Properties

Label 84.9.m
Level $84$
Weight $9$
Character orbit 84.m
Rep. character $\chi_{84}(61,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $22$
Newform subspaces $2$
Sturm bound $144$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.m (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(144\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(84, [\chi])\).

Total New Old
Modular forms 268 22 246
Cusp forms 244 22 222
Eisenstein series 24 0 24

Trace form

\( 22 q + 81 q^{3} + 1674 q^{5} + 1415 q^{7} + 24057 q^{9} + O(q^{10}) \) \( 22 q + 81 q^{3} + 1674 q^{5} + 1415 q^{7} + 24057 q^{9} - 18798 q^{11} - 59616 q^{15} - 219456 q^{17} + 45045 q^{19} - 81972 q^{21} + 249900 q^{23} + 856627 q^{25} - 865248 q^{29} + 1685739 q^{31} - 1380240 q^{33} - 2780142 q^{35} - 145313 q^{37} - 2163267 q^{39} + 6859474 q^{43} + 3661038 q^{45} + 11782746 q^{47} - 7143035 q^{49} - 5186916 q^{51} - 16169976 q^{53} + 5593374 q^{57} + 25922772 q^{59} - 51153384 q^{61} - 2871531 q^{63} - 9919686 q^{65} - 57547375 q^{67} + 83021028 q^{71} + 138516507 q^{73} + 72959211 q^{75} - 137157132 q^{77} - 33675181 q^{79} - 52612659 q^{81} + 162723648 q^{85} - 11569230 q^{87} + 150210468 q^{89} - 444629901 q^{91} + 32404455 q^{93} - 362255118 q^{95} - 82222452 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.9.m.a 84.m 7.d $10$ $34.220$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-405\) \(1389\) \(1217\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-54+3^{3}\beta _{1})q^{3}+(92+93\beta _{1}+\beta _{5}+\cdots)q^{5}+\cdots\)
84.9.m.b 84.m 7.d $12$ $34.220$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(486\) \(285\) \(198\) $\mathrm{SU}(2)[C_{6}]$ \(q+(54-3^{3}\beta _{1})q^{3}+(2^{4}+2^{4}\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)