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Properties

Label 230.2.a
Level $230$
Weight $2$
Character orbit 230.a
Rep. character $\chi_{230}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $4$
Sturm bound $72$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(230))\).

Total New Old
Modular forms 40 9 31
Cusp forms 33 9 24
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.

\(2\)\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(9\)

Trace form

\( 9 q + q^{2} + 4 q^{3} + 9 q^{4} - q^{5} + 8 q^{7} + q^{8} + 17 q^{9} + O(q^{10}) \) \( 9 q + q^{2} + 4 q^{3} + 9 q^{4} - q^{5} + 8 q^{7} + q^{8} + 17 q^{9} - q^{10} + 4 q^{12} + 6 q^{13} + 4 q^{15} + 9 q^{16} - 6 q^{17} - 3 q^{18} + 8 q^{19} - q^{20} - 16 q^{21} + 8 q^{22} - q^{23} + 9 q^{25} - 14 q^{26} - 8 q^{27} + 8 q^{28} - 10 q^{29} - 4 q^{30} + 4 q^{31} + q^{32} - 24 q^{33} - 6 q^{34} + 17 q^{36} + 10 q^{37} - 8 q^{38} - 16 q^{39} - q^{40} - 26 q^{41} - 32 q^{42} + 24 q^{43} - 13 q^{45} - q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + q^{50} - 32 q^{51} + 6 q^{52} - 22 q^{53} - 24 q^{54} - 12 q^{55} - 48 q^{57} - 26 q^{58} - 28 q^{59} + 4 q^{60} + 10 q^{61} - 16 q^{63} + 9 q^{64} - 6 q^{65} - 8 q^{66} + 24 q^{67} - 6 q^{68} - 4 q^{69} - 4 q^{70} - 12 q^{71} - 3 q^{72} + 2 q^{73} - 6 q^{74} + 4 q^{75} + 8 q^{76} - 8 q^{77} + 16 q^{78} + 24 q^{79} - q^{80} + 33 q^{81} + 10 q^{82} + 8 q^{83} - 16 q^{84} + 14 q^{85} + 24 q^{86} + 16 q^{87} + 8 q^{88} + 18 q^{89} - 13 q^{90} + 16 q^{91} - q^{92} + 40 q^{93} + 8 q^{94} - 12 q^{95} + 10 q^{97} + 25 q^{98} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 23
230.2.a.a 230.a 1.a $2$ $1.837$ \(\Q(\sqrt{21}) \) None \(-2\) \(-1\) \(-2\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
230.2.a.b 230.a 1.a $2$ $1.837$ \(\Q(\sqrt{13}) \) None \(-2\) \(3\) \(2\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
230.2.a.c 230.a 1.a $2$ $1.837$ \(\Q(\sqrt{5}) \) None \(2\) \(1\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
230.2.a.d 230.a 1.a $3$ $1.837$ 3.3.1101.1 None \(3\) \(1\) \(-3\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(230))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(230)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)