Properties

Label 13.4.b
Level 13
Weight 4
Character orbit b
Rep. character \(\chi_{13}(12,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

Total New Old
Modular forms 4 4 0
Cusp forms 2 2 0
Eisenstein series 2 2 0

Trace form

\( 2q - 2q^{3} - 2q^{4} - 52q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} - 52q^{9} + 54q^{10} + 2q^{12} + 52q^{13} + 90q^{14} - 142q^{16} - 90q^{17} - 288q^{22} + 324q^{23} + 88q^{25} + 234q^{26} + 106q^{27} - 288q^{29} - 54q^{30} - 270q^{35} + 52q^{36} - 36q^{38} - 52q^{39} + 378q^{40} - 90q^{42} - 194q^{43} + 142q^{48} + 236q^{49} + 90q^{51} - 52q^{52} - 828q^{53} + 864q^{55} + 630q^{56} + 752q^{61} - 1584q^{62} - 866q^{64} - 702q^{65} + 288q^{66} + 90q^{68} - 324q^{69} + 1818q^{74} - 88q^{75} + 1440q^{77} - 234q^{78} - 1660q^{79} + 1298q^{81} + 1152q^{82} + 288q^{87} - 2016q^{88} - 1404q^{90} - 1170q^{91} - 324q^{92} + 666q^{94} + 108q^{95} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.4.b.a \(2\) \(0.767\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) \(q+iq^{2}-q^{3}-q^{4}-3iq^{5}-iq^{6}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 7 T^{2} + 64 T^{4} \)
$3$ \( ( 1 + T + 27 T^{2} )^{2} \)
$5$ \( 1 - 169 T^{2} + 15625 T^{4} \)
$7$ \( 1 - 461 T^{2} + 117649 T^{4} \)
$11$ \( 1 - 358 T^{2} + 1771561 T^{4} \)
$13$ \( 1 - 52 T + 2197 T^{2} \)
$17$ \( ( 1 + 45 T + 4913 T^{2} )^{2} \)
$19$ \( 1 - 13682 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 - 162 T + 12167 T^{2} )^{2} \)
$29$ \( ( 1 + 144 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 10114 T^{2} + 887503681 T^{4} \)
$37$ \( 1 - 9497 T^{2} + 2565726409 T^{4} \)
$41$ \( 1 - 100978 T^{2} + 4750104241 T^{4} \)
$43$ \( ( 1 + 97 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 195325 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 + 414 T + 148877 T^{2} )^{2} \)
$59$ \( 1 - 138274 T^{2} + 42180533641 T^{4} \)
$61$ \( ( 1 - 376 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 600230 T^{2} + 90458382169 T^{4} \)
$71$ \( 1 - 588373 T^{2} + 128100283921 T^{4} \)
$73$ \( ( 1 - 592 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} ) \)
$79$ \( ( 1 + 830 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 951730 T^{2} + 326940373369 T^{4} \)
$89$ \( 1 - 1218094 T^{2} + 496981290961 T^{4} \)
$97$ \( 1 - 1099442 T^{2} + 832972004929 T^{4} \)
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