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Properties

Label 98.4.a
Level $98$
Weight $4$
Character orbit 98.a
Rep. character $\chi_{98}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $8$
Sturm bound $56$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(98))\).

Total New Old
Modular forms 50 10 40
Cusp forms 34 10 24
Eisenstein series 16 0 16

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

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

Trace form

\( 10q - 6q^{3} + 40q^{4} + 26q^{5} + 20q^{6} + 126q^{9} + O(q^{10}) \) \( 10q - 6q^{3} + 40q^{4} + 26q^{5} + 20q^{6} + 126q^{9} - 4q^{10} - 12q^{11} - 24q^{12} - 74q^{13} + 92q^{15} + 160q^{16} + 40q^{17} + 128q^{18} - 82q^{19} + 104q^{20} - 80q^{22} + 268q^{23} + 80q^{24} + 310q^{25} - 76q^{26} - 180q^{27} - 328q^{29} - 432q^{30} - 308q^{31} + 320q^{33} + 376q^{34} + 504q^{36} - 916q^{37} + 156q^{38} - 1352q^{39} - 16q^{40} - 288q^{41} + 288q^{43} - 48q^{44} + 242q^{45} - 80q^{46} - 12q^{47} - 96q^{48} - 1192q^{50} - 884q^{51} - 296q^{52} - 1152q^{53} - 40q^{54} + 184q^{55} + 156q^{57} + 24q^{58} + 62q^{59} + 368q^{60} - 182q^{61} - 328q^{62} + 640q^{64} + 476q^{65} - 256q^{66} + 3044q^{67} + 160q^{68} + 656q^{69} - 704q^{71} + 512q^{72} + 612q^{73} - 232q^{74} - 530q^{75} - 328q^{76} + 128q^{78} + 2028q^{79} + 416q^{80} + 1066q^{81} + 72q^{82} + 694q^{83} + 1240q^{85} + 416q^{86} - 1628q^{87} - 320q^{88} - 420q^{89} - 1588q^{90} + 1072q^{92} + 4916q^{93} + 72q^{94} + 916q^{95} + 320q^{96} - 24q^{97} - 1936q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
98.4.a.a \(1\) \(5.782\) \(\Q\) None \(-2\) \(-8\) \(14\) \(0\) \(+\) \(-\) \(q-2q^{2}-8q^{3}+4q^{4}+14q^{5}+2^{4}q^{6}+\cdots\)
98.4.a.b \(1\) \(5.782\) \(\Q\) None \(-2\) \(-1\) \(7\) \(0\) \(+\) \(+\) \(q-2q^{2}-q^{3}+4q^{4}+7q^{5}+2q^{6}+\cdots\)
98.4.a.c \(1\) \(5.782\) \(\Q\) None \(-2\) \(1\) \(-7\) \(0\) \(+\) \(-\) \(q-2q^{2}+q^{3}+4q^{4}-7q^{5}-2q^{6}+\cdots\)
98.4.a.d \(1\) \(5.782\) \(\Q\) None \(2\) \(-5\) \(-9\) \(0\) \(-\) \(+\) \(q+2q^{2}-5q^{3}+4q^{4}-9q^{5}-10q^{6}+\cdots\)
98.4.a.e \(1\) \(5.782\) \(\Q\) None \(2\) \(2\) \(12\) \(0\) \(-\) \(-\) \(q+2q^{2}+2q^{3}+4q^{4}+12q^{5}+4q^{6}+\cdots\)
98.4.a.f \(1\) \(5.782\) \(\Q\) None \(2\) \(5\) \(9\) \(0\) \(-\) \(-\) \(q+2q^{2}+5q^{3}+4q^{4}+9q^{5}+10q^{6}+\cdots\)
98.4.a.g \(2\) \(5.782\) \(\Q(\sqrt{2}) \) None \(-4\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-2q^{2}+5\beta q^{3}+4q^{4}+14\beta q^{5}-10\beta q^{6}+\cdots\)
98.4.a.h \(2\) \(5.782\) \(\Q(\sqrt{22}) \) None \(4\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}+2\beta q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(98)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)