Properties

Label 1024.2.e
Level $1024$
Weight $2$
Character orbit 1024.e
Rep. character $\chi_{1024}(257,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $56$
Newform subspaces $16$
Sturm bound $256$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 16 \)
Sturm bound: \(256\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(3\), \(5\), \(47\)

Dimensions

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

Total New Old
Modular forms 304 72 232
Cusp forms 208 56 152
Eisenstein series 96 16 80

Trace form

\( 56q + O(q^{10}) \) \( 56q + 16q^{17} - 16q^{33} + 8q^{49} - 16q^{65} + 24q^{81} - 16q^{97} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1024.2.e.a \(2\) \(8.177\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \) \(0\) \(-4\) \(0\) \(0\) \(q+(-2+2i)q^{3}-5iq^{9}+(2+2i)q^{11}+\cdots\)
1024.2.e.b \(2\) \(8.177\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(-4\) \(0\) \(q+(-1+i)q^{3}+(-2-2i)q^{5}-4iq^{7}+\cdots\)
1024.2.e.c \(2\) \(8.177\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(4\) \(0\) \(q+(-1+i)q^{3}+(2+2i)q^{5}+4iq^{7}+\cdots\)
1024.2.e.d \(2\) \(8.177\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(-4\) \(0\) \(q+(1-i)q^{3}+(-2-2i)q^{5}+4iq^{7}+\cdots\)
1024.2.e.e \(2\) \(8.177\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(4\) \(0\) \(q+(1-i)q^{3}+(2+2i)q^{5}-4iq^{7}+iq^{9}+\cdots\)
1024.2.e.f \(2\) \(8.177\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) \(q+(2-2i)q^{3}-5iq^{9}+(-2-2i)q^{11}+\cdots\)
1024.2.e.g \(4\) \(8.177\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(-4\) \(0\) \(0\) \(q+(-1+\zeta_{8}-\zeta_{8}^{2})q^{3}+(-2\zeta_{8}+3\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1024.2.e.h \(4\) \(8.177\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(0\) \(q+(-1+\zeta_{8}^{2})q^{3}-\zeta_{8}q^{5}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}+\cdots\)
1024.2.e.i \(4\) \(8.177\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}+\zeta_{8}^{3}q^{5}+4\zeta_{8}^{2}q^{7}+\zeta_{8}^{2}q^{9}+\cdots\)
1024.2.e.j \(4\) \(8.177\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}q^{5}+3\zeta_{8}^{2}q^{9}-3\zeta_{8}^{3}q^{13}-2q^{17}+\cdots\)
1024.2.e.k \(4\) \(8.177\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{5}+3\zeta_{8}^{2}q^{9}-\zeta_{8}^{3}q^{13}+2q^{17}+\cdots\)
1024.2.e.l \(4\) \(8.177\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}+\zeta_{8}^{2}q^{9}-3\zeta_{8}^{3}q^{11}+6q^{17}+\cdots\)
1024.2.e.m \(4\) \(8.177\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}-\zeta_{8}^{3}q^{5}-4\zeta_{8}^{2}q^{7}+\zeta_{8}^{2}q^{9}+\cdots\)
1024.2.e.n \(4\) \(8.177\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(0\) \(q+(1-\zeta_{8}^{2})q^{3}-\zeta_{8}q^{5}+(\zeta_{8}+\zeta_{8}^{3})q^{7}+\cdots\)
1024.2.e.o \(4\) \(8.177\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) \(q+(1+\zeta_{8}+\zeta_{8}^{2})q^{3}+(2\zeta_{8}+3\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1024.2.e.p \(8\) \(8.177\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{7}q^{3}+\zeta_{24}q^{5}-\zeta_{24}^{6}q^{7}-3\zeta_{24}^{2}q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1024, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)