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Properties

Label 75.4.g
Level $75$
Weight $4$
Character orbit 75.g
Rep. character $\chi_{75}(16,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $56$
Newform subspaces $2$
Sturm bound $40$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.g (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(75, [\chi])\).

Total New Old
Modular forms 128 56 72
Cusp forms 112 56 56
Eisenstein series 16 0 16

Trace form

\( 56 q - 4 q^{2} - 48 q^{4} + 12 q^{6} + 4 q^{7} - 174 q^{8} - 126 q^{9} + O(q^{10}) \) \( 56 q - 4 q^{2} - 48 q^{4} + 12 q^{6} + 4 q^{7} - 174 q^{8} - 126 q^{9} + 10 q^{10} + 84 q^{11} + 24 q^{12} - 96 q^{13} - 132 q^{14} - 30 q^{15} - 20 q^{16} + 280 q^{17} + 144 q^{18} + 188 q^{19} + 560 q^{20} + 84 q^{21} - 990 q^{22} - 432 q^{23} - 576 q^{24} + 190 q^{25} + 92 q^{26} - 548 q^{28} - 344 q^{29} - 780 q^{30} + 786 q^{31} + 2124 q^{32} + 252 q^{33} + 644 q^{34} + 1080 q^{35} - 432 q^{36} + 144 q^{37} + 2882 q^{38} + 312 q^{39} + 580 q^{40} - 1344 q^{41} + 1272 q^{42} - 464 q^{43} - 1694 q^{44} + 1436 q^{46} - 1952 q^{47} - 336 q^{48} + 660 q^{49} - 4580 q^{50} - 1632 q^{51} - 3940 q^{52} + 1728 q^{53} + 108 q^{54} - 2520 q^{55} + 2700 q^{56} - 312 q^{57} - 1806 q^{58} + 1392 q^{59} + 2610 q^{60} - 1420 q^{61} + 1558 q^{62} + 216 q^{63} + 4698 q^{64} - 3360 q^{65} - 744 q^{66} + 4540 q^{67} - 152 q^{68} - 72 q^{69} - 1800 q^{70} + 1432 q^{71} - 1566 q^{72} - 3252 q^{73} + 468 q^{74} - 8832 q^{76} + 608 q^{77} + 600 q^{78} - 320 q^{79} - 560 q^{80} - 1134 q^{81} + 5728 q^{82} + 3092 q^{83} + 4392 q^{84} + 6220 q^{85} - 3996 q^{86} + 1356 q^{87} + 10846 q^{88} + 3768 q^{89} - 180 q^{90} - 1484 q^{91} + 1574 q^{92} - 5376 q^{93} - 2158 q^{94} - 6060 q^{95} + 6084 q^{96} - 9730 q^{97} - 20544 q^{98} - 504 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.4.g.a 75.g 25.d $28$ $4.425$ None \(-4\) \(21\) \(15\) \(58\) $\mathrm{SU}(2)[C_{5}]$
75.4.g.b 75.g 25.d $28$ $4.425$ None \(0\) \(-21\) \(-15\) \(-54\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{4}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)