Properties

Label 28.2.f
Level 28
Weight 2
Character orbit f
Rep. character \(\chi_{28}(3,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 4
Newform subspaces 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 28.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(28, [\chi])\).

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

Trace form

\( 4q - 2q^{2} - 6q^{5} - 8q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 6q^{5} - 8q^{8} + 6q^{10} + 12q^{12} + 2q^{14} + 8q^{16} - 6q^{17} + 6q^{21} - 4q^{22} - 12q^{24} - 4q^{25} - 12q^{26} - 20q^{28} + 16q^{29} - 6q^{30} + 8q^{32} + 6q^{33} - 6q^{37} + 18q^{38} + 12q^{40} + 12q^{42} + 4q^{44} - 2q^{46} - 4q^{49} + 8q^{50} + 2q^{53} - 18q^{54} + 16q^{56} - 36q^{57} - 8q^{58} - 12q^{60} - 18q^{61} + 12q^{65} - 6q^{66} + 6q^{70} + 30q^{73} - 6q^{74} - 10q^{77} + 24q^{78} - 24q^{80} + 18q^{81} + 12q^{82} + 12q^{85} - 4q^{86} + 4q^{88} + 54q^{89} + 8q^{92} + 6q^{93} - 30q^{94} - 24q^{96} - 22q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(28, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
28.2.f.a \(4\) \(0.224\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-6\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \)
$5$ \( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \)
$23$ \( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 46 T^{2} + 961 T^{4} )( 1 - 13 T^{2} + 961 T^{4} ) \)
$37$ \( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 54 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 27 T + 332 T^{2} - 2403 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 106 T^{2} + 9409 T^{4} )^{2} \)
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