Properties

Label 360.4.a
Level $360$
Weight $4$
Character orbit 360.a
Rep. character $\chi_{360}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $15$
Sturm bound $288$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(360))\).

Total New Old
Modular forms 232 15 217
Cusp forms 200 15 185
Eisenstein series 32 0 32

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.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(9\)
Minus space\(-\)\(6\)

Trace form

\( 15 q + 5 q^{5} + 36 q^{7} + O(q^{10}) \) \( 15 q + 5 q^{5} + 36 q^{7} - 64 q^{11} - 62 q^{13} + 178 q^{17} + 124 q^{19} - 76 q^{23} + 375 q^{25} + 182 q^{29} + 248 q^{31} - 80 q^{35} + 346 q^{37} - 114 q^{41} + 744 q^{43} + 716 q^{47} + 695 q^{49} + 222 q^{53} - 180 q^{55} + 40 q^{59} + 1218 q^{61} + 50 q^{65} + 1544 q^{67} + 2328 q^{71} + 1414 q^{73} - 416 q^{77} - 1072 q^{79} - 2312 q^{83} - 170 q^{85} + 318 q^{89} - 2416 q^{91} + 20 q^{95} + 62 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(360))\) 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
360.4.a.a 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(-18\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-18q^{7}-34q^{11}+12q^{13}+\cdots\)
360.4.a.b 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(-16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-2^{4}q^{7}+28q^{11}-26q^{13}+\cdots\)
360.4.a.c 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-4q^{11}+54q^{13}-114q^{17}+\cdots\)
360.4.a.d 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+2q^{7}+34q^{11}-68q^{13}+\cdots\)
360.4.a.e 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+8q^{7}-20q^{11}+22q^{13}+\cdots\)
360.4.a.f 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+2^{4}q^{7}-6^{2}q^{11}-42q^{13}+\cdots\)
360.4.a.g 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(-5\) \(34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}+34q^{7}+18q^{11}+12q^{13}+\cdots\)
360.4.a.h 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(-34\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-34q^{7}-2^{4}q^{11}+58q^{13}+\cdots\)
360.4.a.i 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(-18\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-18q^{7}+2^{4}q^{11}-6q^{13}+\cdots\)
360.4.a.j 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(-18\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}-18q^{7}+34q^{11}+12q^{13}+\cdots\)
360.4.a.k 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+2q^{7}-34q^{11}-68q^{13}+\cdots\)
360.4.a.l 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+4q^{7}-72q^{11}-6q^{13}-38q^{17}+\cdots\)
360.4.a.m 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+20q^{7}-2^{4}q^{11}+58q^{13}+\cdots\)
360.4.a.n 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+20q^{7}+56q^{11}-86q^{13}+\cdots\)
360.4.a.o 360.a 1.a $1$ $21.241$ \(\Q\) None \(0\) \(0\) \(5\) \(34\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{5}+34q^{7}-18q^{11}+12q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(360))\) into lower level spaces

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