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Properties

Label 393.2.a
Level $393$
Weight $2$
Character orbit 393.a
Rep. character $\chi_{393}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $5$
Sturm bound $88$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 393 = 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 393.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(88\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(393))\).

Total New Old
Modular forms 46 21 25
Cusp forms 43 21 22
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.

\(3\)\(131\)FrickeDim
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(13\)

Trace form

\( 21 q - 3 q^{2} - q^{3} + 15 q^{4} - 2 q^{5} - q^{6} - 3 q^{8} + 21 q^{9} + O(q^{10}) \) \( 21 q - 3 q^{2} - q^{3} + 15 q^{4} - 2 q^{5} - q^{6} - 3 q^{8} + 21 q^{9} + 2 q^{10} - 7 q^{12} - 10 q^{13} - 4 q^{14} + 2 q^{15} - 9 q^{16} - 10 q^{17} - 3 q^{18} + 4 q^{19} + 6 q^{20} - 8 q^{21} + 4 q^{22} + 4 q^{23} + 3 q^{24} + 11 q^{25} + 14 q^{26} - q^{27} + 32 q^{28} - 18 q^{29} + 2 q^{30} + 8 q^{31} + 21 q^{32} + 4 q^{33} + 2 q^{34} + 24 q^{35} + 15 q^{36} - 18 q^{37} + 8 q^{38} - 6 q^{39} - 6 q^{40} + 14 q^{41} - 4 q^{42} - 24 q^{43} - 24 q^{44} - 2 q^{45} - 20 q^{46} + 16 q^{47} - 15 q^{48} + 21 q^{49} - 25 q^{50} - 2 q^{51} - 14 q^{52} - 34 q^{53} - q^{54} + 4 q^{55} + 12 q^{56} - 8 q^{57} - 18 q^{58} - 24 q^{59} - 14 q^{60} - 30 q^{61} + 40 q^{62} - 33 q^{64} - 24 q^{65} + 12 q^{66} + 36 q^{67} - 22 q^{68} - 12 q^{69} - 8 q^{70} + 4 q^{71} - 3 q^{72} - 6 q^{73} - 30 q^{74} + q^{75} + 20 q^{76} - 2 q^{78} - 8 q^{79} + 6 q^{80} + 21 q^{81} + 26 q^{82} + 20 q^{83} - 16 q^{84} + 16 q^{85} + 10 q^{87} - 72 q^{88} + 38 q^{89} + 2 q^{90} - 4 q^{91} + 32 q^{92} - 12 q^{93} - 44 q^{94} + 8 q^{95} - 25 q^{96} - 18 q^{97} - 63 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(393))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 131
393.2.a.a 393.a 1.a $2$ $3.138$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots\)
393.2.a.b 393.a 1.a $4$ $3.138$ 4.4.725.1 None \(-3\) \(4\) \(-8\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
393.2.a.c 393.a 1.a $4$ $3.138$ 4.4.1957.1 None \(-1\) \(-4\) \(0\) \(-8\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{2})q^{2}-q^{3}-\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
393.2.a.d 393.a 1.a $5$ $3.138$ 5.5.535221.1 None \(2\) \(-5\) \(-2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-q^{3}+(2-\beta _{2}+\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots\)
393.2.a.e 393.a 1.a $6$ $3.138$ 6.6.12062776.1 None \(1\) \(6\) \(8\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(393))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(393)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(131))\)\(^{\oplus 2}\)