Properties

Label 128.9.d
Level $128$
Weight $9$
Character orbit 128.d
Rep. character $\chi_{128}(63,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $6$
Sturm bound $144$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(144\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(128, [\chi])\).

Total New Old
Modular forms 136 32 104
Cusp forms 120 32 88
Eisenstein series 16 0 16

Trace form

\( 32 q + 69984 q^{9} + O(q^{10}) \) \( 32 q + 69984 q^{9} - 154560 q^{17} - 1791712 q^{25} + 3305344 q^{33} - 1127616 q^{41} - 2084832 q^{49} + 7916416 q^{57} - 14649600 q^{65} + 102896960 q^{73} - 57824352 q^{81} - 300504768 q^{89} + 157386816 q^{97} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(128, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
128.9.d.a 128.d 8.d $2$ $52.144$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 128.9.d.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{5}-3^{8}q^{9}-85iq^{13}+63358q^{17}+\cdots\)
128.9.d.b 128.d 8.d $2$ $52.144$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) 128.9.d.b \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{3}+18527q^{9}+69\beta q^{11}-162434q^{17}+\cdots\)
128.9.d.c 128.d 8.d $4$ $52.144$ \(\Q(i, \sqrt{14})\) None 128.9.d.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}-71\beta _{1}q^{5}+9\beta _{3}q^{7}-4545q^{9}+\cdots\)
128.9.d.d 128.d 8.d $4$ $52.144$ \(\Q(\sqrt{-70}, \sqrt{102})\) None 128.9.d.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{3}q^{5}-\beta _{2}q^{7}-33q^{9}+\cdots\)
128.9.d.e 128.d 8.d $4$ $52.144$ \(\Q(i, \sqrt{30})\) None 128.9.d.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-5\beta _{2}q^{3}-85\beta _{1}q^{5}-7\beta _{3}q^{7}+5439q^{9}+\cdots\)
128.9.d.f 128.d 8.d $16$ $52.144$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 128.9.d.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{6}q^{3}-\beta _{4}q^{5}-\beta _{9}q^{7}+(2663+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(128, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(128, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 2}\)