Properties

Label 1600.3.g
Level $1600$
Weight $3$
Character orbit 1600.g
Rep. character $\chi_{1600}(351,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $11$
Sturm bound $720$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(720\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1600, [\chi])\).

Total New Old
Modular forms 516 76 440
Cusp forms 444 76 368
Eisenstein series 72 0 72

Trace form

\( 76 q + 228 q^{9} + O(q^{10}) \) \( 76 q + 228 q^{9} + 24 q^{17} + 48 q^{33} + 24 q^{41} - 564 q^{49} - 336 q^{57} + 88 q^{73} + 444 q^{81} + 408 q^{89} + 344 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1600, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1600.3.g.a 1600.g 8.d $4$ $43.597$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{2}q^{7}+2q^{9}+\beta _{1}q^{11}+\cdots\)
1600.3.g.b 1600.g 8.d $4$ $43.597$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{2}q^{7}+2q^{9}-\beta _{1}q^{11}+\cdots\)
1600.3.g.c 1600.g 8.d $4$ $43.597$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}q^{3}+\zeta_{12}^{3}q^{7}+3q^{9}+\zeta_{12}q^{11}+\cdots\)
1600.3.g.d 1600.g 8.d $4$ $43.597$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3\zeta_{12}q^{3}-\zeta_{12}^{3}q^{7}+18q^{9}+3\zeta_{12}q^{11}+\cdots\)
1600.3.g.e 1600.g 8.d $4$ $43.597$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3\zeta_{12}q^{3}+\zeta_{12}^{3}q^{7}+18q^{9}-3\zeta_{12}q^{11}+\cdots\)
1600.3.g.f 1600.g 8.d $8$ $43.597$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{24}^{2}q^{3}-\zeta_{24}^{6}q^{7}+(-4-\zeta_{24}^{3}+\cdots)q^{9}+\cdots\)
1600.3.g.g 1600.g 8.d $8$ $43.597$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{24}^{2}q^{3}+\zeta_{24}^{6}q^{7}+(-4-\zeta_{24}^{3}+\cdots)q^{9}+\cdots\)
1600.3.g.h 1600.g 8.d $8$ $43.597$ 8.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}+\beta _{2}q^{7}-3q^{9}-\beta _{7}q^{11}+\cdots\)
1600.3.g.i 1600.g 8.d $8$ $43.597$ 8.0.12960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+(-\beta _{4}+\beta _{6})q^{7}+(-1-\beta _{5}+\cdots)q^{9}+\cdots\)
1600.3.g.j 1600.g 8.d $8$ $43.597$ 8.0.2342560000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(\beta _{5}+\beta _{6})q^{7}+(7-\beta _{2})q^{9}+\cdots\)
1600.3.g.k 1600.g 8.d $16$ $43.597$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{2}q^{7}+(6+\beta _{3})q^{9}+(-2\beta _{11}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1600, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)