Properties

Label 224.2.f
Level 224
Weight 2
Character orbit f
Rep. character \(\chi_{224}(223,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 64
Trace bound 0

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Defining parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(64\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(224, [\chi])\).

Total New Old
Modular forms 40 8 32
Cusp forms 24 8 16
Eisenstein series 16 0 16

Trace form

\( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} + 8q^{25} - 16q^{29} - 16q^{37} + 8q^{49} - 16q^{53} - 32q^{57} - 32q^{65} - 16q^{77} - 24q^{81} + 64q^{85} + 64q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.2.f.a \(8\) \(1.789\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{16}^{2}q^{3}-\zeta_{16}^{5}q^{5}-\zeta_{16}^{6}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 4 T^{2} + 14 T^{4} + 36 T^{6} + 81 T^{8} )^{2} \)
$5$ \( ( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2} \)
$7$ \( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 18 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 4 T + 30 T^{2} + 116 T^{3} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 2 T + 53 T^{2} )^{8} \)
$59$ \( ( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2} \)
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