Properties

Label 28.2.e
Level 28
Weight 2
Character orbit e
Rep. character \(\chi_{28}(9,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 2
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.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
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 14 2 12
Cusp forms 2 2 0
Eisenstein series 12 0 12

Trace form

\( 2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} + 3q^{11} + 4q^{13} + 6q^{15} - 3q^{17} + q^{19} - q^{21} - 3q^{23} - 4q^{25} - 10q^{27} - 12q^{29} + 7q^{31} + 3q^{33} + 15q^{35} + q^{37} - 2q^{39} + 12q^{41} - 8q^{43} + 6q^{45} + 9q^{47} + 2q^{49} - 3q^{51} - 3q^{53} - 18q^{55} - 2q^{57} - 9q^{59} + q^{61} - 10q^{63} - 6q^{65} + 7q^{67} + 6q^{69} + q^{73} - 4q^{75} + 3q^{77} + 13q^{79} - q^{81} + 24q^{83} + 18q^{85} + 6q^{87} - 15q^{89} - 8q^{91} + 7q^{93} + 3q^{95} - 20q^{97} + 12q^{99} + O(q^{100}) \)

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.e.a \(2\) \(0.224\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-3\) \(-4\) \(q-\zeta_{6}q^{3}+(-3+3\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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