Properties

Label 84.3.m
Level $84$
Weight $3$
Character orbit 84.m
Rep. character $\chi_{84}(61,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $6$
Newform subspaces $2$
Sturm bound $48$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(48\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(84, [\chi])\).

Total New Old
Modular forms 76 6 70
Cusp forms 52 6 46
Eisenstein series 24 0 24

Trace form

\( 6 q - 3 q^{3} - 6 q^{5} + 15 q^{7} + 9 q^{9} + O(q^{10}) \) \( 6 q - 3 q^{3} - 6 q^{5} + 15 q^{7} + 9 q^{9} + 18 q^{11} + 24 q^{15} - 48 q^{17} - 63 q^{19} + 12 q^{21} + 12 q^{23} + 39 q^{25} - 96 q^{29} + 27 q^{31} - 36 q^{33} - 78 q^{35} + 51 q^{37} - 15 q^{39} + 90 q^{43} - 18 q^{45} + 66 q^{47} - 123 q^{49} - 12 q^{51} + 48 q^{53} + 198 q^{57} + 372 q^{59} - 72 q^{61} + 9 q^{63} + 114 q^{65} - 75 q^{67} - 204 q^{71} - 81 q^{73} - 249 q^{75} - 252 q^{77} - 69 q^{79} - 27 q^{81} + 384 q^{85} - 162 q^{87} + 324 q^{89} + 159 q^{91} - 69 q^{93} + 42 q^{95} + 108 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.3.m.a 84.m 7.d $2$ $2.289$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(3\) \(13\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(5+3\zeta_{6})q^{7}+\cdots\)
84.3.m.b 84.m 7.d $4$ $2.289$ \(\Q(\sqrt{-3}, \sqrt{65})\) None \(0\) \(-6\) \(-9\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+\beta _{1})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

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