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Properties

Label 1600.2.l
Level $1600$
Weight $2$
Character orbit 1600.l
Rep. character $\chi_{1600}(401,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $70$
Newform subspaces $9$
Sturm bound $480$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 9 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

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

Total New Old
Modular forms 528 82 446
Cusp forms 432 70 362
Eisenstein series 96 12 84

Trace form

\( 70 q - 2 q^{3} + O(q^{10}) \) \( 70 q - 2 q^{3} + 2 q^{11} + 2 q^{13} + 4 q^{17} + 6 q^{19} - 12 q^{21} + 16 q^{27} + 10 q^{29} + 4 q^{33} + 10 q^{37} + 18 q^{43} - 24 q^{47} - 30 q^{49} - 28 q^{51} - 6 q^{53} - 30 q^{59} + 10 q^{61} + 44 q^{63} + 30 q^{67} + 36 q^{69} - 20 q^{77} - 34 q^{81} + 38 q^{83} + 60 q^{91} + 32 q^{93} + 4 q^{97} - 18 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.l.a 1600.l 16.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{3}-2iq^{7}+iq^{9}+(-1+\cdots)q^{11}+\cdots\)
1600.2.l.b 1600.l 16.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{3}+iq^{9}+(3+3i)q^{11}+\cdots\)
1600.2.l.c 1600.l 16.e $2$ $12.776$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{3}+iq^{9}+(3+3i)q^{11}+(-3+\cdots)q^{13}+\cdots\)
1600.2.l.d 1600.l 16.e $4$ $12.776$ \(\Q(i, \sqrt{11})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{3})q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1600.2.l.e 1600.l 16.e $4$ $12.776$ \(\Q(i, \sqrt{11})\) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1600.2.l.f 1600.l 16.e $12$ $12.776$ 12.0.\(\cdots\).1 None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{4}q^{3}+(\beta _{2}+\beta _{3})q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots\)
1600.2.l.g 1600.l 16.e $12$ $12.776$ 12.0.\(\cdots\).1 None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{3}+(-\beta _{2}-\beta _{3})q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots\)
1600.2.l.h 1600.l 16.e $16$ $12.776$ 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{14}q^{3}+(-\beta _{2}+\beta _{3}+\beta _{5}-\beta _{8}+\cdots)q^{7}+\cdots\)
1600.2.l.i 1600.l 16.e $16$ $12.776$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{6}q^{3}+(-\beta _{3}-\beta _{6}-\beta _{8}+\beta _{9}-\beta _{10}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)