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Properties

Label 170.2.a
Level $170$
Weight $2$
Character orbit 170.a
Rep. character $\chi_{170}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $6$
Sturm bound $54$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(54\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(170))\).

Total New Old
Modular forms 30 7 23
Cusp forms 23 7 16
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.

\(2\)\(5\)\(17\)FrickeDim
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(6\)

Trace form

\( 7 q - q^{2} + 7 q^{4} + q^{5} + 8 q^{7} - q^{8} + 7 q^{9} + O(q^{10}) \) \( 7 q - q^{2} + 7 q^{4} + q^{5} + 8 q^{7} - q^{8} + 7 q^{9} + q^{10} - 8 q^{11} + 2 q^{13} - 4 q^{15} + 7 q^{16} + 3 q^{17} - 5 q^{18} + q^{20} - 8 q^{21} - 8 q^{22} - 16 q^{23} + 7 q^{25} + 6 q^{26} + 8 q^{28} - 2 q^{29} - 8 q^{31} - q^{32} - 16 q^{33} - q^{34} - 4 q^{35} + 7 q^{36} - 2 q^{37} - 4 q^{38} - 8 q^{39} + q^{40} - 18 q^{41} - 24 q^{42} - 4 q^{43} - 8 q^{44} - 3 q^{45} + 7 q^{49} - q^{50} - 4 q^{51} + 2 q^{52} - 22 q^{53} - 24 q^{54} - 12 q^{55} + 16 q^{57} - 2 q^{58} + 8 q^{59} - 4 q^{60} + 22 q^{61} + 24 q^{63} + 7 q^{64} + 6 q^{65} + 24 q^{66} + 20 q^{67} + 3 q^{68} + 16 q^{69} + 4 q^{70} + 8 q^{71} - 5 q^{72} + 38 q^{73} + 6 q^{74} - 16 q^{78} - 8 q^{79} + q^{80} - 25 q^{81} + 14 q^{82} + 12 q^{83} - 8 q^{84} + q^{85} - 4 q^{86} + 40 q^{87} - 8 q^{88} + 2 q^{89} + 13 q^{90} + 40 q^{91} - 16 q^{92} + 8 q^{93} + 12 q^{94} - 20 q^{95} - 10 q^{97} + 31 q^{98} - 32 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(170))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 17
170.2.a.a 170.a 1.a $1$ $1.357$ \(\Q\) None 170.2.a.a \(-1\) \(-2\) \(-1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+2q^{7}+\cdots\)
170.2.a.b 170.a 1.a $1$ $1.357$ \(\Q\) None 170.2.a.b \(-1\) \(-2\) \(1\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-2q^{7}+\cdots\)
170.2.a.c 170.a 1.a $1$ $1.357$ \(\Q\) None 170.2.a.c \(-1\) \(1\) \(1\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
170.2.a.d 170.a 1.a $1$ $1.357$ \(\Q\) None 170.2.a.d \(-1\) \(3\) \(-1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+2q^{7}+\cdots\)
170.2.a.e 170.a 1.a $1$ $1.357$ \(\Q\) None 170.2.a.e \(1\) \(1\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
170.2.a.f 170.a 1.a $2$ $1.357$ \(\Q(\sqrt{17}) \) None 170.2.a.f \(2\) \(-1\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+2\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(170))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(170)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)