Properties

Label 300.3.g
Level $300$
Weight $3$
Character orbit 300.g
Rep. character $\chi_{300}(101,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $8$
Sturm bound $180$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 138 13 125
Cusp forms 102 13 89
Eisenstein series 36 0 36

Trace form

\( 13 q - q^{3} - 6 q^{7} - 13 q^{9} + O(q^{10}) \) \( 13 q - q^{3} - 6 q^{7} - 13 q^{9} + 6 q^{13} + 48 q^{19} + 32 q^{21} + 71 q^{27} + 52 q^{31} - 60 q^{33} - 138 q^{37} - 132 q^{39} + 6 q^{43} + 225 q^{49} - 10 q^{51} + 214 q^{57} - 28 q^{61} - 14 q^{63} - 186 q^{67} - 280 q^{69} + 258 q^{73} + 114 q^{79} + 23 q^{81} + 180 q^{87} - 162 q^{91} - 194 q^{93} - 246 q^{97} - 290 q^{99} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
300.3.g.a 300.g 3.b $1$ $8.174$ \(\Q\) \(\Q(\sqrt{-3}) \) 300.3.g.a \(0\) \(-3\) \(0\) \(-13\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}-13q^{7}+9q^{9}+23q^{13}+11q^{19}+\cdots\)
300.3.g.b 300.g 3.b $1$ $8.174$ \(\Q\) \(\Q(\sqrt{-3}) \) 12.3.c.a \(0\) \(3\) \(0\) \(-2\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}-2q^{7}+9q^{9}+22q^{13}+26q^{19}+\cdots\)
300.3.g.c 300.g 3.b $1$ $8.174$ \(\Q\) \(\Q(\sqrt{-3}) \) 300.3.g.a \(0\) \(3\) \(0\) \(13\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}+13q^{7}+9q^{9}-23q^{13}+11q^{19}+\cdots\)
300.3.g.d 300.g 3.b $2$ $8.174$ \(\Q(\sqrt{-5}) \) None 60.3.g.a \(0\) \(-4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2+\beta )q^{3}-2q^{7}+(-1-4\beta )q^{9}+\cdots\)
300.3.g.e 300.g 3.b $2$ $8.174$ \(\Q(\sqrt{-5}) \) None 60.3.b.a \(0\) \(-4\) \(0\) \(16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2+\beta )q^{3}+8q^{7}+(-1-4\beta )q^{9}+\cdots\)
300.3.g.f 300.g 3.b $2$ $8.174$ \(\Q(\sqrt{-35}) \) None 300.3.g.f \(0\) \(-1\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta q^{3}-8q^{7}+(-9+\beta )q^{9}+(-3+\cdots)q^{11}+\cdots\)
300.3.g.g 300.g 3.b $2$ $8.174$ \(\Q(\sqrt{-35}) \) None 300.3.g.f \(0\) \(1\) \(0\) \(16\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}+8q^{7}+(-9+\beta )q^{9}+(-3+\cdots)q^{11}+\cdots\)
300.3.g.h 300.g 3.b $2$ $8.174$ \(\Q(\sqrt{-5}) \) None 60.3.b.a \(0\) \(4\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2+\beta )q^{3}-8q^{7}+(-1+4\beta )q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(300, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)