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 $11$

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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

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(240))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
240.4.a.a \(1\) \(14.160\) \(\Q\) None \(0\) \(-3\) \(-5\) \(-20\) \(+\) \(+\) \(+\) \(q-3q^{3}-5q^{5}-20q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
240.4.a.b \(1\) \(14.160\) \(\Q\) None \(0\) \(-3\) \(-5\) \(4\) \(-\) \(+\) \(+\) \(q-3q^{3}-5q^{5}+4q^{7}+9q^{9}+48q^{11}+\cdots\)
240.4.a.c \(1\) \(14.160\) \(\Q\) None \(0\) \(-3\) \(5\) \(-32\) \(-\) \(+\) \(-\) \(q-3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}+60q^{11}+\cdots\)
240.4.a.d \(1\) \(14.160\) \(\Q\) None \(0\) \(-3\) \(5\) \(-8\) \(+\) \(+\) \(-\) \(q-3q^{3}+5q^{5}-8q^{7}+9q^{9}-20q^{11}+\cdots\)
240.4.a.e \(1\) \(14.160\) \(\Q\) None \(0\) \(-3\) \(5\) \(24\) \(-\) \(+\) \(-\) \(q-3q^{3}+5q^{5}+24q^{7}+9q^{9}-52q^{11}+\cdots\)
240.4.a.f \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(-5\) \(-20\) \(-\) \(-\) \(+\) \(q+3q^{3}-5q^{5}-20q^{7}+9q^{9}+24q^{11}+\cdots\)
240.4.a.g \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(-5\) \(-20\) \(+\) \(-\) \(+\) \(q+3q^{3}-5q^{5}-20q^{7}+9q^{9}+56q^{11}+\cdots\)
240.4.a.h \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(-5\) \(-4\) \(+\) \(-\) \(+\) \(q+3q^{3}-5q^{5}-4q^{7}+9q^{9}-72q^{11}+\cdots\)
240.4.a.i \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(-5\) \(28\) \(-\) \(-\) \(+\) \(q+3q^{3}-5q^{5}+28q^{7}+9q^{9}+24q^{11}+\cdots\)
240.4.a.j \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(5\) \(-32\) \(-\) \(-\) \(-\) \(q+3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
240.4.a.k \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(5\) \(0\) \(+\) \(-\) \(-\) \(q+3q^{3}+5q^{5}+9q^{9}-4q^{11}+54q^{13}+\cdots\)
240.4.a.l \(1\) \(14.160\) \(\Q\) None \(0\) \(3\) \(5\) \(16\) \(+\) \(-\) \(-\) \(q+3q^{3}+5q^{5}+2^{4}q^{7}+9q^{9}+28q^{11}+\cdots\)

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

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