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Properties

Label 84.4.i
Level $84$
Weight $4$
Character orbit 84.i
Rep. character $\chi_{84}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $2$
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.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 108 8 100
Cusp forms 84 8 76
Eisenstein series 24 0 24

Trace form

\( 8 q - 8 q^{5} - 14 q^{7} - 36 q^{9} + O(q^{10}) \) \( 8 q - 8 q^{5} - 14 q^{7} - 36 q^{9} + 56 q^{11} + 132 q^{13} + 84 q^{15} - 124 q^{17} - 236 q^{19} - 12 q^{21} + 116 q^{23} + 82 q^{25} + 472 q^{29} - 270 q^{31} - 138 q^{33} - 760 q^{35} + 178 q^{37} + 168 q^{39} + 24 q^{41} + 448 q^{43} - 72 q^{45} + 36 q^{47} + 44 q^{49} - 228 q^{51} - 780 q^{53} - 740 q^{55} - 612 q^{57} + 1096 q^{59} + 204 q^{61} + 252 q^{63} + 864 q^{65} + 764 q^{67} + 1608 q^{69} + 392 q^{71} + 146 q^{73} + 312 q^{75} + 244 q^{77} - 1278 q^{79} - 324 q^{81} - 3616 q^{83} - 3848 q^{85} - 942 q^{87} + 1152 q^{89} + 3900 q^{91} - 1362 q^{93} - 148 q^{95} + 2804 q^{97} - 1008 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.4.i.a 84.i 7.c $4$ $4.956$ \(\Q(\sqrt{-3}, \sqrt{193})\) None \(0\) \(-6\) \(-11\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\beta _{2})q^{3}+(\beta _{1}-6\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
84.4.i.b 84.i 7.c $4$ $4.956$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(6\) \(3\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{2}q^{3}+(1+2\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)