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Properties

Label 1540.2.a
Level $1540$
Weight $2$
Character orbit 1540.a
Rep. character $\chi_{1540}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $9$
Sturm bound $576$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1540 = 2^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1540.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(576\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1540))\).

Total New Old
Modular forms 300 20 280
Cusp forms 277 20 257
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.

\(2\)\(5\)\(7\)\(11\)FrickeDim
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(2\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(4\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(8\)
Minus space\(-\)\(12\)

Trace form

\( 20 q + 20 q^{9} + O(q^{10}) \) \( 20 q + 20 q^{9} + 8 q^{17} + 8 q^{19} + 20 q^{25} + 8 q^{29} - 8 q^{31} + 16 q^{39} + 8 q^{41} - 8 q^{43} + 16 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{51} - 16 q^{53} - 32 q^{57} + 16 q^{59} + 16 q^{61} + 16 q^{63} + 8 q^{65} - 24 q^{67} - 16 q^{69} + 8 q^{73} - 8 q^{77} - 16 q^{79} + 68 q^{81} + 8 q^{83} + 16 q^{87} - 40 q^{89} - 16 q^{91} + 16 q^{93} + 8 q^{95} - 16 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1540))\) 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 7 11
1540.2.a.a 1540.a 1.a $1$ $12.297$ \(\Q\) None \(0\) \(-2\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
1540.2.a.b 1540.a 1.a $1$ $12.297$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}-3q^{9}-q^{11}+2q^{13}+\cdots\)
1540.2.a.c 1540.a 1.a $1$ $12.297$ \(\Q\) None \(0\) \(2\) \(1\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
1540.2.a.d 1540.a 1.a $2$ $12.297$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}-q^{7}-q^{9}-q^{11}+(2+\cdots)q^{13}+\cdots\)
1540.2.a.e 1540.a 1.a $2$ $12.297$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+q^{7}+(1+2\beta )q^{9}+\cdots\)
1540.2.a.f 1540.a 1.a $3$ $12.297$ 3.3.564.1 None \(0\) \(-2\) \(-3\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}-q^{5}+q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
1540.2.a.g 1540.a 1.a $3$ $12.297$ 3.3.148.1 None \(0\) \(0\) \(-3\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}-q^{5}-q^{7}+(-\beta _{1}-\beta _{2})q^{9}+\cdots\)
1540.2.a.h 1540.a 1.a $3$ $12.297$ 3.3.3028.1 None \(0\) \(0\) \(3\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+q^{5}+q^{7}+(4+\beta _{1}+\beta _{2})q^{9}+\cdots\)
1540.2.a.i 1540.a 1.a $4$ $12.297$ 4.4.111028.1 None \(0\) \(0\) \(4\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{5}-q^{7}+(2+\beta _{2})q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1540))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1540)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(385))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(770))\)\(^{\oplus 2}\)