Properties

Label 399.2.a
Level $399$
Weight $2$
Character orbit 399.a
Rep. character $\chi_{399}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $7$
Sturm bound $106$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(106\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(399))\).

Total New Old
Modular forms 56 19 37
Cusp forms 49 19 30
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.

\(3\)\(7\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(2\)
Minus space\(-\)\(17\)

Trace form

\( 19 q + 5 q^{2} - q^{3} + 21 q^{4} + 10 q^{5} - 3 q^{6} + 3 q^{7} + 9 q^{8} + 19 q^{9} + O(q^{10}) \) \( 19 q + 5 q^{2} - q^{3} + 21 q^{4} + 10 q^{5} - 3 q^{6} + 3 q^{7} + 9 q^{8} + 19 q^{9} + 6 q^{10} + 4 q^{11} + 9 q^{12} + 10 q^{13} + q^{14} - 6 q^{15} + 29 q^{16} + 22 q^{17} + 5 q^{18} - q^{19} - 2 q^{20} - q^{21} - 12 q^{22} - 8 q^{23} - 15 q^{24} + 29 q^{25} - 18 q^{26} - q^{27} + 5 q^{28} + 18 q^{29} + 6 q^{30} - 7 q^{32} - 4 q^{33} - 14 q^{34} - 6 q^{35} + 21 q^{36} + 2 q^{37} - 3 q^{38} + 18 q^{39} - 18 q^{40} + 22 q^{41} + q^{42} + 36 q^{43} - 4 q^{44} + 10 q^{45} - 16 q^{46} - 8 q^{47} + q^{48} + 19 q^{49} - 37 q^{50} - 2 q^{51} - 10 q^{52} - 6 q^{53} - 3 q^{54} - 24 q^{55} + 21 q^{56} + 3 q^{57} - 50 q^{58} + 4 q^{59} - 42 q^{60} + 10 q^{61} - 56 q^{62} + 3 q^{63} + 29 q^{64} + 28 q^{65} + 4 q^{66} - 20 q^{67} + 2 q^{68} - 16 q^{69} - 18 q^{70} - 40 q^{71} + 9 q^{72} + 30 q^{73} - 18 q^{74} + 17 q^{75} - 7 q^{76} + 4 q^{77} - 34 q^{78} + 16 q^{79} - 98 q^{80} + 19 q^{81} - 70 q^{82} - 52 q^{83} - 7 q^{84} - 44 q^{85} - 68 q^{86} - 14 q^{87} - 108 q^{88} + 38 q^{89} + 6 q^{90} - 6 q^{91} - 88 q^{92} - 48 q^{94} - 6 q^{95} - 63 q^{96} - 26 q^{97} + 5 q^{98} + 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(399))\) 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 7 19
399.2.a.a 399.a 1.a $1$ $3.186$ \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+q^{7}+3q^{8}+\cdots\)
399.2.a.b 399.a 1.a $1$ $3.186$ \(\Q\) None \(-1\) \(1\) \(4\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
399.2.a.c 399.a 1.a $1$ $3.186$ \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-q^{7}-3q^{8}+\cdots\)
399.2.a.d 399.a 1.a $3$ $3.186$ 3.3.148.1 None \(1\) \(-3\) \(4\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.e 399.a 1.a $3$ $3.186$ 3.3.404.1 None \(1\) \(3\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.f 399.a 1.a $5$ $3.186$ 5.5.1240016.1 None \(1\) \(5\) \(-2\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+q^{3}+(1+\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots\)
399.2.a.g 399.a 1.a $5$ $3.186$ 5.5.368464.1 None \(3\) \(-5\) \(4\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(399))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(399)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 2}\)