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Properties

Label 50.4.d
Level $50$
Weight $4$
Character orbit 50.d
Rep. character $\chi_{50}(11,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $28$
Newform subspaces $2$
Sturm bound $30$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(30\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(50, [\chi])\).

Total New Old
Modular forms 100 28 72
Cusp forms 84 28 56
Eisenstein series 16 0 16

Trace form

\( 28 q - 2 q^{2} + 8 q^{3} - 28 q^{4} + 35 q^{5} + 12 q^{6} + 4 q^{7} - 8 q^{8} - 109 q^{9} + O(q^{10}) \) \( 28 q - 2 q^{2} + 8 q^{3} - 28 q^{4} + 35 q^{5} + 12 q^{6} + 4 q^{7} - 8 q^{8} - 109 q^{9} - 10 q^{10} + 106 q^{11} - 48 q^{12} + 58 q^{13} + 56 q^{14} - 180 q^{15} - 112 q^{16} + 254 q^{17} + 376 q^{18} + 110 q^{19} + 80 q^{20} + 36 q^{21} - 384 q^{22} - 642 q^{23} - 192 q^{24} - 265 q^{25} - 588 q^{26} + 200 q^{27} - 104 q^{28} + 120 q^{29} + 200 q^{30} + 486 q^{31} + 128 q^{32} - 544 q^{33} + 646 q^{34} + 880 q^{35} - 436 q^{36} - 111 q^{37} + 200 q^{38} + 1012 q^{39} - 40 q^{40} - 134 q^{41} + 956 q^{42} - 312 q^{43} - 136 q^{44} - 775 q^{45} + 832 q^{46} - 2556 q^{47} - 192 q^{48} + 864 q^{49} - 50 q^{50} - 884 q^{51} + 232 q^{52} + 663 q^{53} - 120 q^{54} - 1820 q^{55} + 224 q^{56} - 240 q^{57} + 180 q^{58} + 2250 q^{59} + 2360 q^{60} - 1764 q^{61} + 416 q^{62} + 6608 q^{63} - 448 q^{64} + 1415 q^{65} - 16 q^{66} + 3364 q^{67} - 2184 q^{68} - 4708 q^{69} - 2360 q^{70} + 5096 q^{71} - 296 q^{72} - 5522 q^{73} - 3804 q^{74} - 7540 q^{75} - 800 q^{76} - 6512 q^{77} - 488 q^{78} - 2800 q^{79} - 80 q^{80} + 5333 q^{81} - 564 q^{82} - 942 q^{83} + 624 q^{84} + 8025 q^{85} + 692 q^{86} + 370 q^{87} + 1584 q^{88} + 7345 q^{89} - 550 q^{90} - 3764 q^{91} + 2832 q^{92} + 4296 q^{93} + 656 q^{94} + 2800 q^{95} + 192 q^{96} - 4676 q^{97} - 2626 q^{98} - 2988 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.4.d.a 50.d 25.d $12$ $2.950$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(6\) \(1\) \(20\) \(58\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{5}q^{2}+(-\beta _{2}+\beta _{3}-\beta _{4}+\beta _{5}+\cdots)q^{3}+\cdots\)
50.4.d.b 50.d 25.d $16$ $2.950$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-8\) \(7\) \(15\) \(-54\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{6}q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)