Properties

Label 1024.2.b
Level $1024$
Weight $2$
Character orbit 1024.b
Rep. character $\chi_{1024}(513,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $8$
Sturm bound $256$
Trace bound $33$

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Defining parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(256\)
Trace bound: \(33\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1024, [\chi])\).

Total New Old
Modular forms 152 36 116
Cusp forms 104 28 76
Eisenstein series 48 8 40

Trace form

\( 28 q - 20 q^{9} + O(q^{10}) \) \( 28 q - 20 q^{9} + 8 q^{17} - 12 q^{25} - 8 q^{33} + 12 q^{49} - 24 q^{57} - 8 q^{65} + 4 q^{81} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1024, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1024.2.b.a 1024.b 8.b $2$ $8.177$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta q^{3}+\beta q^{5}-4q^{7}-5q^{9}-2\beta q^{11}+\cdots\)
1024.2.b.b 1024.b 8.b $2$ $8.177$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}-\beta q^{5}-2q^{7}+q^{9}+\beta q^{11}+\cdots\)
1024.2.b.c 1024.b 8.b $2$ $8.177$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{5}+3q^{9}-5\beta q^{13}-8q^{17}+3q^{25}+\cdots\)
1024.2.b.d 1024.b 8.b $2$ $8.177$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3\beta q^{5}+3q^{9}+\beta q^{13}+8q^{17}-13q^{25}+\cdots\)
1024.2.b.e 1024.b 8.b $2$ $8.177$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}+\beta q^{5}+2q^{7}+q^{9}+\beta q^{11}+\cdots\)
1024.2.b.f 1024.b 8.b $2$ $8.177$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta q^{3}-\beta q^{5}+4q^{7}-5q^{9}-2\beta q^{11}+\cdots\)
1024.2.b.g 1024.b 8.b $8$ $8.177$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{16}^{2}q^{3}+\zeta_{16}q^{5}-\zeta_{16}^{3}q^{7}+(-1+\cdots)q^{9}+\cdots\)
1024.2.b.h 1024.b 8.b $8$ $8.177$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{24}q^{3}+\zeta_{24}^{5}q^{5}+\zeta_{24}^{4}q^{7}+(-1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1024, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1024, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)