Properties

Label 4056.2.a
Level $4056$
Weight $2$
Character orbit 4056.a
Rep. character $\chi_{4056}(1,\cdot)$
Character field $\Q$
Dimension $77$
Newform subspaces $35$
Sturm bound $1456$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 35 \)
Sturm bound: \(1456\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4056))\).

Total New Old
Modular forms 784 77 707
Cusp forms 673 77 596
Eisenstein series 111 0 111

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(11\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(12\)
Plus space\(+\)\(32\)
Minus space\(-\)\(45\)

Trace form

\( 77 q + q^{3} + 2 q^{5} + 4 q^{7} + 77 q^{9} + O(q^{10}) \) \( 77 q + q^{3} + 2 q^{5} + 4 q^{7} + 77 q^{9} - 4 q^{11} + 2 q^{15} - 2 q^{17} - 8 q^{19} + 4 q^{21} - 16 q^{23} + 75 q^{25} + q^{27} - 6 q^{29} - 4 q^{31} - 4 q^{33} - 8 q^{35} + 14 q^{37} + 14 q^{41} + 12 q^{43} + 2 q^{45} + 16 q^{47} + 81 q^{49} + 2 q^{51} + 10 q^{53} + 8 q^{55} + 8 q^{57} + 20 q^{59} + 10 q^{61} + 4 q^{63} + 2 q^{73} - q^{75} - 8 q^{77} + 28 q^{79} + 77 q^{81} - 20 q^{83} - 28 q^{85} + 10 q^{87} + 38 q^{89} - 4 q^{93} + 24 q^{95} + 10 q^{97} - 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 13
4056.2.a.a 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}+q^{9}+2q^{11}+4q^{15}+\cdots\)
4056.2.a.b 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(-3\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
4056.2.a.c 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(-2\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{7}+q^{9}-6q^{11}+2q^{15}+\cdots\)
4056.2.a.d 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{17}+\cdots\)
4056.2.a.e 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{17}+\cdots\)
4056.2.a.f 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}+2q^{11}-6q^{17}+\cdots\)
4056.2.a.g 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(2\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-4q^{7}+q^{9}-2q^{15}+\cdots\)
4056.2.a.h 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}+q^{9}+6q^{11}-2q^{15}+\cdots\)
4056.2.a.i 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{9}-4q^{11}-2q^{15}+\cdots\)
4056.2.a.j 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(-1\) \(3\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
4056.2.a.k 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(-4\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-2q^{7}+q^{9}+2q^{11}+\cdots\)
4056.2.a.l 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(-3\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}+q^{9}-3q^{15}-q^{17}+\cdots\)
4056.2.a.m 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}-2q^{15}+2q^{17}+\cdots\)
4056.2.a.n 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(-2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
4056.2.a.o 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}-6q^{11}+2q^{17}+4q^{23}+\cdots\)
4056.2.a.p 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(2\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
4056.2.a.q 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}+q^{9}+3q^{15}-q^{17}+\cdots\)
4056.2.a.r 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
4056.2.a.s 4056.a 1.a $1$ $32.387$ \(\Q\) None \(0\) \(1\) \(4\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
4056.2.a.t 4056.a 1.a $2$ $32.387$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+(-1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
4056.2.a.u 4056.a 1.a $2$ $32.387$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+(1+\beta )q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
4056.2.a.v 4056.a 1.a $2$ $32.387$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(-2\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(1-\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
4056.2.a.w 4056.a 1.a $2$ $32.387$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(2\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+(-1-\beta )q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
4056.2.a.x 4056.a 1.a $3$ $32.387$ \(\Q(\zeta_{14})^+\) None \(0\) \(-3\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{2})q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
4056.2.a.y 4056.a 1.a $3$ $32.387$ 3.3.837.1 None \(0\) \(-3\) \(0\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{2}q^{5}+(-1+\beta _{1})q^{7}+q^{9}+\cdots\)
4056.2.a.z 4056.a 1.a $3$ $32.387$ 3.3.837.1 None \(0\) \(-3\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{2}q^{5}+(1-\beta _{1})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
4056.2.a.ba 4056.a 1.a $3$ $32.387$ \(\Q(\zeta_{14})^+\) None \(0\) \(-3\) \(2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1})q^{5}+(-1+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
4056.2.a.bb 4056.a 1.a $3$ $32.387$ \(\Q(\zeta_{14})^+\) None \(0\) \(3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1-2\beta _{1}+\beta _{2})q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
4056.2.a.bc 4056.a 1.a $3$ $32.387$ \(\Q(\zeta_{14})^+\) None \(0\) \(3\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)
4056.2.a.bd 4056.a 1.a $4$ $32.387$ 4.4.25488.1 None \(0\) \(4\) \(-4\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
4056.2.a.be 4056.a 1.a $4$ $32.387$ 4.4.25488.1 None \(0\) \(4\) \(4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(1+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
4056.2.a.bf 4056.a 1.a $6$ $32.387$ 6.6.27700337.1 None \(0\) \(-6\) \(-1\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{5}q^{5}+(-1-\beta _{1})q^{7}+q^{9}+\cdots\)
4056.2.a.bg 4056.a 1.a $6$ $32.387$ 6.6.27700337.1 None \(0\) \(-6\) \(1\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{5}q^{5}+(1+\beta _{1})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
4056.2.a.bh 4056.a 1.a $6$ $32.387$ 6.6.27700337.1 None \(0\) \(6\) \(-1\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(\beta _{1}-\beta _{3})q^{5}+(1-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
4056.2.a.bi 4056.a 1.a $6$ $32.387$ 6.6.27700337.1 None \(0\) \(6\) \(1\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4056))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4056)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1014))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2028))\)\(^{\oplus 2}\)