Properties

Label 1536.2.j
Level $1536$
Weight $2$
Character orbit 1536.j
Rep. character $\chi_{1536}(385,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $64$
Newform subspaces $10$
Sturm bound $512$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 10 \)
Sturm bound: \(512\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(7\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1536, [\chi])\).

Total New Old
Modular forms 576 64 512
Cusp forms 448 64 384
Eisenstein series 128 0 128

Trace form

\( 64 q + O(q^{10}) \) \( 64 q - 64 q^{49} - 128 q^{65} - 64 q^{81} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1536, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1536.2.j.a 1536.j 16.e $4$ $12.265$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{3}q^{3}+(-1-\zeta_{8}^{2})q^{5}-\zeta_{8}^{2}q^{9}+\cdots\)
1536.2.j.b 1536.j 16.e $4$ $12.265$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{3}+(-1+\zeta_{8}^{2})q^{5}+(2\zeta_{8}+2\zeta_{8}^{3})q^{7}+\cdots\)
1536.2.j.c 1536.j 16.e $4$ $12.265$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{3}+(1-\zeta_{8}^{2})q^{5}+(-2\zeta_{8}-2\zeta_{8}^{3})q^{7}+\cdots\)
1536.2.j.d 1536.j 16.e $4$ $12.265$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{3}q^{3}+(1+\zeta_{8}^{2})q^{5}-\zeta_{8}^{2}q^{9}+\cdots\)
1536.2.j.e 1536.j 16.e $8$ $12.265$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{16}^{4}q^{3}+(-1-\zeta_{16}^{3}+\zeta_{16}^{7})q^{5}+\cdots\)
1536.2.j.f 1536.j 16.e $8$ $12.265$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{16}^{4}q^{3}+(-1-\zeta_{16}^{3}+\zeta_{16}^{7})q^{5}+\cdots\)
1536.2.j.g 1536.j 16.e $8$ $12.265$ 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{3}-\beta _{5}q^{5}+(\beta _{1}+\beta _{4}-\beta _{6})q^{7}+\cdots\)
1536.2.j.h 1536.j 16.e $8$ $12.265$ 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{3}+\beta _{5}q^{5}+(-\beta _{1}-\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
1536.2.j.i 1536.j 16.e $8$ $12.265$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{16}q^{3}+(1-\zeta_{16}^{2}-\zeta_{16}^{3}+\zeta_{16}^{5}+\cdots)q^{5}+\cdots\)
1536.2.j.j 1536.j 16.e $8$ $12.265$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{16}q^{3}+(1-\zeta_{16}^{2}-\zeta_{16}^{3}+\zeta_{16}^{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1536, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1536, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(768, [\chi])\)\(^{\oplus 2}\)