Properties

Label 402.2.a
Level 402
Weight 2
Character orbit a
Rep. character \(\chi_{402}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 7
Sturm bound 136
Trace bound 5

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Defining parameters

Level: \( N \) = \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 402.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(136\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(402))\).

Total New Old
Modular forms 72 11 61
Cusp forms 65 11 54
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.

\(2\)\(3\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(9\)

Trace form

\( 11q + q^{2} - q^{3} + 11q^{4} + 6q^{5} + q^{6} + 4q^{7} + q^{8} + 11q^{9} + O(q^{10}) \) \( 11q + q^{2} - q^{3} + 11q^{4} + 6q^{5} + q^{6} + 4q^{7} + q^{8} + 11q^{9} + 6q^{10} + 4q^{11} - q^{12} - 10q^{13} - 2q^{15} + 11q^{16} - 2q^{17} + q^{18} - 4q^{19} + 6q^{20} - 4q^{21} + 4q^{22} + 12q^{23} + q^{24} + 25q^{25} + 14q^{26} - q^{27} + 4q^{28} + 2q^{29} + 2q^{30} - 12q^{31} + q^{32} + 4q^{33} + 2q^{34} - 8q^{35} + 11q^{36} - 2q^{37} - 4q^{38} - 14q^{39} + 6q^{40} - 6q^{41} + 4q^{42} - 4q^{43} + 4q^{44} + 6q^{45} - 20q^{47} - q^{48} - 9q^{49} - q^{50} - 10q^{51} - 10q^{52} + 22q^{53} + q^{54} - 24q^{55} - 12q^{57} + 10q^{58} - 24q^{59} - 2q^{60} - 26q^{61} - 24q^{62} + 4q^{63} + 11q^{64} - 12q^{65} - 12q^{66} + q^{67} - 2q^{68} - 16q^{69} - 32q^{70} - 28q^{71} + q^{72} + 34q^{73} + 14q^{74} - 15q^{75} - 4q^{76} - 16q^{77} - 14q^{78} - 20q^{79} + 6q^{80} + 11q^{81} - 6q^{82} + 24q^{83} - 4q^{84} - 12q^{85} - 12q^{86} + 6q^{87} + 4q^{88} - 18q^{89} + 6q^{90} - 24q^{91} + 12q^{92} - 4q^{93} - 32q^{94} - 56q^{95} + q^{96} - 10q^{97} - 7q^{98} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(402))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 67
402.2.a.a \(1\) \(3.210\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-3\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
402.2.a.b \(1\) \(3.210\) \(\Q\) None \(-1\) \(1\) \(-3\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-q^{7}+\cdots\)
402.2.a.c \(1\) \(3.210\) \(\Q\) None \(-1\) \(1\) \(2\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{8}+\cdots\)
402.2.a.d \(1\) \(3.210\) \(\Q\) None \(1\) \(-1\) \(2\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+2q^{7}+\cdots\)
402.2.a.e \(2\) \(3.210\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(6\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2\beta q^{5}+q^{6}+(3+\cdots)q^{7}+\cdots\)
402.2.a.f \(2\) \(3.210\) \(\Q(\sqrt{41}) \) None \(2\) \(-2\) \(1\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+\beta q^{5}-q^{6}-\beta q^{7}+\cdots\)
402.2.a.g \(3\) \(3.210\) 3.3.316.1 None \(3\) \(3\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{2})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(402))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(402)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 2}\)