Properties

Label 448.2.f
Level 448
Weight 2
Character orbit f
Rep. character \(\chi_{448}(447,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 4
Sturm bound 128
Trace bound 3

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Defining parameters

Level: \( N \) = \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 448.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(128\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(448, [\chi])\).

Total New Old
Modular forms 76 18 58
Cusp forms 52 14 38
Eisenstein series 24 4 20

Trace form

\( 14q + 6q^{9} + O(q^{10}) \) \( 14q + 6q^{9} + 16q^{21} - 10q^{25} - 4q^{29} + 12q^{37} - 2q^{49} + 12q^{53} - 16q^{57} + 16q^{65} + 12q^{77} - 50q^{81} - 64q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(448, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
448.2.f.a \(2\) \(3.577\) \(\Q(\sqrt{-3}) \) None \(0\) \(-4\) \(0\) \(-4\) \(q-2q^{3}-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+q^{9}+\cdots\)
448.2.f.b \(2\) \(3.577\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{7}-3q^{9}-2\beta q^{11}-2\beta q^{23}+\cdots\)
448.2.f.c \(2\) \(3.577\) \(\Q(\sqrt{-3}) \) None \(0\) \(4\) \(0\) \(4\) \(q+2q^{3}+2\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}+q^{9}+\cdots\)
448.2.f.d \(8\) \(3.577\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{16}^{2}q^{3}-\zeta_{16}^{5}q^{5}-\zeta_{16}^{4}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(448, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))
$3$ (\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 14 T^{4} + 36 T^{6} + 81 T^{8} )^{2} \))
$5$ (\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))
$11$ (\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 18 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 34 T^{2} + 529 T^{4} \))(\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( ( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 116 T^{3} + 841 T^{4} )^{4} \))
$31$ (\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{8} \))
$59$ (\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 110 T^{2} + 3721 T^{4} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( ( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 - 146 T^{2} + 6241 T^{4} \))(\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( ( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2} \))
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