17 q^{43} + 10 q^{44} - 8 q^{46} - 3 q^{47} + 13 q^{49} + q^{50} + 14 q^{52} - 10 q^{53} - 3 q^{55} + 14 q^{58} + 26 q^{59} + 17 q^{61} - 16 q^{62} + 53 q^{64} + 36 q^{65} - 8 q^{67} - 16 q^{68} + 24 q^{70} - 6 q^{71} - 5 q^{73} + 26 q^{74} + 5 q^{76} + 17 q^{77} + 8 q^{79} - 22 q^{80} - 22 q^{82} - 44 q^{83} + 39 q^{85} - 12 q^{88} - 14 q^{89} - 20 q^{91} + 4 q^{92} + 24 q^{94} - 7 q^{95} - 10 q^{97} + 47 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(171))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 19
171.2.a.a 171.a 1.a $1$ $1.365$ \(\Q\) None 57.2.a.c \(-1\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{5}+3q^{8}-2q^{10}+\cdots\)
171.2.a.b 171.a 1.a $1$ $1.365$ \(\Q\) None 19.2.a.a \(0\) \(0\) \(-3\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-3q^{5}-q^{7}-3q^{11}-4q^{13}+\cdots\)
171.2.a.c 171.a 1.a $1$ $1.365$ \(\Q\) None 57.2.a.b \(2\) \(0\) \(-1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{5}+3q^{7}-2q^{10}+\cdots\)
171.2.a.d 171.a 1.a $1$ $1.365$ \(\Q\) None 57.2.a.a \(2\) \(0\) \(3\) \(-5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+3q^{5}-5q^{7}+6q^{10}+\cdots\)
171.2.a.e 171.a 1.a $4$ $1.365$ 4.4.13068.1 None 171.2.a.e \(0\) \(0\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{3})q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(171))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(171)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)