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

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

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7
49.4.a.a 49.a 1.a $1$ $2.891$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-5\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-5q^{2}+17q^{4}-45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
49.4.a.b 49.a 1.a $1$ $2.891$ \(\Q\) None \(-1\) \(2\) \(-16\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-7q^{4}-2^{4}q^{5}-2q^{6}+\cdots\)
49.4.a.c 49.a 1.a $1$ $2.891$ \(\Q\) None \(2\) \(-7\) \(-7\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}-4q^{4}-7q^{5}-14q^{6}+\cdots\)
49.4.a.d 49.a 1.a $1$ $2.891$ \(\Q\) None \(2\) \(7\) \(7\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+7q^{3}-4q^{4}+7q^{5}+14q^{6}+\cdots\)
49.4.a.e 49.a 1.a $4$ $2.891$ \(\Q(\sqrt{2}, \sqrt{65})\) None \(2\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)

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

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