Properties

Label 84.2.a
Level 84
Weight 2
Character orbit 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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 - T \))
$5$ (\( 1 - 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))
$71$ (\( 1 - 14 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))
$79$ (\( 1 - 12 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))
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