Properties

Label 720.4.a
Level $720$
Weight $4$
Character orbit 720.a
Rep. character $\chi_{720}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $30$
Sturm bound $576$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(576\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(720))\).

Total New Old
Modular forms 456 30 426
Cusp forms 408 30 378
Eisenstein series 48 0 48

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.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(16\)
Minus space\(-\)\(14\)

Trace form

\( 30 q - 22 q^{7} + O(q^{10}) \) \( 30 q - 22 q^{7} - 20 q^{11} + 76 q^{17} - 192 q^{19} - 302 q^{23} + 750 q^{25} - 172 q^{29} + 12 q^{31} + 210 q^{35} - 8 q^{37} + 168 q^{41} - 494 q^{43} - 842 q^{47} + 1882 q^{49} + 768 q^{53} + 220 q^{55} + 1424 q^{59} + 656 q^{61} - 140 q^{65} - 954 q^{67} - 2220 q^{71} + 612 q^{73} + 112 q^{77} + 1800 q^{79} + 2486 q^{83} - 120 q^{85} + 1404 q^{89} - 796 q^{91} - 760 q^{95} - 340 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\) 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
720.4.a.a 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-34q^{7}-18q^{11}+12q^{13}+\cdots\)
720.4.a.b 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2^{5}q^{7}-60q^{11}-34q^{13}+\cdots\)
720.4.a.c 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2^{5}q^{7}+6^{2}q^{11}-10q^{13}+\cdots\)
720.4.a.d 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2^{4}q^{7}+6^{2}q^{11}-42q^{13}+\cdots\)
720.4.a.e 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-14q^{7}-6q^{11}+68q^{13}+\cdots\)
720.4.a.f 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-8q^{7}+20q^{11}+22q^{13}+\cdots\)
720.4.a.g 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2q^{7}-34q^{11}-68q^{13}+\cdots\)
720.4.a.h 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2q^{7}+30q^{11}-4q^{13}-90q^{17}+\cdots\)
720.4.a.i 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+4q^{11}+54q^{13}-114q^{17}+\cdots\)
720.4.a.j 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+4q^{7}+12q^{11}-58q^{13}+\cdots\)
720.4.a.k 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+2^{4}q^{7}-60q^{11}+86q^{13}+\cdots\)
720.4.a.l 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+2^{4}q^{7}-28q^{11}-26q^{13}+\cdots\)
720.4.a.m 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(18\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+18q^{7}+34q^{11}+12q^{13}+\cdots\)
720.4.a.n 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(24\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+24q^{7}+52q^{11}+22q^{13}+\cdots\)
720.4.a.o 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(-5\) \(30\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+30q^{7}-50q^{11}-20q^{13}+\cdots\)
720.4.a.p 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-34\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-34q^{7}+18q^{11}+12q^{13}+\cdots\)
720.4.a.q 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-20q^{7}-56q^{11}-86q^{13}+\cdots\)
720.4.a.r 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-20\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-20q^{7}-24q^{11}+74q^{13}+\cdots\)
720.4.a.s 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-20q^{7}+2^{4}q^{11}+58q^{13}+\cdots\)
720.4.a.t 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-14q^{7}+6q^{11}+68q^{13}+\cdots\)
720.4.a.u 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-6q^{7}+2^{5}q^{11}-38q^{13}+\cdots\)
720.4.a.v 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-4q^{7}+72q^{11}-6q^{13}-38q^{17}+\cdots\)
720.4.a.w 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-2q^{7}-30q^{11}-4q^{13}+90q^{17}+\cdots\)
720.4.a.x 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-2q^{7}+34q^{11}-68q^{13}+\cdots\)
720.4.a.y 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+4q^{7}-48q^{11}+2q^{13}+114q^{17}+\cdots\)
720.4.a.z 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(18\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+18q^{7}-34q^{11}+12q^{13}+\cdots\)
720.4.a.ba 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(18\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+18q^{7}-2^{4}q^{11}-6q^{13}+\cdots\)
720.4.a.bb 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(28\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+28q^{7}-24q^{11}-70q^{13}+\cdots\)
720.4.a.bc 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+30q^{7}+50q^{11}-20q^{13}+\cdots\)
720.4.a.bd 720.a 1.a $1$ $42.481$ \(\Q\) None \(0\) \(0\) \(5\) \(34\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+34q^{7}+2^{4}q^{11}+58q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(720))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(720)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)