Properties

Label 240.4.a
Level $240$
Weight $4$
Character orbit 240.a
Rep. character $\chi_{240}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $12$
Sturm bound $192$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(192\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(240))\).

Total New Old
Modular forms 156 12 144
Cusp forms 132 12 120
Eisenstein series 24 0 24

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

\(2\)\(3\)\(5\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(1\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12 q + 6 q^{3} - 64 q^{7} + 108 q^{9} + O(q^{10}) \) \( 12 q + 6 q^{3} - 64 q^{7} + 108 q^{9} + 40 q^{11} - 30 q^{15} - 192 q^{19} + 328 q^{23} + 300 q^{25} + 54 q^{27} + 400 q^{29} + 192 q^{31} + 16 q^{37} - 420 q^{39} - 296 q^{41} + 1240 q^{43} + 744 q^{47} + 844 q^{49} + 372 q^{51} - 1536 q^{53} - 440 q^{55} + 168 q^{57} - 808 q^{59} + 56 q^{61} - 576 q^{63} + 280 q^{65} - 360 q^{67} + 264 q^{69} + 2688 q^{71} - 216 q^{73} + 150 q^{75} - 1536 q^{77} - 720 q^{79} + 972 q^{81} + 456 q^{83} - 120 q^{85} - 1044 q^{87} - 504 q^{89} - 1504 q^{91} - 1768 q^{97} + 360 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(240))\) 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 5
240.4.a.a 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(-3\) \(-5\) \(-20\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}-20q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
240.4.a.b 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(-3\) \(-5\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+4q^{7}+9q^{9}+48q^{11}+\cdots\)
240.4.a.c 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(-3\) \(5\) \(-32\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}+60q^{11}+\cdots\)
240.4.a.d 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(-3\) \(5\) \(-8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}-8q^{7}+9q^{9}-20q^{11}+\cdots\)
240.4.a.e 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(-3\) \(5\) \(24\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+24q^{7}+9q^{9}-52q^{11}+\cdots\)
240.4.a.f 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(-5\) \(-20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}-20q^{7}+9q^{9}+24q^{11}+\cdots\)
240.4.a.g 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(-5\) \(-20\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}-20q^{7}+9q^{9}+56q^{11}+\cdots\)
240.4.a.h 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(-5\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}-4q^{7}+9q^{9}-72q^{11}+\cdots\)
240.4.a.i 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(-5\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+28q^{7}+9q^{9}+24q^{11}+\cdots\)
240.4.a.j 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(5\) \(-32\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
240.4.a.k 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+9q^{9}-4q^{11}+54q^{13}+\cdots\)
240.4.a.l 240.a 1.a $1$ $14.160$ \(\Q\) None \(0\) \(3\) \(5\) \(16\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+2^{4}q^{7}+9q^{9}+28q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(240)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)