Properties

Label 300.3.f
Level $300$
Weight $3$
Character orbit 300.f
Rep. character $\chi_{300}(199,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $3$
Sturm bound $180$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(180\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(300, [\chi])\).

Total New Old
Modular forms 132 36 96
Cusp forms 108 36 72
Eisenstein series 24 0 24

Trace form

\( 36 q + 4 q^{4} - 12 q^{6} + 108 q^{9} + O(q^{10}) \) \( 36 q + 4 q^{4} - 12 q^{6} + 108 q^{9} + 44 q^{14} + 116 q^{16} + 36 q^{24} - 212 q^{26} + 40 q^{29} - 24 q^{34} + 12 q^{36} + 168 q^{41} - 328 q^{44} - 40 q^{46} + 76 q^{49} - 36 q^{54} - 112 q^{56} + 24 q^{61} - 68 q^{64} + 48 q^{66} + 192 q^{69} + 56 q^{74} - 96 q^{76} + 324 q^{81} + 456 q^{84} - 308 q^{86} - 744 q^{89} - 312 q^{94} - 348 q^{96} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(300, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
300.3.f.a 300.f 20.d $4$ $8.174$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{3}q^{3}+(2+2\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
300.3.f.b 300.f 20.d $16$ $8.174$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{12}q^{2}-\beta _{9}q^{3}+(-1+\beta _{4})q^{4}+\cdots\)
300.3.f.c 300.f 20.d $16$ $8.174$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{4}q^{3}+(1+\beta _{5})q^{4}+(-1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(300, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)