Properties

Label 1600.2.d
Level $1600$
Weight $2$
Character orbit 1600.d
Rep. character $\chi_{1600}(801,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $9$
Sturm bound $480$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(480\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(7\), \(17\)

Dimensions

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

Total New Old
Modular forms 276 38 238
Cusp forms 204 38 166
Eisenstein series 72 0 72

Trace form

\( 38q - 38q^{9} + O(q^{10}) \) \( 38q - 38q^{9} + 12q^{17} + 24q^{33} - 36q^{41} + 22q^{49} - 8q^{57} + 28q^{73} + 110q^{81} - 36q^{89} - 52q^{97} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1600.2.d.a \(2\) \(12.776\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}-q^{9}+3iq^{11}+6q^{17}-iq^{19}+\cdots\)
1600.2.d.b \(4\) \(12.776\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-12\) \(q-\zeta_{12}^{3}q^{3}+(-3+\zeta_{12})q^{7}+(-1+\cdots)q^{9}+\cdots\)
1600.2.d.c \(4\) \(12.776\) \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{3}+(-4-\beta _{3})q^{9}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
1600.2.d.d \(4\) \(12.776\) \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{3}+(-4-\beta _{3})q^{9}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1600.2.d.e \(4\) \(12.776\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{3}-\zeta_{12}^{3}q^{7}+2q^{9}+3\zeta_{12}q^{11}+\cdots\)
1600.2.d.f \(4\) \(12.776\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{3}+\zeta_{12}^{3}q^{7}+2q^{9}-3\zeta_{12}q^{11}+\cdots\)
1600.2.d.g \(4\) \(12.776\) \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-10}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{7}+3q^{9}-\beta _{1}q^{11}+\beta _{2}q^{13}+\cdots\)
1600.2.d.h \(4\) \(12.776\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(12\) \(q-\zeta_{12}^{3}q^{3}+(3-\zeta_{12})q^{7}+(-1+2\zeta_{12}+\cdots)q^{9}+\cdots\)
1600.2.d.i \(8\) \(12.776\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{3}-\zeta_{24}^{4}q^{7}-3q^{9}-\zeta_{24}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)