Properties

Label 84.2.i
Level 84
Weight 2
Character orbit i
Rep. character \(\chi_{84}(25,\cdot)\)
Character field \(\Q(\zeta_{3})\)
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.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
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 44 2 42
Cusp forms 20 2 18
Eisenstein series 24 0 24

Trace form

\( 2q + q^{3} + 2q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} + q^{7} - q^{9} - 2q^{11} - 6q^{13} + 4q^{15} - 8q^{17} + q^{19} - 4q^{21} - 8q^{23} + q^{25} - 2q^{27} + 8q^{29} - 3q^{31} + 2q^{33} + 10q^{35} + q^{37} - 3q^{39} + 12q^{41} + 22q^{43} + 2q^{45} - 6q^{47} - 13q^{49} + 8q^{51} + 12q^{53} - 8q^{55} + 2q^{57} - 4q^{59} + 6q^{61} - 5q^{63} - 6q^{65} - 13q^{67} - 16q^{69} - 20q^{71} + 11q^{73} - q^{75} + 8q^{77} + 3q^{79} - q^{81} + 4q^{83} - 32q^{85} + 4q^{87} - 3q^{91} + 3q^{93} - 2q^{95} + 20q^{97} + 4q^{99} + O(q^{100}) \)

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.i.a \(2\) \(0.671\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(2\) \(1\) \(q+\zeta_{6}q^{3}+(2-2\zeta_{6})q^{5}+(-1+3\zeta_{6})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 11 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 10 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 2 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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