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Properties

Label 300.9.g
Level $300$
Weight $9$
Character orbit 300.g
Rep. character $\chi_{300}(101,\cdot)$
Character field $\Q$
Dimension $51$
Newform subspaces $8$
Sturm bound $540$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(540\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(300, [\chi])\).

Total New Old
Modular forms 498 51 447
Cusp forms 462 51 411
Eisenstein series 36 0 36

Trace form

\( 51 q - 91 q^{3} - 1994 q^{7} - 1939 q^{9} + O(q^{10}) \) \( 51 q - 91 q^{3} - 1994 q^{7} - 1939 q^{9} + 54466 q^{13} - 323276 q^{19} - 274396 q^{21} + 93869 q^{27} + 1220744 q^{31} - 1440060 q^{33} - 882062 q^{37} + 3798264 q^{39} + 6461026 q^{43} + 44800155 q^{49} + 13072850 q^{51} - 1576694 q^{57} - 40919376 q^{61} + 15229606 q^{63} - 12796814 q^{67} + 13142600 q^{69} + 13092238 q^{73} + 121145662 q^{79} - 28903279 q^{81} + 40636620 q^{87} + 157439246 q^{91} + 62977666 q^{93} - 31762634 q^{97} + 191259250 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(300, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
300.9.g.a 300.g 3.b $1$ $122.214$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-81\) \(0\) \(-4034\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{4}q^{3}-4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots\)
300.9.g.b 300.g 3.b $1$ $122.214$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-81\) \(0\) \(-239\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{4}q^{3}-239q^{7}+3^{8}q^{9}+20641q^{13}+\cdots\)
300.9.g.c 300.g 3.b $1$ $122.214$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(81\) \(0\) \(239\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{4}q^{3}+239q^{7}+3^{8}q^{9}-20641q^{13}+\cdots\)
300.9.g.d 300.g 3.b $2$ $122.214$ \(\Q(\sqrt{-110}) \) None \(0\) \(102\) \(0\) \(6188\) $\mathrm{SU}(2)[C_{2}]$ \(q+(51+\beta )q^{3}+3094q^{7}+(-1359+\cdots)q^{9}+\cdots\)
300.9.g.e 300.g 3.b $10$ $122.214$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-137\) \(0\) \(-1048\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-14-\beta _{1})q^{3}+(-104+2\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
300.9.g.f 300.g 3.b $10$ $122.214$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-112\) \(0\) \(-4148\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-11+\beta _{1})q^{3}+(-417-11\beta _{1}+\cdots)q^{7}+\cdots\)
300.9.g.g 300.g 3.b $10$ $122.214$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(137\) \(0\) \(1048\) $\mathrm{SU}(2)[C_{2}]$ \(q+(14+\beta _{1})q^{3}+(104-2\beta _{1}+\beta _{3})q^{7}+\cdots\)
300.9.g.h 300.g 3.b $16$ $122.214$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{7}q^{7}+(922+\beta _{2})q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(300, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)