Properties

Label 84.12.f
Level $84$
Weight $12$
Character orbit 84.f
Rep. character $\chi_{84}(41,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $2$
Sturm bound $192$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(192\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(84, [\chi])\).

Total New Old
Modular forms 182 30 152
Cusp forms 170 30 140
Eisenstein series 12 0 12

Trace form

\( 30 q + 9900 q^{7} - 86634 q^{9} + O(q^{10}) \) \( 30 q + 9900 q^{7} - 86634 q^{9} - 3434160 q^{15} + 18627078 q^{21} + 300220650 q^{25} - 493621044 q^{37} + 25354392 q^{39} + 3705967392 q^{43} - 2809339074 q^{49} + 1424787216 q^{51} - 14769423420 q^{57} + 16343140860 q^{63} - 17959454640 q^{67} + 64478554440 q^{79} + 145468933326 q^{81} + 197440859760 q^{85} - 91391893896 q^{91} - 77050550988 q^{93} - 81383696064 q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.12.f.a 84.f 21.c $2$ $64.541$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(268\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{5}\zeta_{6}q^{3}+(134-25673\zeta_{6})q^{7}+\cdots\)
84.12.f.b 84.f 21.c $28$ $64.541$ None \(0\) \(0\) \(0\) \(9632\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{12}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{12}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)