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Properties

Label 50.6.d
Level $50$
Weight $6$
Character orbit 50.d
Rep. character $\chi_{50}(11,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $52$
Newform subspaces $2$
Sturm bound $45$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(45\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(50, [\chi])\).

Total New Old
Modular forms 156 52 104
Cusp forms 140 52 88
Eisenstein series 16 0 16

Trace form

\( 52 q + 4 q^{2} - 4 q^{3} - 208 q^{4} - 265 q^{5} - 72 q^{6} + 312 q^{7} + 64 q^{8} - 1339 q^{9} + O(q^{10}) \) \( 52 q + 4 q^{2} - 4 q^{3} - 208 q^{4} - 265 q^{5} - 72 q^{6} + 312 q^{7} + 64 q^{8} - 1339 q^{9} + 100 q^{10} - 716 q^{11} + 96 q^{12} - 114 q^{13} - 784 q^{14} + 120 q^{15} - 3328 q^{16} + 992 q^{17} - 4208 q^{18} - 5990 q^{19} + 1280 q^{20} + 4344 q^{21} + 1408 q^{22} + 2886 q^{23} + 4608 q^{24} + 1675 q^{25} + 11448 q^{26} - 17260 q^{27} + 1952 q^{28} - 14220 q^{29} - 15280 q^{30} + 18834 q^{31} - 4096 q^{32} + 17882 q^{33} - 6604 q^{34} + 32410 q^{35} - 21424 q^{36} + 43967 q^{37} + 17360 q^{38} + 13432 q^{39} + 1600 q^{40} - 16196 q^{41} + 32168 q^{42} - 73124 q^{43} + 1184 q^{44} - 73405 q^{45} - 28672 q^{46} + 39822 q^{47} + 1536 q^{48} + 185444 q^{49} + 2500 q^{50} + 85984 q^{51} - 1824 q^{52} + 45501 q^{53} + 16080 q^{54} - 1450 q^{55} - 12544 q^{56} - 254880 q^{57} - 37320 q^{58} + 18000 q^{59} + 33920 q^{60} - 48136 q^{61} + 48128 q^{62} - 302704 q^{63} - 53248 q^{64} - 142045 q^{65} + 52256 q^{66} - 156718 q^{67} - 100128 q^{68} + 60692 q^{69} - 132720 q^{70} - 5326 q^{71} + 62272 q^{72} + 297346 q^{73} + 277656 q^{74} + 219650 q^{75} + 102400 q^{76} + 223384 q^{77} - 23696 q^{78} + 113600 q^{79} + 6400 q^{80} - 162103 q^{81} - 381272 q^{82} + 36186 q^{83} - 174816 q^{84} - 942275 q^{85} + 26888 q^{86} - 491360 q^{87} + 3968 q^{88} + 4645 q^{89} + 526940 q^{90} + 216324 q^{91} - 21024 q^{92} + 643092 q^{93} + 79136 q^{94} + 244540 q^{95} - 18432 q^{96} + 513422 q^{97} + 203588 q^{98} + 695052 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.6.d.a 50.d 25.d $24$ $8.019$ None \(-24\) \(-11\) \(-120\) \(548\) $\mathrm{SU}(2)[C_{5}]$
50.6.d.b 50.d 25.d $28$ $8.019$ None \(28\) \(7\) \(-145\) \(-236\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{6}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)