Properties

Label 50.3.c
Level $50$
Weight $3$
Character orbit 50.c
Rep. character $\chi_{50}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $6$
Newform subspaces $3$
Sturm bound $22$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 50.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(22\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 42 6 36
Cusp forms 18 6 12
Eisenstein series 24 0 24

Trace form

\( 6 q + 2 q^{2} + 4 q^{3} - 16 q^{6} - 4 q^{7} - 4 q^{8} + O(q^{10}) \) \( 6 q + 2 q^{2} + 4 q^{3} - 16 q^{6} - 4 q^{7} - 4 q^{8} + 32 q^{11} + 8 q^{12} - 6 q^{13} - 24 q^{16} - 14 q^{17} - 2 q^{18} - 88 q^{21} - 16 q^{22} + 4 q^{23} + 84 q^{26} + 40 q^{27} + 8 q^{28} + 72 q^{31} - 8 q^{32} - 32 q^{33} + 68 q^{36} + 6 q^{37} + 40 q^{38} - 208 q^{41} - 16 q^{42} + 84 q^{43} - 16 q^{46} + 36 q^{47} - 16 q^{48} + 232 q^{51} - 12 q^{52} - 106 q^{53} - 32 q^{56} - 80 q^{57} - 80 q^{58} + 32 q^{61} + 104 q^{62} + 4 q^{63} - 352 q^{66} - 124 q^{67} + 28 q^{68} - 248 q^{71} - 4 q^{72} + 94 q^{73} - 80 q^{76} + 32 q^{77} - 24 q^{78} + 466 q^{81} - 16 q^{82} - 36 q^{83} + 384 q^{86} + 160 q^{87} + 32 q^{88} + 312 q^{91} + 8 q^{92} + 208 q^{93} + 64 q^{96} + 126 q^{97} + 82 q^{98} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.3.c.a 50.c 5.c $2$ $1.362$ \(\Q(\sqrt{-1}) \) None 50.3.c.a \(-2\) \(6\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+(3-3i)q^{3}+2iq^{4}+\cdots\)
50.3.c.b 50.c 5.c $2$ $1.362$ \(\Q(\sqrt{-1}) \) None 50.3.c.a \(2\) \(-6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+(-3+3i)q^{3}+2iq^{4}+\cdots\)
50.3.c.c 50.c 5.c $2$ $1.362$ \(\Q(\sqrt{-1}) \) None 10.3.c.a \(2\) \(4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+(2-2i)q^{3}+2iq^{4}+4q^{6}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)