Properties

Label 228.2.a
Level $228$
Weight $2$
Character orbit 228.a
Rep. character $\chi_{228}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
Sturm bound $80$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(80\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(228))\).

Total New Old
Modular forms 46 4 42
Cusp forms 35 4 31
Eisenstein series 11 0 11

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

\(2\)\(3\)\(19\)FrickeDim
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q + 2 q^{5} + 2 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{5} + 2 q^{7} + 4 q^{9} - 6 q^{11} + 4 q^{15} - 2 q^{17} + 2 q^{19} + 14 q^{25} + 4 q^{29} - 4 q^{31} - 18 q^{35} - 12 q^{37} + 8 q^{39} - 16 q^{41} + 2 q^{43} + 2 q^{45} - 18 q^{47} - 10 q^{49} - 4 q^{51} + 4 q^{53} - 2 q^{55} + 2 q^{57} - 8 q^{59} + 2 q^{61} + 2 q^{63} + 28 q^{65} + 16 q^{67} - 12 q^{69} - 32 q^{71} - 10 q^{73} + 8 q^{75} + 10 q^{77} - 8 q^{79} + 4 q^{81} + 8 q^{83} + 6 q^{85} - 16 q^{87} + 28 q^{89} - 4 q^{91} - 2 q^{95} + 16 q^{97} - 6 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(228))\) 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 3 19
228.2.a.a 228.a 1.a $1$ $1.821$ \(\Q\) None \(0\) \(-1\) \(-3\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}+q^{7}+q^{9}-5q^{11}-6q^{13}+\cdots\)
228.2.a.b 228.a 1.a $1$ $1.821$ \(\Q\) None \(0\) \(-1\) \(2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{9}+2q^{11}+2q^{13}+\cdots\)
228.2.a.c 228.a 1.a $2$ $1.821$ \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(3\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(228))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(228)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)