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Properties

Label 84.4.a
Level $84$
Weight $4$
Character orbit 84.a
Rep. character $\chi_{84}(1,\cdot)$
Character field $\Q$
Dimension $2$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(84))\).

Total New Old
Modular forms 54 2 52
Cusp forms 42 2 40
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(0\)

Trace form

\( 2 q + 20 q^{5} + 18 q^{9} + O(q^{10}) \) \( 2 q + 20 q^{5} + 18 q^{9} + 40 q^{11} + 116 q^{13} + 24 q^{15} + 100 q^{17} + 16 q^{19} - 42 q^{21} - 176 q^{23} - 18 q^{25} - 52 q^{29} - 256 q^{31} - 96 q^{33} - 56 q^{35} - 20 q^{37} - 24 q^{39} - 12 q^{41} - 680 q^{43} + 180 q^{45} - 720 q^{47} + 98 q^{49} - 384 q^{51} + 204 q^{53} + 272 q^{55} + 504 q^{57} + 32 q^{59} + 932 q^{61} + 1128 q^{65} - 168 q^{67} - 384 q^{69} + 832 q^{71} + 468 q^{73} + 480 q^{75} + 224 q^{77} + 464 q^{79} + 162 q^{81} - 656 q^{83} + 488 q^{85} - 480 q^{87} - 444 q^{89} + 56 q^{91} - 96 q^{93} + 832 q^{95} - 492 q^{97} + 360 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(84))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
84.4.a.a 84.a 1.a $1$ $4.956$ \(\Q\) None 84.4.a.a \(0\) \(-3\) \(6\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+6q^{5}+7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
84.4.a.b 84.a 1.a $1$ $4.956$ \(\Q\) None 84.4.a.b \(0\) \(3\) \(14\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+14q^{5}-7q^{7}+9q^{9}+4q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(84)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)