Properties

Label 49.4.a
Level $49$
Weight $4$
Character orbit 49.a
Rep. character $\chi_{49}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $5$
Sturm bound $18$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(49))\).

Total New Old
Modular forms 18 13 5
Cusp forms 10 8 2
Eisenstein series 8 5 3

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

\(7\)Dim.
\(+\)\(5\)
\(-\)\(3\)

Trace form

\( 8q + 2q^{3} + 36q^{4} - 16q^{5} - 2q^{6} - 12q^{8} + 34q^{9} + O(q^{10}) \) \( 8q + 2q^{3} + 36q^{4} - 16q^{5} - 2q^{6} - 12q^{8} + 34q^{9} + 16q^{10} + 14q^{11} - 14q^{12} - 28q^{13} + 2q^{15} - 76q^{16} - 54q^{17} - 124q^{18} + 110q^{19} + 112q^{20} - 12q^{22} + 42q^{23} + 30q^{24} - 274q^{25} + 28q^{26} - 100q^{27} + 100q^{29} - 324q^{30} - 12q^{31} + 44q^{32} - 16q^{33} + 54q^{34} - 524q^{36} + 854q^{37} - 110q^{38} + 700q^{39} - 240q^{40} - 182q^{41} + 240q^{43} - 600q^{44} + 368q^{45} + 212q^{46} - 324q^{47} + 82q^{48} + 1556q^{50} + 26q^{51} + 196q^{52} + 1050q^{53} + 100q^{54} + 128q^{55} - 978q^{57} + 392q^{58} - 810q^{59} - 1232q^{60} + 488q^{61} + 12q^{62} - 1236q^{64} - 504q^{65} + 16q^{66} - 1610q^{67} + 378q^{68} + 96q^{69} + 1304q^{71} + 84q^{72} + 702q^{73} - 1412q^{74} + 262q^{75} - 770q^{76} - 1680q^{78} - 2198q^{79} - 656q^{80} - 1784q^{81} + 182q^{82} + 1302q^{83} - 2714q^{85} + 5096q^{86} - 220q^{87} + 4440q^{88} - 730q^{89} - 368q^{90} + 3768q^{92} - 3334q^{93} + 324q^{94} + 1118q^{95} - 322q^{96} - 294q^{97} + 5140q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7
49.4.a.a \(1\) \(2.891\) \(\Q\) \(\Q(\sqrt{-7}) \) \(-5\) \(0\) \(0\) \(0\) \(-\) \(q-5q^{2}+17q^{4}-45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
49.4.a.b \(1\) \(2.891\) \(\Q\) None \(-1\) \(2\) \(-16\) \(0\) \(-\) \(q-q^{2}+2q^{3}-7q^{4}-2^{4}q^{5}-2q^{6}+\cdots\)
49.4.a.c \(1\) \(2.891\) \(\Q\) None \(2\) \(-7\) \(-7\) \(0\) \(-\) \(q+2q^{2}-7q^{3}-4q^{4}-7q^{5}-14q^{6}+\cdots\)
49.4.a.d \(1\) \(2.891\) \(\Q\) None \(2\) \(7\) \(7\) \(0\) \(+\) \(q+2q^{2}+7q^{3}-4q^{4}+7q^{5}+14q^{6}+\cdots\)
49.4.a.e \(4\) \(2.891\) \(\Q(\sqrt{2}, \sqrt{65})\) None \(2\) \(0\) \(0\) \(0\) \(+\) \(q+(1+\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

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