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Properties

Label 12.3.d
Level 12
Weight 3
Character orbit d
Rep. character \(\chi_{12}(7,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 12.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(12, [\chi])\).

Total New Old
Modular forms 6 2 4
Cusp forms 2 2 0
Eisenstein series 4 0 4

Trace form

\( 2q - 2q^{2} - 4q^{4} - 4q^{5} + 6q^{6} + 16q^{8} - 6q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} - 4q^{5} + 6q^{6} + 16q^{8} - 6q^{9} + 4q^{10} - 12q^{12} + 4q^{13} - 24q^{14} - 16q^{16} + 20q^{17} + 6q^{18} + 8q^{20} + 24q^{21} + 24q^{22} - 42q^{25} - 4q^{26} + 48q^{28} - 52q^{29} - 12q^{30} - 32q^{32} - 24q^{33} - 20q^{34} + 12q^{36} + 52q^{37} + 72q^{38} - 32q^{40} + 116q^{41} - 24q^{42} - 48q^{44} + 12q^{45} - 96q^{46} + 48q^{48} + 2q^{49} + 42q^{50} - 8q^{52} - 148q^{53} - 18q^{54} - 72q^{57} + 52q^{58} + 24q^{60} + 52q^{61} + 24q^{62} + 128q^{64} - 8q^{65} + 24q^{66} - 40q^{68} + 96q^{69} + 48q^{70} - 48q^{72} - 92q^{73} - 52q^{74} - 144q^{76} + 96q^{77} + 12q^{78} + 32q^{80} + 18q^{81} - 116q^{82} - 48q^{84} - 40q^{85} - 168q^{86} + 164q^{89} - 12q^{90} + 192q^{92} - 24q^{93} + 240q^{94} - 96q^{96} + 4q^{97} - 2q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(12, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
12.3.d.a \(2\) \(0.327\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-4\) \(0\) \(q+(-1-\zeta_{6})q^{2}+\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( ( 1 + 2 T + 25 T^{2} )^{2} \)
$7$ \( 1 - 50 T^{2} + 2401 T^{4} \)
$11$ \( 1 - 194 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 - 2 T + 169 T^{2} )^{2} \)
$17$ \( ( 1 - 10 T + 289 T^{2} )^{2} \)
$19$ \( 1 - 290 T^{2} + 130321 T^{4} \)
$23$ \( 1 - 290 T^{2} + 279841 T^{4} \)
$29$ \( ( 1 + 26 T + 841 T^{2} )^{2} \)
$31$ \( 1 - 1874 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 - 26 T + 1369 T^{2} )^{2} \)
$41$ \( ( 1 - 58 T + 1681 T^{2} )^{2} \)
$43$ \( 1 - 1346 T^{2} + 3418801 T^{4} \)
$47$ \( 1 + 382 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 + 74 T + 2809 T^{2} )^{2} \)
$59$ \( 1 + 1150 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 - 26 T + 3721 T^{2} )^{2} \)
$67$ \( 1 - 8930 T^{2} + 20151121 T^{4} \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 + 46 T + 5329 T^{2} )^{2} \)
$79$ \( 1 + 1390 T^{2} + 38950081 T^{4} \)
$83$ \( 1 - 11426 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 82 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 - 2 T + 9409 T^{2} )^{2} \)
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