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Properties

Label 751.1.b
Level $751$
Weight $1$
Character orbit 751.b
Rep. character $\chi_{751}(750,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $4$
Sturm bound $62$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 751 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 751.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 751 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(62\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(751, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 9 9 0
Eisenstein series 1 1 0

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 7 0 2 0

Trace form

\( 9 q - 3 q^{2} + 6 q^{4} - q^{5} + 5 q^{9} + O(q^{10}) \) \( 9 q - 3 q^{2} + 6 q^{4} - q^{5} + 5 q^{9} - 2 q^{10} + q^{13} + 3 q^{16} + q^{18} + q^{19} - 3 q^{20} + 4 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{26} - 3 q^{32} + 4 q^{33} + 6 q^{36} + q^{37} - 4 q^{38} - 4 q^{40} - 4 q^{42} - q^{43} - q^{45} + q^{47} + 5 q^{49} - q^{50} + 4 q^{51} - 3 q^{52} - 3 q^{53} - 3 q^{59} + q^{61} + 6 q^{64} - 2 q^{65} - 4 q^{66} - q^{71} - 4 q^{72} - 4 q^{74} - 3 q^{76} - 4 q^{77} - 5 q^{80} + 5 q^{81} - 2 q^{86} + q^{89} - 2 q^{90} - 3 q^{92} - 4 q^{94} - 2 q^{95} + q^{97} + q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{1}^{\mathrm{new}}(751, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
751.1.b.a 751.b 751.b $1$ $0.375$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-751}) \) None \(-1\) \(0\) \(2\) \(0\) \(q-q^{2}+2q^{5}+q^{8}+q^{9}-2q^{10}-q^{13}+\cdots\)
751.1.b.b 751.b 751.b $2$ $0.375$ \(\Q(\sqrt{-2}) \) $S_{4}$ None None \(-2\) \(0\) \(0\) \(0\) \(q-q^{2}-\beta q^{3}+\beta q^{6}+\beta q^{7}+q^{8}-q^{9}+\cdots\)
751.1.b.c 751.b 751.b $2$ $0.375$ \(\Q(\sqrt{5}) \) $D_{5}$ \(\Q(\sqrt{-751}) \) None \(-1\) \(0\) \(-1\) \(0\) \(q+(-1+\beta )q^{2}+(1-\beta )q^{4}+(-1+\beta )q^{5}+\cdots\)
751.1.b.d 751.b 751.b $4$ $0.375$ \(\Q(\zeta_{15})^+\) $D_{15}$ \(\Q(\sqrt{-751}) \) None \(1\) \(0\) \(-2\) \(0\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{3})q^{5}+\cdots\)