Properties

Label 3200.2.f
Level $3200$
Weight $2$
Character orbit 3200.f
Rep. character $\chi_{3200}(449,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $19$
Sturm bound $960$
Trace bound $49$

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Defining parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(960\)
Trace bound: \(49\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(13\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3200, [\chi])\).

Total New Old
Modular forms 528 72 456
Cusp forms 432 72 360
Eisenstein series 96 0 96

Trace form

\( 72 q + 72 q^{9} + O(q^{10}) \) \( 72 q + 72 q^{9} - 16 q^{41} - 72 q^{49} + 8 q^{81} - 16 q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3200, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3200.2.f.a $2$ $25.552$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-q^{3}+4iq^{7}-2q^{9}+3iq^{11}-iq^{17}+\cdots\)
3200.2.f.b $2$ $25.552$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-q^{3}-4iq^{7}-2q^{9}+3iq^{11}-iq^{17}+\cdots\)
3200.2.f.c $2$ $25.552$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-3q^{9}-4q^{13}+iq^{17}-2iq^{29}+\cdots\)
3200.2.f.d $2$ $25.552$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-3q^{9}+4q^{13}+iq^{17}+2iq^{29}+\cdots\)
3200.2.f.e $2$ $25.552$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+4iq^{7}-2q^{9}-3iq^{11}-iq^{17}+\cdots\)
3200.2.f.f $2$ $25.552$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+4iq^{7}-2q^{9}+3iq^{11}+iq^{17}+\cdots\)
3200.2.f.g $4$ $25.552$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}+2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.h $4$ $25.552$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{3}q^{3}+\zeta_{8}^{2}q^{7}-q^{9}+2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.i $4$ $25.552$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.j $4$ $25.552$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{3}+3\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.k $4$ $25.552$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{3}q^{3}+\zeta_{8}^{2}q^{7}-q^{9}-2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.l $4$ $25.552$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}-2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.m $4$ $25.552$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+2q^{9}+\beta _{2}q^{11}-4q^{13}+\cdots\)
3200.2.f.n $4$ $25.552$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+2q^{9}-\beta _{2}q^{11}+4q^{13}+\cdots\)
3200.2.f.o $4$ $25.552$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{3}q^{3}+5q^{9}-\zeta_{8}^{2}q^{11}-3\zeta_{8}q^{17}+\cdots\)
3200.2.f.p $4$ $25.552$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}-\beta _{2}q^{7}+7q^{9}-6q^{13}-\beta _{1}q^{17}+\cdots\)
3200.2.f.q $4$ $25.552$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+\beta _{2}q^{7}+7q^{9}+6q^{13}-\beta _{1}q^{17}+\cdots\)
3200.2.f.r $8$ $25.552$ 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}-\beta _{2}q^{7}+2q^{9}+\beta _{3}q^{11}+\cdots\)
3200.2.f.s $8$ $25.552$ \(\Q(\zeta_{24})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}q^{3}+(2-\zeta_{24}^{3})q^{9}+\zeta_{24}^{6}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3200, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(640, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1600, [\chi])\)\(^{\oplus 2}\)