Properties

Label 50.4.b
Level $50$
Weight $4$
Character orbit 50.b
Rep. character $\chi_{50}(49,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $2$
Sturm bound $30$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(30\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 28 4 24
Cusp forms 16 4 12
Eisenstein series 12 0 12

Trace form

\( 4 q - 16 q^{4} - 4 q^{6} - 118 q^{9} + O(q^{10}) \) \( 4 q - 16 q^{4} - 4 q^{6} - 118 q^{9} + 78 q^{11} + 152 q^{14} + 64 q^{16} + 130 q^{19} - 412 q^{21} + 16 q^{24} - 344 q^{26} + 420 q^{29} + 668 q^{31} - 348 q^{34} + 472 q^{36} - 536 q^{39} - 162 q^{41} - 312 q^{44} + 216 q^{46} - 972 q^{49} - 762 q^{51} + 460 q^{54} - 608 q^{56} + 840 q^{59} + 1328 q^{61} - 256 q^{64} + 372 q^{66} + 3204 q^{69} - 552 q^{71} - 448 q^{74} - 520 q^{76} - 980 q^{79} - 2396 q^{81} + 1648 q^{84} - 464 q^{86} - 3090 q^{89} + 2368 q^{91} + 1152 q^{94} - 64 q^{96} - 2076 q^{99} + O(q^{100}) \)

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.b.a 50.b 5.b $2$ $2.950$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+4iq^{3}-4q^{4}-2^{4}q^{6}-2iq^{7}+\cdots\)
50.4.b.b 50.b 5.b $2$ $2.950$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-7iq^{3}-4q^{4}+14q^{6}-34iq^{7}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)