Properties

Label 1600.2.o
Level $1600$
Weight $2$
Character orbit 1600.o
Rep. character $\chi_{1600}(543,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $72$
Newform subspaces $12$
Sturm bound $480$
Trace bound $39$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 12 \)
Sturm bound: \(480\)
Trace bound: \(39\)
Distinguishing \(T_p\): \(3\), \(7\), \(79\)

Dimensions

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

Total New Old
Modular forms 552 72 480
Cusp forms 408 72 336
Eisenstein series 144 0 144

Trace form

\( 72q + O(q^{10}) \) \( 72q - 24q^{17} + 24q^{73} + 120q^{81} + 120q^{97} + O(q^{100}) \)

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.o.a \(2\) \(12.776\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-2\) \(q+(-1+i)q^{3}+(-1+i)q^{7}+iq^{9}+\cdots\)
1600.2.o.b \(2\) \(12.776\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(2\) \(q+(-1+i)q^{3}+(1-i)q^{7}+iq^{9}+4q^{11}+\cdots\)
1600.2.o.c \(2\) \(12.776\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-2\) \(q+(1-i)q^{3}+(-1+i)q^{7}+iq^{9}-4q^{11}+\cdots\)
1600.2.o.d \(2\) \(12.776\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(2\) \(q+(1-i)q^{3}+(1-i)q^{7}+iq^{9}-4q^{11}+\cdots\)
1600.2.o.e \(8\) \(12.776\) 8.0.49787136.1 None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{2}q^{3}+(-\beta _{1}+\beta _{4})q^{7}+(2\beta _{1}+\beta _{4}+\cdots)q^{9}+\cdots\)
1600.2.o.f \(8\) \(12.776\) 8.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{3}+2\beta _{6}q^{7}-4\beta _{3}q^{9}-\beta _{4}q^{11}+\cdots\)
1600.2.o.g \(8\) \(12.776\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}q^{3}+\zeta_{24}^{2}q^{7}+\zeta_{24}^{3}q^{9}-\zeta_{24}^{7}q^{11}+\cdots\)
1600.2.o.h \(8\) \(12.776\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{3}+2\zeta_{24}^{3}q^{9}-\zeta_{24}^{6}q^{11}+\cdots\)
1600.2.o.i \(8\) \(12.776\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{3}+2\zeta_{24}^{3}q^{9}-\zeta_{24}^{6}q^{11}+\cdots\)
1600.2.o.j \(8\) \(12.776\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}q^{3}+\zeta_{24}^{2}q^{7}+\zeta_{24}^{3}q^{9}+\zeta_{24}^{7}q^{11}+\cdots\)
1600.2.o.k \(8\) \(12.776\) 8.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{3}-2\beta _{6}q^{7}-4\beta _{3}q^{9}-\beta _{4}q^{11}+\cdots\)
1600.2.o.l \(8\) \(12.776\) 8.0.49787136.1 None \(0\) \(0\) \(0\) \(4\) \(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{4})q^{7}+(2\beta _{1}+\beta _{4}+\cdots)q^{9}+\cdots\)

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

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