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Properties

Label 10.3.c
Level 10
Weight 3
Character orbit c
Rep. character \(\chi_{10}(3,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 2
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(10, [\chi])\).

Total New Old
Modular forms 10 2 8
Cusp forms 2 2 0
Eisenstein series 8 0 8

Trace form

\( 2q - 2q^{2} - 4q^{3} + 8q^{6} + 4q^{7} + 4q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{3} + 8q^{6} + 4q^{7} + 4q^{8} - 10q^{10} - 16q^{11} - 8q^{12} + 6q^{13} + 20q^{15} - 8q^{16} + 14q^{17} + 2q^{18} + 20q^{20} - 16q^{21} + 16q^{22} - 4q^{23} - 50q^{25} - 12q^{26} - 40q^{27} - 8q^{28} + 104q^{31} + 8q^{32} + 32q^{33} + 20q^{35} - 4q^{36} - 6q^{37} - 40q^{38} - 20q^{40} - 16q^{41} + 16q^{42} - 84q^{43} + 10q^{45} + 8q^{46} - 36q^{47} + 16q^{48} + 50q^{50} - 56q^{51} + 12q^{52} + 106q^{53} + 16q^{56} + 80q^{57} + 80q^{58} - 40q^{60} - 96q^{61} - 104q^{62} - 4q^{63} - 30q^{65} - 64q^{66} + 124q^{67} - 28q^{68} - 40q^{70} - 56q^{71} + 4q^{72} - 94q^{73} + 100q^{75} + 80q^{76} - 32q^{77} + 24q^{78} + 142q^{81} + 16q^{82} + 36q^{83} + 70q^{85} + 168q^{86} - 160q^{87} - 32q^{88} - 10q^{90} + 24q^{91} - 8q^{92} - 208q^{93} - 200q^{95} - 32q^{96} - 126q^{97} - 82q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(10, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
10.3.c.a \(2\) \(0.272\) \(\Q(\sqrt{-1}) \) None \(-2\) \(-4\) \(0\) \(4\) \(q+(-1-i)q^{2}+(-2+2i)q^{3}+2iq^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( 1 + 4 T + 8 T^{2} + 36 T^{3} + 81 T^{4} \)
$5$ \( 1 + 25 T^{2} \)
$7$ \( 1 - 4 T + 8 T^{2} - 196 T^{3} + 2401 T^{4} \)
$11$ \( ( 1 + 8 T + 121 T^{2} )^{2} \)
$13$ \( 1 - 6 T + 18 T^{2} - 1014 T^{3} + 28561 T^{4} \)
$17$ \( ( 1 - 30 T + 289 T^{2} )( 1 + 16 T + 289 T^{2} ) \)
$19$ \( 1 - 322 T^{2} + 130321 T^{4} \)
$23$ \( 1 + 4 T + 8 T^{2} + 2116 T^{3} + 279841 T^{4} \)
$29$ \( ( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} ) \)
$31$ \( ( 1 - 52 T + 961 T^{2} )^{2} \)
$37$ \( 1 + 6 T + 18 T^{2} + 8214 T^{3} + 1874161 T^{4} \)
$41$ \( ( 1 + 8 T + 1681 T^{2} )^{2} \)
$43$ \( 1 + 84 T + 3528 T^{2} + 155316 T^{3} + 3418801 T^{4} \)
$47$ \( 1 + 36 T + 648 T^{2} + 79524 T^{3} + 4879681 T^{4} \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 2809 T^{2} ) \)
$59$ \( 1 - 6562 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 48 T + 3721 T^{2} )^{2} \)
$67$ \( 1 - 124 T + 7688 T^{2} - 556636 T^{3} + 20151121 T^{4} \)
$71$ \( ( 1 + 28 T + 5041 T^{2} )^{2} \)
$73$ \( 1 + 94 T + 4418 T^{2} + 500926 T^{3} + 28398241 T^{4} \)
$79$ \( ( 1 - 79 T )^{2}( 1 + 79 T )^{2} \)
$83$ \( 1 - 36 T + 648 T^{2} - 248004 T^{3} + 47458321 T^{4} \)
$89$ \( 1 - 9442 T^{2} + 62742241 T^{4} \)
$97$ \( 1 + 126 T + 7938 T^{2} + 1185534 T^{3} + 88529281 T^{4} \)
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