Properties

Label 1600.2.f
Level $1600$
Weight $2$
Character orbit 1600.f
Rep. character $\chi_{1600}(1249,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $10$
Sturm bound $480$
Trace bound $31$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(480\)
Trace bound: \(31\)
Distinguishing \(T_p\): \(3\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1600, [\chi])\).

Total New Old
Modular forms 276 36 240
Cusp forms 204 36 168
Eisenstein series 72 0 72

Trace form

\( 36 q + 36 q^{9} + O(q^{10}) \) \( 36 q + 36 q^{9} + 24 q^{41} - 36 q^{49} + 132 q^{81} + 120 q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1600, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1600.2.f.a 1600.f 40.f $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \) \(0\) \(-4\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{3}+q^{9}+3iq^{11}-3iq^{17}+iq^{19}+\cdots\)
1600.2.f.b 1600.f 40.f $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{3}+q^{9}-3iq^{11}-3iq^{17}-iq^{19}+\cdots\)
1600.2.f.c 1600.f 40.f $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}-\zeta_{12}^{2}q^{7}-2q^{9}+3\zeta_{12}q^{11}+\cdots\)
1600.2.f.d 1600.f 40.f $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\zeta_{12})q^{3}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{7}+\cdots\)
1600.2.f.e 1600.f 40.f $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\zeta_{12})q^{3}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{7}+\cdots\)
1600.2.f.f 1600.f 40.f $4$ $12.776$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-2}) \) \(0\) \(-4\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1+\beta _{3})q^{3}+(4-2\beta _{3})q^{9}+(3\beta _{1}+\cdots)q^{11}+\cdots\)
1600.2.f.g 1600.f 40.f $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+\zeta_{12}^{2}q^{7}-2q^{9}+3\zeta_{12}q^{11}+\cdots\)
1600.2.f.h 1600.f 40.f $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\zeta_{12})q^{3}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+\cdots\)
1600.2.f.i 1600.f 40.f $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\zeta_{12})q^{3}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+\cdots\)
1600.2.f.j 1600.f 40.f $4$ $12.776$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1-\beta _{3})q^{3}+(4-2\beta _{3})q^{9}+(3\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)