Properties

Label 84.2.a
Level $84$
Weight $2$
Character orbit 84.a
Rep. character $\chi_{84}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(84))\).

Total New Old
Modular forms 22 2 20
Cusp forms 11 2 9
Eisenstein series 11 0 11

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\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2q + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} + 2q^{9} - 4q^{11} - 4q^{13} - 4q^{15} - 4q^{17} - 8q^{19} + 2q^{21} - 4q^{23} + 6q^{25} + 4q^{29} + 8q^{31} - 8q^{33} - 4q^{35} + 4q^{37} + 8q^{39} + 12q^{41} - 8q^{43} + 4q^{45} + 24q^{47} + 2q^{49} + 4q^{51} - 12q^{53} + 8q^{55} - 8q^{59} - 4q^{61} - 24q^{65} - 8q^{69} + 20q^{71} - 12q^{73} - 16q^{75} - 8q^{77} + 8q^{79} + 2q^{81} - 16q^{83} - 16q^{85} + 8q^{87} + 12q^{89} + 8q^{91} + 8q^{93} - 16q^{95} - 12q^{97} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(84))\) 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
84.2.a.a \(1\) \(0.671\) \(\Q\) None \(0\) \(-1\) \(4\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{3}+4q^{5}-q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
84.2.a.b \(1\) \(0.671\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}-6q^{11}+2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

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