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Properties

Label 89.2.a
Level $89$
Weight $2$
Character orbit 89.a
Rep. character $\chi_{89}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $3$
Sturm bound $15$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 89.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(15\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(89))\).

Total New Old
Modular forms 8 8 0
Cusp forms 7 7 0
Eisenstein series 1 1 0

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

\(89\)Dim
\(+\)\(1\)
\(-\)\(6\)

Trace form

\( 7 q - q^{2} - 2 q^{3} + 9 q^{4} - 4 q^{5} + 6 q^{7} + 3 q^{8} + q^{9} + O(q^{10}) \) \( 7 q - q^{2} - 2 q^{3} + 9 q^{4} - 4 q^{5} + 6 q^{7} + 3 q^{8} + q^{9} - 4 q^{10} - 16 q^{12} + 4 q^{13} - 8 q^{14} + 6 q^{15} + 9 q^{16} - 4 q^{17} - 15 q^{18} + 6 q^{19} - 28 q^{20} - 4 q^{21} - 16 q^{22} + 10 q^{23} - 14 q^{24} - q^{25} - 4 q^{26} + 4 q^{27} + 14 q^{28} - 4 q^{29} + 16 q^{31} + 15 q^{32} - 10 q^{33} - 8 q^{34} + 4 q^{35} + 31 q^{36} - 6 q^{37} + 26 q^{38} + 24 q^{39} - 8 q^{40} - 8 q^{41} + 16 q^{42} - 4 q^{43} + 12 q^{44} - 26 q^{45} + 18 q^{46} - 4 q^{47} - 46 q^{48} + 15 q^{49} + 19 q^{50} - 10 q^{51} + 8 q^{52} - 20 q^{53} + 16 q^{54} + 16 q^{55} - 8 q^{56} - 6 q^{57} - 20 q^{58} - 6 q^{59} + 60 q^{60} + 4 q^{61} + 2 q^{62} + 20 q^{63} + 25 q^{64} - 18 q^{65} + 38 q^{66} + 28 q^{67} - 24 q^{68} - 42 q^{69} - 18 q^{70} - 8 q^{71} - 31 q^{72} - 8 q^{73} + 6 q^{74} - 22 q^{75} + 28 q^{76} - 8 q^{77} - 44 q^{78} + 36 q^{79} - 52 q^{80} - 17 q^{81} + 32 q^{82} - 14 q^{83} - 48 q^{84} - 26 q^{85} + 12 q^{86} - 72 q^{88} + 5 q^{89} + 18 q^{90} - 24 q^{91} + 24 q^{92} - 6 q^{93} - 4 q^{94} + 14 q^{95} + 8 q^{96} + 4 q^{97} - 41 q^{98} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(89))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 89
89.2.a.a 89.a 1.a $1$ $0.711$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-q^{5}+q^{6}-4q^{7}+\cdots\)
89.2.a.b 89.a 1.a $1$ $0.711$ \(\Q\) None \(1\) \(2\) \(-2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}-q^{4}-2q^{5}+2q^{6}+2q^{7}+\cdots\)
89.2.a.c 89.a 1.a $5$ $0.711$ 5.5.535120.1 None \(-1\) \(-3\) \(-1\) \(8\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(-1+\beta _{1})q^{3}+(3-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)