Properties

Label 84.2.f
Level 84
Weight 2
Character orbit f
Rep. character \(\chi_{84}(41,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 1
Sturm bound 32
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(32\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(84, [\chi])\).

Total New Old
Modular forms 22 2 20
Cusp forms 10 2 8
Eisenstein series 12 0 12

Trace form

\( 2q + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 4q^{7} - 6q^{9} - 6q^{21} - 10q^{25} + 20q^{37} + 24q^{39} - 16q^{43} + 2q^{49} + 12q^{57} - 12q^{63} - 32q^{67} - 8q^{79} + 18q^{81} + 24q^{91} - 36q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
84.2.f.a \(2\) \(0.671\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{6}q^{3}+(2-\zeta_{6})q^{7}-3q^{9}+4\zeta_{6}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 - 10 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( ( 1 + 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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