Properties

Label 84.2.b
Level 84
Weight 2
Character orbit b
Rep. character \(\chi_{84}(55,\cdot)\)
Character field \(\Q\)
Dimension 8
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.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(19\)

Dimensions

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

Total New Old
Modular forms 20 8 12
Cusp forms 12 8 4
Eisenstein series 8 0 8

Trace form

\( 8q - 2q^{2} + 2q^{4} - 14q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 2q^{2} + 2q^{4} - 14q^{8} + 8q^{9} - 2q^{14} - 14q^{16} - 2q^{18} - 4q^{21} + 12q^{22} - 16q^{25} + 18q^{28} - 16q^{29} + 16q^{30} + 18q^{32} + 2q^{36} - 24q^{37} - 16q^{42} + 28q^{44} - 12q^{46} + 16q^{49} + 38q^{50} + 32q^{53} - 22q^{56} - 24q^{57} + 4q^{58} - 8q^{60} + 2q^{64} + 16q^{65} - 8q^{70} - 14q^{72} - 28q^{74} - 16q^{77} - 24q^{78} + 8q^{81} + 16q^{84} + 40q^{85} - 60q^{86} - 4q^{88} - 28q^{92} + 30q^{98} + O(q^{100}) \)

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.b.a \(4\) \(0.671\) 4.0.2312.1 None \(-1\) \(-4\) \(0\) \(2\) \(q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
84.2.b.b \(4\) \(0.671\) 4.0.2312.1 None \(-1\) \(4\) \(0\) \(-2\) \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))
$3$ (\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))
$5$ (\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))
$7$ (\( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \))(\( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \))(\( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \))
$17$ (\( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \))(\( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 2 T + 29 T^{2} )^{4} \))(\( ( 1 + 2 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 31 T^{2} )^{4} \))
$37$ (\( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \))(\( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \))(\( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \))
$47$ (\( ( 1 + 4 T + 30 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 - 4 T + 59 T^{2} )^{4} \))(\( ( 1 + 4 T + 59 T^{2} )^{4} \))
$61$ (\( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \))(\( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \))
$67$ (\( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \))(\( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \))
$71$ (\( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \))(\( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \))(\( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \))
$83$ (\( ( 1 + 4 T + 102 T^{2} + 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \))(\( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \))(\( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \))
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