Properties

Label 224.2.x
Level 224
Weight 2
Character orbit x
Rep. character \(\chi_{224}(27,\cdot)\)
Character field \(\Q(\zeta_{8})\)
Dimension 120
Newform subspaces 2
Sturm bound 64
Trace bound 1

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Defining parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.x (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 224 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 2 \)
Sturm bound: \(64\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 136 136 0
Cusp forms 120 120 0
Eisenstein series 16 16 0

Trace form

\( 120q - 8q^{2} - 8q^{4} - 4q^{7} - 8q^{8} - 8q^{9} + O(q^{10}) \) \( 120q - 8q^{2} - 8q^{4} - 4q^{7} - 8q^{8} - 8q^{9} - 8q^{11} - 20q^{14} - 16q^{15} + 12q^{16} - 28q^{18} - 4q^{21} - 4q^{22} - 16q^{23} - 8q^{25} - 24q^{28} - 8q^{29} + 16q^{30} - 8q^{32} + 20q^{35} + 32q^{36} - 8q^{37} - 8q^{39} + 16q^{42} - 16q^{43} - 92q^{44} - 8q^{46} + 16q^{50} - 32q^{51} + 8q^{53} + 36q^{56} - 8q^{57} - 72q^{58} - 88q^{60} + 112q^{64} - 16q^{65} - 48q^{67} - 40q^{70} + 56q^{71} - 8q^{72} - 124q^{74} - 4q^{77} + 192q^{78} - 16q^{79} - 24q^{84} - 48q^{85} - 8q^{86} - 96q^{88} - 52q^{91} + 68q^{92} - 32q^{93} + 88q^{98} + 16q^{99} + O(q^{100}) \)

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.x.a \(8\) \(1.789\) 8.0.157351936.1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{2}+\beta _{4})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(2\beta _{2}+\cdots)q^{7}+\cdots\)
224.2.x.b \(112\) \(1.789\) None \(-8\) \(0\) \(0\) \(-4\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{4} + 16 T^{8} \))
$3$ (\( ( 1 + 81 T^{8} )^{2} \))
$5$ (\( ( 1 + 625 T^{8} )^{2} \))
$7$ (\( ( 1 + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 6 T^{2} + 121 T^{4} )^{2}( 1 - 206 T^{4} + 14641 T^{8} ) \))
$13$ (\( ( 1 + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 17 T^{2} )^{8} \))
$19$ (\( ( 1 + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 8 T + 23 T^{2} )^{4}( 1 + 18 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 54 T^{2} + 841 T^{4} )^{2}( 1 + 1234 T^{4} + 707281 T^{8} ) \))
$31$ (\( ( 1 + 31 T^{2} )^{8} \))
$37$ (\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2}( 1 - 1294 T^{4} + 1874161 T^{8} ) \))
$41$ (\( ( 1 + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 + 12 T + 43 T^{2} )^{4}( 1 - 334 T^{4} + 3418801 T^{8} ) \))
$47$ (\( ( 1 - 47 T^{2} )^{8} \))
$53$ (\( ( 1 - 10 T + 53 T^{2} )^{4}( 1 - 5582 T^{4} + 7890481 T^{8} ) \))
$59$ (\( ( 1 + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4946 T^{4} + 20151121 T^{8} ) \))
$71$ (\( ( 1 + 2914 T^{4} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 3646 T^{4} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 7921 T^{4} )^{4} \))
$97$ (\( ( 1 - 97 T^{2} )^{8} \))
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