Properties

Label 800.4.c
Level $800$
Weight $4$
Character orbit 800.c
Rep. character $\chi_{800}(449,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $15$
Sturm bound $480$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(800, [\chi])\).

Total New Old
Modular forms 384 54 330
Cusp forms 336 54 282
Eisenstein series 48 0 48

Trace form

\( 54 q - 486 q^{9} + O(q^{10}) \) \( 54 q - 486 q^{9} - 272 q^{21} + 172 q^{29} + 116 q^{41} - 1622 q^{49} - 780 q^{61} - 3520 q^{69} + 7462 q^{81} - 4388 q^{89} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(800, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
800.4.c.a $2$ $47.202$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+4iq^{3}-8iq^{7}-37q^{9}-40q^{11}+\cdots\)
800.4.c.b $2$ $47.202$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+4iq^{3}-8iq^{7}-37q^{9}+40q^{11}+\cdots\)
800.4.c.c $2$ $47.202$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+5iq^{3}+10iq^{7}+2q^{9}-15q^{11}+\cdots\)
800.4.c.d $2$ $47.202$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+5iq^{3}+10iq^{7}+2q^{9}+15q^{11}+\cdots\)
800.4.c.e $2$ $47.202$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}+3iq^{7}+23q^{9}-60q^{11}+\cdots\)
800.4.c.f $2$ $47.202$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}+3iq^{7}+23q^{9}+60q^{11}+\cdots\)
800.4.c.g $2$ $47.202$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+3^{3}q^{9}-9iq^{13}+47iq^{17}+130q^{29}+\cdots\)
800.4.c.h $4$ $47.202$ \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{3}+\beta _{2}q^{7}-5^{2}q^{9}+3\beta _{3}q^{11}+\cdots\)
800.4.c.i $4$ $47.202$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}-5\beta _{2})q^{7}+\cdots\)
800.4.c.j $4$ $47.202$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{3}+3\beta _{2}q^{7}-13q^{9}+\beta _{3}q^{11}+\cdots\)
800.4.c.k $4$ $47.202$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}-5\beta _{2})q^{7}+\cdots\)
800.4.c.l $4$ $47.202$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{3}+7\beta _{2}q^{7}+7q^{9}-\beta _{3}q^{11}+\cdots\)
800.4.c.m $6$ $47.202$ 6.0.2068430400.1 None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+2\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots\)
800.4.c.n $6$ $47.202$ 6.0.2068430400.1 None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+2\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots\)
800.4.c.o $8$ $47.202$ 8.0.\(\cdots\).12 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{3}+(\beta _{3}+\beta _{7})q^{7}+(-24+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)