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Properties

Label 84.9.l
Level $84$
Weight $9$
Character orbit 84.l
Rep. character $\chi_{84}(67,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $128$
Newform subspaces $2$
Sturm bound $144$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.l (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(144\)
Trace bound: \(3\)

Dimensions

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

Total New Old
Modular forms 264 128 136
Cusp forms 248 128 120
Eisenstein series 16 0 16

Trace form

\( 128 q - 6 q^{2} - 186 q^{4} + 9000 q^{8} + 139968 q^{9} + O(q^{10}) \) \( 128 q - 6 q^{2} - 186 q^{4} + 9000 q^{8} + 139968 q^{9} + 15642 q^{10} - 51392 q^{13} + 180054 q^{14} + 37222 q^{16} + 13122 q^{18} + 397128 q^{20} + 243648 q^{21} - 1219388 q^{22} + 259038 q^{24} - 5318688 q^{25} + 185742 q^{26} - 1477430 q^{28} + 4264704 q^{29} - 980424 q^{30} + 2533704 q^{32} - 920160 q^{33} + 3808616 q^{34} - 813564 q^{36} - 2576032 q^{37} - 10724130 q^{38} + 5371418 q^{40} + 6794112 q^{41} + 5288328 q^{42} + 6921744 q^{44} + 8216820 q^{46} - 9467280 q^{48} + 3782528 q^{49} - 13120860 q^{50} + 42647828 q^{52} + 13169472 q^{53} - 50614560 q^{56} + 4650048 q^{57} - 55382046 q^{58} - 530226 q^{60} + 16484096 q^{61} + 48695220 q^{62} + 128073684 q^{64} + 18120000 q^{65} - 28414476 q^{66} - 33033492 q^{68} + 98071914 q^{70} + 9841500 q^{72} + 1467040 q^{73} + 56813202 q^{74} - 114253416 q^{76} - 37992960 q^{77} - 14001660 q^{78} - 20180508 q^{80} - 306110016 q^{81} - 56766604 q^{82} + 23196456 q^{84} + 204922624 q^{85} + 47247198 q^{86} - 30146002 q^{88} - 62568576 q^{89} + 68418108 q^{90} - 600311544 q^{92} + 20106144 q^{93} + 93173388 q^{94} + 85847850 q^{96} + 287972416 q^{97} - 694806852 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.l.a 84.l 28.g $64$ $34.220$ None \(-3\) \(-2592\) \(0\) \(-2256\) $\mathrm{SU}(2)[C_{6}]$
84.9.l.b 84.l 28.g $64$ $34.220$ None \(-3\) \(2592\) \(0\) \(2256\) $\mathrm{SU}(2)[C_{6}]$

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

\( S_{9}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)