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Properties

Label 84.2.k
Level $84$
Weight $2$
Character orbit 84.k
Rep. character $\chi_{84}(5,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $6$
Newform subspaces $3$
Sturm bound $32$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.k (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(84, [\chi])\).

Total New Old
Modular forms 44 6 38
Cusp forms 20 6 14
Eisenstein series 24 0 24

Trace form

\( 6 q + 3 q^{7} + O(q^{10}) \) \( 6 q + 3 q^{7} - 18 q^{15} - 9 q^{19} - 9 q^{21} - 3 q^{25} - 9 q^{31} + 27 q^{33} - 3 q^{37} + 9 q^{39} + 42 q^{43} + 27 q^{45} + 15 q^{49} + 9 q^{51} - 36 q^{57} - 54 q^{61} - 15 q^{67} - 45 q^{73} + 27 q^{75} - 15 q^{79} + 36 q^{85} - 9 q^{91} - 54 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.2.k.a 84.k 21.g $2$ $0.671$ \(\Q(\sqrt{-3}) \) None 84.2.k.a \(0\) \(-3\) \(3\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
84.2.k.b 84.k 21.g $2$ $0.671$ \(\Q(\sqrt{-3}) \) None 84.2.k.a \(0\) \(0\) \(-3\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
84.2.k.c 84.k 21.g $2$ $0.671$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 84.2.k.c \(0\) \(3\) \(0\) \(-5\) $\mathrm{U}(1)[D_{6}]$ \(q+(1+\zeta_{6})q^{3}+(-2-\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

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