Properties

Label 1600.2.n
Level $1600$
Weight $2$
Character orbit 1600.n
Rep. character $\chi_{1600}(1343,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $68$
Newform subspaces $22$
Sturm bound $480$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 22 \)
Sturm bound: \(480\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 552 76 476
Cusp forms 408 68 340
Eisenstein series 144 8 136

Trace form

\( 68 q + O(q^{10}) \) \( 68 q - 4 q^{13} + 12 q^{17} + 8 q^{21} - 8 q^{33} - 4 q^{37} - 8 q^{41} - 52 q^{53} + 16 q^{57} + 72 q^{61} + 12 q^{73} + 72 q^{77} - 108 q^{81} + 56 q^{93} + 28 q^{97} + O(q^{100}) \)

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.n.a 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-2i)q^{3}+(2-2i)q^{7}+5iq^{9}+\cdots\)
1600.2.n.b 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(0\) \(-2\) \(0\) \(-6\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1-i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots\)
1600.2.n.c 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots\)
1600.2.n.d 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(1-i)q^{7}-iq^{9}+4iq^{11}+\cdots\)
1600.2.n.e 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(1-i)q^{7}-iq^{9}+6iq^{11}+\cdots\)
1600.2.n.f 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(1-i)q^{7}-iq^{9}-4iq^{11}+\cdots\)
1600.2.n.g 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-3iq^{9}+(-5+5i)q^{13}+(5+5i)q^{17}+\cdots\)
1600.2.n.h 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-3iq^{9}+(-1+i)q^{13}+(-3-3i)q^{17}+\cdots\)
1600.2.n.i 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(-1+i)q^{7}-iq^{9}-4iq^{11}+\cdots\)
1600.2.n.j 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(-1+i)q^{7}-iq^{9}-6iq^{11}+\cdots\)
1600.2.n.k 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(-1+i)q^{7}-iq^{9}+4iq^{11}+\cdots\)
1600.2.n.l 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(0\) \(2\) \(0\) \(6\) $\mathrm{U}(1)[D_{4}]$ \(q+(1+i)q^{3}+(3-3i)q^{7}-iq^{9}+6q^{21}+\cdots\)
1600.2.n.m 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(3-3i)q^{7}-iq^{9}+2iq^{11}+\cdots\)
1600.2.n.n 1600.n 20.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{3}+(-2+2i)q^{7}+5iq^{9}+\cdots\)
1600.2.n.o 1600.n 20.e $4$ $12.776$ \(\Q(i, \sqrt{6})\) None \(0\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+2\beta _{2}+\cdots)q^{7}+\cdots\)
1600.2.n.p 1600.n 20.e $4$ $12.776$ \(\Q(i, \sqrt{6})\) None \(0\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+2\beta _{2}+\cdots)q^{7}+\cdots\)
1600.2.n.q 1600.n 20.e $4$ $12.776$ \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-\beta _{3}q^{3}-\beta _{2}q^{7}-7\beta _{1}q^{9}-10q^{21}+\cdots\)
1600.2.n.r 1600.n 20.e $4$ $12.776$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{12}^{2}q^{3}-\zeta_{12}^{3}q^{7}+3\zeta_{12}q^{9}+\cdots\)
1600.2.n.s 1600.n 20.e $4$ $12.776$ \(\Q(i, \sqrt{6})\) None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}+\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots\)
1600.2.n.t 1600.n 20.e $4$ $12.776$ \(\Q(i, \sqrt{6})\) None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}+\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots\)
1600.2.n.u 1600.n 20.e $8$ $12.776$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{24}^{5}q^{3}-4\zeta_{24}q^{7}+2\zeta_{24}^{3}q^{9}+\cdots\)
1600.2.n.v 1600.n 20.e $8$ $12.776$ 8.0.3317760000.5 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{3}+2\beta _{3}q^{9}+\beta _{7}q^{11}+2\beta _{4}q^{13}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)