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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
84.4.a.a \(1\) \(4.956\) \(\Q\) None \(0\) \(-3\) \(6\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}+6q^{5}+7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
84.4.a.b \(1\) \(4.956\) \(\Q\) None \(0\) \(3\) \(14\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}+14q^{5}-7q^{7}+9q^{9}+4q^{11}+\cdots\)

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

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