Properties

Label 4000.1.h
Level $4000$
Weight $1$
Character orbit 4000.h
Rep. character $\chi_{4000}(3999,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
Sturm bound $600$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(600\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(4000, [\chi])\).

Total New Old
Modular forms 78 8 70
Cusp forms 38 8 30
Eisenstein series 40 0 40

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 0 0 0 8

Trace form

\( 8 q + 4 q^{9} + O(q^{10}) \) \( 8 q + 4 q^{9} - 4 q^{21} + 8 q^{29} + 8 q^{41} - 4 q^{61} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(4000, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
4000.1.h.a 4000.h 20.d $4$ $1.996$ \(\Q(i, \sqrt{5})\) $A_{5}$ None None \(0\) \(-2\) \(0\) \(4\) \(q+(-1-\beta _{2})q^{3}+q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4000.1.h.b 4000.h 20.d $4$ $1.996$ \(\Q(i, \sqrt{5})\) $A_{5}$ None None \(0\) \(2\) \(0\) \(-4\) \(q+(1+\beta _{2})q^{3}-q^{7}+(1+\beta _{2})q^{9}+\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(4000, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(4000, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(500, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(2000, [\chi])\)\(^{\oplus 2}\)