Properties

Label 1600.2.c
Level $1600$
Weight $2$
Character orbit 1600.c
Rep. character $\chi_{1600}(449,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $15$
Sturm bound $480$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(480\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(19\)

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 34 170
Eisenstein series 72 4 68

Trace form

\( 34 q - 26 q^{9} + O(q^{10}) \) \( 34 q - 26 q^{9} + 16 q^{21} - 4 q^{29} + 4 q^{41} - 18 q^{49} - 28 q^{61} + 64 q^{69} - 30 q^{81} + 44 q^{89} + O(q^{100}) \)

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.c.a 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}-2iq^{7}-6q^{9}-q^{11}-4iq^{13}+\cdots\)
1600.2.c.b 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}-2iq^{7}-6q^{9}+q^{11}+4iq^{13}+\cdots\)
1600.2.c.c 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+iq^{7}-q^{9}-4q^{11}-3iq^{13}+\cdots\)
1600.2.c.d 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-iq^{7}-q^{9}-iq^{13}+3iq^{17}+\cdots\)
1600.2.c.e 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-iq^{7}-q^{9}+iq^{13}-3iq^{17}+\cdots\)
1600.2.c.f 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+iq^{7}-q^{9}+4q^{11}+3iq^{13}+\cdots\)
1600.2.c.g 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2iq^{7}+2q^{9}-5q^{11}+5iq^{17}+\cdots\)
1600.2.c.h 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+2iq^{7}+2q^{9}-3q^{11}+4iq^{13}+\cdots\)
1600.2.c.i 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+2iq^{7}+2q^{9}+3q^{11}-4iq^{13}+\cdots\)
1600.2.c.j 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2iq^{7}+2q^{9}+5q^{11}-5iq^{17}+\cdots\)
1600.2.c.k 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{7}+3q^{9}-4q^{11}-iq^{13}+iq^{17}+\cdots\)
1600.2.c.l 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{9}-3iq^{13}-iq^{17}-10q^{29}+\cdots\)
1600.2.c.m 1600.c 5.b $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}+3q^{9}+4q^{11}-iq^{13}+iq^{17}+\cdots\)
1600.2.c.n 1600.c 5.b $4$ $12.776$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{3}-\zeta_{8}^{2}q^{7}-5q^{9}-\zeta_{8}^{3}q^{11}+\cdots\)
1600.2.c.o 1600.c 5.b $4$ $12.776$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+2\beta _{2}q^{7}-2q^{9}+\beta _{3}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)