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Properties

Label 84.4.k
Level $84$
Weight $4$
Character orbit 84.k
Rep. character $\chi_{84}(5,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $3$
Sturm bound $64$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(84, [\chi])\).

Total New Old
Modular forms 108 16 92
Cusp forms 84 16 68
Eisenstein series 24 0 24

Trace form

\( 16 q - 22 q^{7} - 30 q^{9} + O(q^{10}) \) \( 16 q - 22 q^{7} - 30 q^{9} + 132 q^{15} + 36 q^{19} - 54 q^{21} - 194 q^{25} + 534 q^{31} - 108 q^{33} + 130 q^{37} - 108 q^{39} - 1040 q^{43} - 342 q^{45} - 932 q^{49} - 300 q^{51} + 1800 q^{57} + 2148 q^{61} + 1056 q^{63} + 1100 q^{67} + 486 q^{73} - 3384 q^{75} + 446 q^{79} - 450 q^{81} - 6144 q^{85} - 2898 q^{87} + 708 q^{91} + 72 q^{93} + 9216 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.4.k.a 84.k 21.g $2$ $4.956$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 84.4.k.a \(0\) \(-9\) \(0\) \(-17\) $\mathrm{U}(1)[D_{6}]$ \(q+(-3-3\zeta_{6})q^{3}+(-18+19\zeta_{6})q^{7}+\cdots\)
84.4.k.b 84.k 21.g $2$ $4.956$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 84.4.k.b \(0\) \(9\) \(0\) \(37\) $\mathrm{U}(1)[D_{6}]$ \(q+(3+3\zeta_{6})q^{3}+(18+\zeta_{6})q^{7}+3^{3}\zeta_{6}q^{9}+\cdots\)
84.4.k.c 84.k 21.g $12$ $4.956$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 84.4.k.c \(0\) \(0\) \(0\) \(-42\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{3}-\beta _{9}q^{5}+(-1+2\beta _{1}-2\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

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