Properties

Label 5220.2.a
Level $5220$
Weight $2$
Character orbit 5220.a
Rep. character $\chi_{5220}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $26$
Sturm bound $2160$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5220.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(2160\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5220))\).

Total New Old
Modular forms 1104 48 1056
Cusp forms 1057 48 1009
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(29\)FrickeDim
\(-\)\(+\)\(+\)\(+\)\(-\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(18\)
Minus space\(-\)\(30\)

Trace form

\( 48 q - 4 q^{7} + O(q^{10}) \) \( 48 q - 4 q^{7} - 4 q^{11} - 8 q^{13} - 16 q^{17} + 12 q^{19} - 4 q^{23} + 48 q^{25} + 4 q^{31} + 4 q^{35} + 8 q^{37} + 8 q^{41} + 28 q^{43} + 4 q^{47} + 64 q^{49} - 8 q^{53} + 8 q^{55} + 16 q^{59} - 16 q^{61} + 4 q^{67} + 8 q^{71} + 12 q^{73} - 28 q^{77} + 12 q^{79} - 20 q^{83} + 4 q^{85} + 8 q^{89} + 24 q^{91} + 16 q^{95} + 12 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5220))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 29
5220.2.a.a 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.f \(0\) \(0\) \(-1\) \(-5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-5q^{7}-5q^{11}-5q^{13}-7q^{17}+\cdots\)
5220.2.a.b 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.g \(0\) \(0\) \(-1\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}+3q^{11}+q^{13}+3q^{17}+\cdots\)
5220.2.a.c 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.b \(0\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-3q^{11}-2q^{13}+4q^{17}+\cdots\)
5220.2.a.d 5220.a 1.a $1$ $41.682$ \(\Q\) None 5220.2.a.d \(0\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+q^{11}-2q^{17}-2q^{19}+\cdots\)
5220.2.a.e 5220.a 1.a $1$ $41.682$ \(\Q\) None 580.2.a.b \(0\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+4q^{11}-6q^{13}+4q^{17}+\cdots\)
5220.2.a.f 5220.a 1.a $1$ $41.682$ \(\Q\) None 5220.2.a.f \(0\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{7}-3q^{11}-q^{13}+3q^{17}+\cdots\)
5220.2.a.g 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.h \(0\) \(0\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-4q^{11}+2q^{13}+4q^{19}+\cdots\)
5220.2.a.h 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.c \(0\) \(0\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}+4q^{11}+2q^{13}+8q^{17}+\cdots\)
5220.2.a.i 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.d \(0\) \(0\) \(-1\) \(3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}-3q^{11}+3q^{13}-q^{17}+\cdots\)
5220.2.a.j 5220.a 1.a $1$ $41.682$ \(\Q\) None 5220.2.a.j \(0\) \(0\) \(-1\) \(3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}+q^{11}-5q^{13}+3q^{17}+\cdots\)
5220.2.a.k 5220.a 1.a $1$ $41.682$ \(\Q\) None 5220.2.a.d \(0\) \(0\) \(1\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-q^{11}+2q^{17}-2q^{19}+\cdots\)
5220.2.a.l 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.e \(0\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}+3q^{11}-q^{13}-3q^{17}+\cdots\)
5220.2.a.m 5220.a 1.a $1$ $41.682$ \(\Q\) None 5220.2.a.f \(0\) \(0\) \(1\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}+3q^{11}-q^{13}-3q^{17}+\cdots\)
5220.2.a.n 5220.a 1.a $1$ $41.682$ \(\Q\) None 580.2.a.a \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{11}-2q^{13}-2q^{19}+8q^{23}+\cdots\)
5220.2.a.o 5220.a 1.a $1$ $41.682$ \(\Q\) None 1740.2.a.a \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+6q^{11}+6q^{13}-4q^{17}+2q^{19}+\cdots\)
5220.2.a.p 5220.a 1.a $1$ $41.682$ \(\Q\) None 5220.2.a.j \(0\) \(0\) \(1\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+3q^{7}-q^{11}-5q^{13}-3q^{17}+\cdots\)
5220.2.a.q 5220.a 1.a $2$ $41.682$ \(\Q(\sqrt{17}) \) None 1740.2.a.k \(0\) \(0\) \(-2\) \(-5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(-2-\beta )q^{7}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
5220.2.a.r 5220.a 1.a $2$ $41.682$ \(\Q(\sqrt{57}) \) None 1740.2.a.n \(0\) \(0\) \(-2\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}+(2-\beta )q^{11}+2q^{13}+\cdots\)
5220.2.a.s 5220.a 1.a $2$ $41.682$ \(\Q(\sqrt{33}) \) None 1740.2.a.i \(0\) \(0\) \(2\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-\beta q^{7}+(-2-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
5220.2.a.t 5220.a 1.a $2$ $41.682$ \(\Q(\sqrt{33}) \) None 1740.2.a.l \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta q^{7}+(-2+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
5220.2.a.u 5220.a 1.a $2$ $41.682$ \(\Q(\sqrt{17}) \) None 1740.2.a.m \(0\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2\beta q^{7}+(2-\beta )q^{11}+6q^{13}+\cdots\)
5220.2.a.v 5220.a 1.a $2$ $41.682$ \(\Q(\sqrt{17}) \) None 1740.2.a.j \(0\) \(0\) \(2\) \(3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta )q^{7}+(1-2\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
5220.2.a.w 5220.a 1.a $3$ $41.682$ 3.3.148.1 None 580.2.a.d \(0\) \(0\) \(-3\) \(2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1-\beta _{2})q^{7}+(-1+\beta _{2})q^{11}+\cdots\)
5220.2.a.x 5220.a 1.a $3$ $41.682$ 3.3.564.1 None 580.2.a.c \(0\) \(0\) \(3\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+\beta _{2})q^{7}+(-3-\beta _{2})q^{11}+\cdots\)
5220.2.a.y 5220.a 1.a $7$ $41.682$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 5220.2.a.y \(0\) \(0\) \(-7\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta _{2}q^{7}-\beta _{5}q^{11}-\beta _{6}q^{13}+\cdots\)
5220.2.a.z 5220.a 1.a $7$ $41.682$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 5220.2.a.y \(0\) \(0\) \(7\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta _{2}q^{7}+\beta _{5}q^{11}-\beta _{6}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5220))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5220)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(522))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(870))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1044))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1740))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2610))\)\(^{\oplus 2}\)