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Properties

Label 1300.1.bc
Level $1300$
Weight $1$
Character orbit 1300.bc
Rep. character $\chi_{1300}(451,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $2$
Newform subspaces $1$
Sturm bound $210$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1300.bc (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 52 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(210\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1300, [\chi])\).

Total New Old
Modular forms 32 14 18
Cusp forms 8 2 6
Eisenstein series 24 12 12

The following table gives the dimensions of subspaces with specified projective image type.

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 2 0 0 0

Trace form

\( 2 q + q^{2} - q^{4} - 2 q^{8} - q^{9} + O(q^{10}) \) \( 2 q + q^{2} - q^{4} - 2 q^{8} - q^{9} + q^{13} - q^{16} - q^{17} - 2 q^{18} - q^{26} + q^{29} + q^{32} - 2 q^{34} - q^{36} - q^{37} + q^{41} - q^{49} - 2 q^{52} + 2 q^{53} - q^{58} + q^{61} + 2 q^{64} - q^{68} + q^{72} + 2 q^{73} + q^{74} - q^{81} - q^{82} - 2 q^{89} + 2 q^{97} + q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{1}^{\mathrm{new}}(1300, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1300.1.bc.a 1300.bc 52.j $2$ $0.649$ \(\Q(\sqrt{-3}) \) $D_{3}$ \(\Q(\sqrt{-1}) \) None \(1\) \(0\) \(0\) \(0\) \(q+\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}-q^{8}+\zeta_{6}^{2}q^{9}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1300, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1300, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)