Properties

Label 403.2.f
Level 403
Weight 2
Character orbit f
Rep. character \(\chi_{403}(94,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 72
Newform subspaces 3
Sturm bound 74
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(74\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(403, [\chi])\).

Total New Old
Modular forms 80 72 8
Cusp forms 72 72 0
Eisenstein series 8 0 8

Trace form

\( 72q - 2q^{2} - 38q^{4} - 8q^{5} + 12q^{8} - 36q^{9} + O(q^{10}) \) \( 72q - 2q^{2} - 38q^{4} - 8q^{5} + 12q^{8} - 36q^{9} - 6q^{10} - 4q^{11} + 16q^{12} + 8q^{14} - 8q^{15} - 50q^{16} + 8q^{17} + 4q^{18} - 4q^{19} + 10q^{20} - 8q^{21} + 4q^{22} + 8q^{23} - 8q^{24} + 48q^{25} + 6q^{26} - 12q^{27} + 22q^{28} + 18q^{29} + 16q^{30} - 34q^{32} - 10q^{33} - 44q^{34} - 2q^{35} - 38q^{36} + 16q^{37} + 44q^{38} - 18q^{39} - 4q^{40} - 8q^{41} + 24q^{42} - 12q^{43} + 8q^{44} + 22q^{45} - 10q^{46} + 6q^{48} - 40q^{49} - 44q^{50} + 24q^{51} + 28q^{52} - 56q^{53} + 6q^{54} - 10q^{55} - 22q^{56} - 60q^{57} + 54q^{58} - 6q^{59} + 12q^{60} + 16q^{61} - 8q^{62} + 2q^{63} + 132q^{64} - 22q^{65} + 104q^{66} - 16q^{67} + 24q^{68} - 28q^{69} - 8q^{70} + 34q^{71} - 62q^{72} + 36q^{73} + 74q^{74} - 36q^{75} - 36q^{76} + 20q^{77} + 62q^{78} + 44q^{80} - 36q^{81} - 20q^{82} - 16q^{83} + 4q^{84} + 4q^{85} - 72q^{86} - 12q^{87} - 32q^{88} - 6q^{89} + 16q^{90} - 86q^{91} - 64q^{92} + 8q^{94} - 24q^{95} + 104q^{96} - 48q^{97} - 18q^{98} + 120q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(403, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
403.2.f.a \(2\) \(3.218\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-6\) \(-2\) \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}-3q^{5}-2\zeta_{6}q^{7}+\cdots\)
403.2.f.b \(34\) \(3.218\) None \(4\) \(0\) \(-14\) \(6\)
403.2.f.c \(36\) \(3.218\) None \(-5\) \(0\) \(12\) \(-4\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))
$3$ (\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2} \))
$7$ (\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 5 T + 13 T^{2} \))
$17$ (\( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))
$31$ (\( ( 1 + T )^{2} \))
$37$ (\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 2 T - 39 T^{2} - 86 T^{3} + 1849 T^{4} \))
$47$ (\( ( 1 + 6 T + 47 T^{2} )^{2} \))
$53$ (\( ( 1 + 9 T + 53 T^{2} )^{2} \))
$59$ (\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))
$67$ (\( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 4 T - 55 T^{2} - 284 T^{3} + 5041 T^{4} \))
$73$ (\( ( 1 + 3 T + 73 T^{2} )^{2} \))
$79$ (\( ( 1 - 2 T + 79 T^{2} )^{2} \))
$83$ (\( ( 1 + 2 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} \))
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