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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

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

Display 100 coefficients 10 coefficients

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

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

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