Properties

Label 85.2.a
Level $85$
Weight $2$
Character orbit 85.a
Rep. character $\chi_{85}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $3$
Sturm bound $18$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(85))\).

Total New Old
Modular forms 10 5 5
Cusp forms 7 5 2
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.

\(5\)\(17\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(3\)

Trace form

\( 5q - q^{2} + 3q^{4} - q^{5} - 4q^{6} - 8q^{7} - 9q^{8} + 9q^{9} + O(q^{10}) \) \( 5q - q^{2} + 3q^{4} - q^{5} - 4q^{6} - 8q^{7} - 9q^{8} + 9q^{9} + q^{10} + 4q^{12} - 6q^{13} + 12q^{14} + 4q^{15} - 5q^{16} - 3q^{17} - q^{18} + 4q^{19} + q^{20} - 8q^{21} + 8q^{22} - 4q^{23} + 8q^{24} + 5q^{25} - 6q^{26} - 12q^{27} - 12q^{28} - 10q^{29} - 8q^{30} + 11q^{32} + 28q^{33} + 3q^{34} + 4q^{35} - 25q^{36} - 10q^{37} + 4q^{38} + 4q^{39} + 9q^{40} + 14q^{41} + 4q^{42} - 12q^{44} - 5q^{45} + 24q^{46} + 20q^{47} - 24q^{48} - 11q^{49} - q^{50} + 4q^{51} + 6q^{52} + 14q^{53} - 20q^{54} + 12q^{55} + 16q^{56} + 2q^{58} - 4q^{59} - 6q^{61} + 8q^{62} - 12q^{63} - 5q^{64} - 10q^{65} - 24q^{66} - 24q^{67} - 5q^{68} + 4q^{71} + 7q^{72} - 26q^{73} + 18q^{74} + 20q^{76} + 4q^{77} + 36q^{78} - 8q^{79} - 15q^{80} + 37q^{81} - 6q^{82} + 24q^{83} - q^{85} + 4q^{86} - 8q^{87} + 28q^{88} - 22q^{89} - 23q^{90} - 4q^{91} - 8q^{92} - 28q^{93} - 16q^{94} + 4q^{95} + 12q^{96} + 2q^{97} - 29q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(85))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 17
85.2.a.a \(1\) \(0.679\) \(\Q\) None \(1\) \(2\) \(-1\) \(-2\) \(+\) \(-\) \(q+q^{2}+2q^{3}-q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots\)
85.2.a.b \(2\) \(0.679\) \(\Q(\sqrt{2}) \) None \(-2\) \(-4\) \(-2\) \(-4\) \(+\) \(+\) \(q+(-1+\beta )q^{2}+(-2-\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
85.2.a.c \(2\) \(0.679\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(-2\) \(-\) \(+\) \(q+\beta q^{2}+(1-\beta )q^{3}+q^{4}+q^{5}+(-3+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(85))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(85)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)