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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(399))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7 19
399.2.a.a \(1\) \(3.186\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}-q^{4}+q^{6}+q^{7}+3q^{8}+\cdots\)
399.2.a.b \(1\) \(3.186\) \(\Q\) None \(-1\) \(1\) \(4\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}-q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
399.2.a.c \(1\) \(3.186\) \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{6}-q^{7}-3q^{8}+\cdots\)
399.2.a.d \(3\) \(3.186\) 3.3.148.1 None \(1\) \(-3\) \(4\) \(-3\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.e \(3\) \(3.186\) 3.3.404.1 None \(1\) \(3\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(q+\beta _{2}q^{2}+q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.f \(5\) \(3.186\) 5.5.1240016.1 None \(1\) \(5\) \(-2\) \(5\) \(-\) \(-\) \(-\) \(q+\beta _{3}q^{2}+q^{3}+(1+\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots\)
399.2.a.g \(5\) \(3.186\) 5.5.368464.1 None \(3\) \(-5\) \(4\) \(5\) \(+\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

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

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