Properties

Label 4056.2.c
Level $4056$
Weight $2$
Character orbit 4056.c
Rep. character $\chi_{4056}(337,\cdot)$
Character field $\Q$
Dimension $78$
Newform subspaces $18$
Sturm bound $1456$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(1456\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4056, [\chi])\).

Total New Old
Modular forms 784 78 706
Cusp forms 672 78 594
Eisenstein series 112 0 112

Trace form

\( 78 q - 2 q^{3} + 78 q^{9} + O(q^{10}) \) \( 78 q - 2 q^{3} + 78 q^{9} - 8 q^{23} - 74 q^{25} - 2 q^{27} - 24 q^{29} - 8 q^{35} + 24 q^{43} - 98 q^{49} + 4 q^{51} + 48 q^{53} + 32 q^{55} + 48 q^{61} - 8 q^{69} + 30 q^{75} + 8 q^{77} - 20 q^{79} + 78 q^{81} - 32 q^{87} - 16 q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4056, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
4056.2.c.a 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+2iq^{5}+iq^{7}+q^{9}-6iq^{11}+\cdots\)
4056.2.c.b 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+3iq^{5}+4iq^{7}+q^{9}-4iq^{11}+\cdots\)
4056.2.c.c 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+iq^{5}+2iq^{7}+q^{9}-iq^{15}+\cdots\)
4056.2.c.d 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+2iq^{5}+q^{9}+iq^{11}-2iq^{15}+\cdots\)
4056.2.c.e 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+iq^{5}+q^{9}+2iq^{11}-iq^{15}+\cdots\)
4056.2.c.f 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+2iq^{7}+q^{9}+iq^{11}+6q^{17}+\cdots\)
4056.2.c.g 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+q^{9}+3iq^{11}-2q^{17}-4q^{23}+\cdots\)
4056.2.c.h 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+iq^{5}+q^{9}+iq^{15}-2q^{17}+\cdots\)
4056.2.c.i 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+3iq^{5}+q^{9}+3iq^{15}+q^{17}+\cdots\)
4056.2.c.j 4056.c 13.b $2$ $32.387$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+2iq^{5}-2iq^{7}+q^{9}-iq^{11}+\cdots\)
4056.2.c.k 4056.c 13.b $4$ $32.387$ \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}-\zeta_{12}^{2}q^{5}+(-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
4056.2.c.l 4056.c 13.b $4$ $32.387$ \(\Q(i, \sqrt{13})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+\beta _{1}q^{5}+(\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
4056.2.c.m 4056.c 13.b $6$ $32.387$ 6.0.44836416.1 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}-\beta _{4}q^{5}+(\beta _{1}+\beta _{5})q^{7}+q^{9}+\cdots\)
4056.2.c.n 4056.c 13.b $6$ $32.387$ 6.0.153664.1 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+(\beta _{3}+\beta _{5})q^{5}+(-2\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
4056.2.c.o 4056.c 13.b $6$ $32.387$ 6.0.153664.1 None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+(\beta _{1}+\beta _{3})q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
4056.2.c.p 4056.c 13.b $8$ $32.387$ 8.0.649638144.4 None \(0\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}-\beta _{1}q^{5}+(-\beta _{1}-\beta _{4})q^{7}+q^{9}+\cdots\)
4056.2.c.q 4056.c 13.b $12$ $32.387$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}-\beta _{10}q^{5}+(-\beta _{1}+\beta _{9})q^{7}+q^{9}+\cdots\)
4056.2.c.r 4056.c 13.b $12$ $32.387$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+\beta _{1}q^{5}+(-\beta _{9}+\beta _{11})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4056, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4056, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(507, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(676, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1014, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1352, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2028, [\chi])\)\(^{\oplus 2}\)