Properties

Label 50.3.f
Level $50$
Weight $3$
Character orbit 50.f
Rep. character $\chi_{50}(3,\cdot)$
Character field $\Q(\zeta_{20})$
Dimension $40$
Newform subspaces $2$
Sturm bound $22$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 50.f (of order \(20\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{20})\)
Newform subspaces: \( 2 \)
Sturm bound: \(22\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(50, [\chi])\).

Total New Old
Modular forms 136 40 96
Cusp forms 104 40 64
Eisenstein series 32 0 32

Trace form

\( 40 q + 2 q^{2} + 4 q^{3} - 4 q^{7} - 4 q^{8} + O(q^{10}) \) \( 40 q + 2 q^{2} + 4 q^{3} - 4 q^{7} - 4 q^{8} + 10 q^{10} + 8 q^{12} - 6 q^{13} - 20 q^{15} + 40 q^{16} - 154 q^{17} - 152 q^{18} - 200 q^{19} - 40 q^{20} - 96 q^{22} - 36 q^{23} + 110 q^{25} - 20 q^{26} + 220 q^{27} + 88 q^{28} + 200 q^{29} + 240 q^{30} + 32 q^{32} + 328 q^{33} + 250 q^{34} + 200 q^{35} - 60 q^{36} - 54 q^{37} + 40 q^{38} - 400 q^{39} + 20 q^{40} + 80 q^{41} - 16 q^{42} - 156 q^{43} - 330 q^{45} - 44 q^{47} - 16 q^{48} - 50 q^{50} - 12 q^{52} + 74 q^{53} + 240 q^{55} + 560 q^{57} - 80 q^{58} + 100 q^{59} - 240 q^{60} - 120 q^{61} - 616 q^{62} - 536 q^{63} - 790 q^{65} - 684 q^{67} - 332 q^{68} - 700 q^{69} - 280 q^{70} + 120 q^{71} - 4 q^{72} + 14 q^{73} + 60 q^{75} + 512 q^{77} + 376 q^{78} + 400 q^{79} - 190 q^{81} + 624 q^{82} + 904 q^{83} + 600 q^{84} + 460 q^{85} + 980 q^{87} + 32 q^{88} + 1450 q^{89} + 970 q^{90} + 368 q^{92} + 348 q^{93} + 40 q^{95} + 86 q^{97} + 82 q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.3.f.a 50.f 25.f $16$ $1.362$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-4\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{20}]$ \(q+(\beta _{1}+\beta _{9})q^{2}+(-1+\beta _{2}-\beta _{3}+\beta _{7}+\cdots)q^{3}+\cdots\)
50.3.f.b 50.f 25.f $24$ $1.362$ None \(6\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{20}]$

Decomposition of \(S_{3}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)