Properties

Label 84.2.e
Level 84
Weight 2
Character orbit e
Rep. character \(\chi_{84}(71,\cdot)\)
Character field \(\Q\)
Dimension 12
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.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 12 \)
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 20 12 8
Cusp forms 12 12 0
Eisenstein series 8 0 8

Trace form

\( 12q - 4q^{4} - 6q^{6} - 4q^{9} + O(q^{10}) \) \( 12q - 4q^{4} - 6q^{6} - 4q^{9} + 4q^{10} - 6q^{12} + 4q^{16} - 8q^{18} - 16q^{22} + 2q^{24} - 12q^{25} + 8q^{28} + 20q^{30} - 16q^{33} + 32q^{34} - 20q^{36} - 16q^{37} + 20q^{40} + 10q^{42} + 24q^{45} + 46q^{48} - 12q^{49} - 28q^{52} + 10q^{54} + 16q^{57} - 32q^{58} + 28q^{60} - 16q^{61} + 20q^{64} - 12q^{66} - 24q^{69} - 12q^{70} - 32q^{72} + 24q^{73} - 60q^{76} + 20q^{78} + 28q^{81} + 8q^{82} - 14q^{84} + 40q^{85} - 56q^{88} - 80q^{90} + 24q^{93} - 34q^{96} - 24q^{97} + O(q^{100}) \)

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.e.a \(12\) \(0.671\) 12.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{2}-\beta _{9}q^{3}+(-\beta _{2}-\beta _{3})q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + T^{4} - 4 T^{6} + 4 T^{8} + 32 T^{10} + 64 T^{12} \)
$3$ \( 1 + 2 T^{2} - 5 T^{4} - 44 T^{6} - 45 T^{8} + 162 T^{10} + 729 T^{12} \)
$5$ \( ( 1 - 12 T^{2} + 83 T^{4} - 472 T^{6} + 2075 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2} \)
$7$ \( ( 1 + T^{2} )^{6} \)
$11$ \( ( 1 + 32 T^{2} + 607 T^{4} + 8240 T^{6} + 73447 T^{8} + 468512 T^{10} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 + 29 T^{2} - 4 T^{3} + 377 T^{4} + 2197 T^{6} )^{4} \)
$17$ \( ( 1 - 68 T^{2} + 2311 T^{4} - 49064 T^{6} + 667879 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 50 T^{2} + 1675 T^{4} - 38132 T^{6} + 604675 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 + 112 T^{2} + 5671 T^{4} + 166576 T^{6} + 2999959 T^{8} + 31342192 T^{10} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 - 30 T^{2} + 1047 T^{4} - 12836 T^{6} + 880527 T^{8} - 21218430 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 - 118 T^{2} + 7327 T^{4} - 280660 T^{6} + 7041247 T^{8} - 108975478 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 + 4 T + 83 T^{2} + 312 T^{3} + 3071 T^{4} + 5476 T^{5} + 50653 T^{6} )^{4} \)
$41$ \( ( 1 - 180 T^{2} + 14903 T^{4} - 753928 T^{6} + 25051943 T^{8} - 508636980 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 74 T^{2} + 4039 T^{4} - 189452 T^{6} + 7468111 T^{8} - 252991274 T^{10} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 + 178 T^{2} + 15631 T^{4} + 882364 T^{6} + 34528879 T^{8} + 868583218 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 - 262 T^{2} + 31111 T^{4} - 2121556 T^{6} + 87390799 T^{8} - 2067306022 T^{10} + 22164361129 T^{12} )^{2} \)
$59$ \( ( 1 + 74 T^{2} + 3563 T^{4} + 111204 T^{6} + 12402803 T^{8} + 896684714 T^{10} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 + 4 T + 37 T^{2} + 436 T^{3} + 2257 T^{4} + 14884 T^{5} + 226981 T^{6} )^{4} \)
$67$ \( ( 1 - 330 T^{2} + 48695 T^{4} - 4163660 T^{6} + 218591855 T^{8} - 6649869930 T^{10} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 + 152 T^{2} + 19111 T^{4} + 1530752 T^{6} + 96338551 T^{8} + 3862575512 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 - 6 T + 191 T^{2} - 772 T^{3} + 13943 T^{4} - 31974 T^{5} + 389017 T^{6} )^{4} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{6} \)
$83$ \( ( 1 + 370 T^{2} + 64699 T^{4} + 6784180 T^{6} + 445711411 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 - 212 T^{2} + 30551 T^{4} - 3163080 T^{6} + 241994471 T^{8} - 13301355092 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{12} \)
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