Properties

Label 1520.2.a
Level $1520$
Weight $2$
Character orbit 1520.a
Rep. character $\chi_{1520}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $20$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1520))\).

Total New Old
Modular forms 252 36 216
Cusp forms 229 36 193
Eisenstein series 23 0 23

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

\(2\)\(5\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(22\)

Trace form

\( 36 q + 4 q^{7} + 36 q^{9} + O(q^{10}) \) \( 36 q + 4 q^{7} + 36 q^{9} + 8 q^{11} + 8 q^{21} + 20 q^{23} + 36 q^{25} + 24 q^{27} - 8 q^{29} - 12 q^{35} - 16 q^{37} - 4 q^{43} + 8 q^{45} - 4 q^{47} + 36 q^{49} + 16 q^{51} - 16 q^{53} + 8 q^{55} + 16 q^{59} + 60 q^{63} + 40 q^{67} + 40 q^{69} + 16 q^{77} + 16 q^{79} + 28 q^{81} + 4 q^{83} - 24 q^{87} - 8 q^{89} + 64 q^{91} - 16 q^{93} + 8 q^{95} - 40 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\) 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 5 19
1520.2.a.a 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(-3\) \(1\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
1520.2.a.b 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(-2\) \(-1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}-2q^{7}+q^{9}+6q^{13}+\cdots\)
1520.2.a.c 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(-2\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
1520.2.a.d 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}-2q^{9}-q^{13}-q^{15}+\cdots\)
1520.2.a.e 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-3q^{9}+4q^{11}-4q^{13}+\cdots\)
1520.2.a.f 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{9}+4q^{11}-6q^{13}-6q^{17}+\cdots\)
1520.2.a.g 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}-2q^{9}-3q^{13}-q^{15}+\cdots\)
1520.2.a.h 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(2\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+\cdots\)
1520.2.a.i 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(2\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}+4q^{11}+4q^{13}+\cdots\)
1520.2.a.j 1520.a 1.a $1$ $12.137$ \(\Q\) None \(0\) \(3\) \(-1\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{5}+5q^{7}+6q^{9}+4q^{11}+\cdots\)
1520.2.a.k 1520.a 1.a $2$ $12.137$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}+(1-2\beta )q^{9}-2q^{11}+\cdots\)
1520.2.a.l 1520.a 1.a $2$ $12.137$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(2\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}-2q^{7}+(1-2\beta )q^{9}+\cdots\)
1520.2.a.m 1520.a 1.a $2$ $12.137$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+2\beta q^{7}-q^{9}+(-2+2\beta )q^{11}+\cdots\)
1520.2.a.n 1520.a 1.a $2$ $12.137$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}+\beta q^{7}+(1+\beta )q^{9}-4q^{11}+\cdots\)
1520.2.a.o 1520.a 1.a $2$ $12.137$ \(\Q(\sqrt{2}) \) None \(0\) \(4\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+q^{5}+(2-2\beta )q^{7}+(3+4\beta )q^{9}+\cdots\)
1520.2.a.p 1520.a 1.a $3$ $12.137$ 3.3.148.1 None \(0\) \(-2\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1520.2.a.q 1520.a 1.a $3$ $12.137$ 3.3.316.1 None \(0\) \(1\) \(-3\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-q^{5}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
1520.2.a.r 1520.a 1.a $3$ $12.137$ 3.3.568.1 None \(0\) \(1\) \(-3\) \(5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-q^{5}+(2+\beta _{2})q^{7}+(1+2\beta _{1}+\cdots)q^{9}+\cdots\)
1520.2.a.s 1520.a 1.a $3$ $12.137$ 3.3.229.1 None \(0\) \(1\) \(3\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+q^{5}-\beta _{1}q^{7}+(1-\beta _{1})q^{9}+\cdots\)
1520.2.a.t 1520.a 1.a $4$ $12.137$ 4.4.11344.1 None \(0\) \(-2\) \(-4\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}-q^{5}+(-1+\beta _{1})q^{7}+(2-\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1520)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 2}\)