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Properties

Label 6.13.b
Level $6$
Weight $13$
Character orbit 6.b
Rep. character $\chi_{6}(5,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $1$
Sturm bound $13$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(13\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(6, [\chi])\).

Total New Old
Modular forms 14 4 10
Cusp forms 10 4 6
Eisenstein series 4 0 4

Trace form

\( 4 q + 780 q^{3} - 8192 q^{4} - 9984 q^{6} + 153080 q^{7} - 1530972 q^{9} + O(q^{10}) \) \( 4 q + 780 q^{3} - 8192 q^{4} - 9984 q^{6} + 153080 q^{7} - 1530972 q^{9} + 1641984 q^{10} - 1597440 q^{12} + 7253000 q^{13} - 17613792 q^{15} + 16777216 q^{16} + 42600960 q^{18} - 120268072 q^{19} + 163232328 q^{21} - 159244800 q^{22} + 20447232 q^{24} + 435605764 q^{25} - 784941300 q^{27} - 313507840 q^{28} + 571258368 q^{30} + 2731727672 q^{31} - 2567489760 q^{33} - 3097810944 q^{34} + 3135430656 q^{36} - 15280120 q^{37} - 2508657000 q^{39} - 3362783232 q^{40} + 20958988800 q^{42} + 1629119960 q^{43} - 15576677568 q^{45} - 29905849344 q^{46} + 3271557120 q^{48} + 72937649100 q^{49} - 63012636288 q^{51} - 14854144000 q^{52} + 38602586880 q^{54} + 6285799872 q^{55} - 424311000 q^{57} - 62351992320 q^{58} + 36073046016 q^{60} + 45477065096 q^{61} + 45447449400 q^{63} - 34359738368 q^{64} - 673085952 q^{66} - 213433609960 q^{67} + 95560926912 q^{69} + 293322353664 q^{70} - 87246766080 q^{72} - 254383625080 q^{73} - 15705158772 q^{75} + 246309011456 q^{76} - 645782208000 q^{78} + 308580159032 q^{79} + 219015659268 q^{81} + 234603709440 q^{82} - 334299807744 q^{84} - 18844054272 q^{85} + 1341091294560 q^{87} + 326133350400 q^{88} - 432622121472 q^{90} - 3323734346000 q^{91} + 871044956040 q^{93} + 775668172800 q^{94} - 41875931136 q^{96} + 1276228475720 q^{97} - 23456841408 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{13}^{\mathrm{new}}(6, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
6.13.b.a 6.b 3.b $4$ $5.484$ \(\Q(\sqrt{-2}, \sqrt{1009})\) None \(0\) \(780\) \(0\) \(153080\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(195-\beta _{1}-\beta _{3})q^{3}-2^{11}q^{4}+\cdots\)

Decomposition of \(S_{13}^{\mathrm{old}}(6, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(6, [\chi]) \cong \) \(S_{13}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 2}\)