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Properties

Label 360.6.a
Level $360$
Weight $6$
Character orbit 360.a
Rep. character $\chi_{360}(1,\cdot)$
Character field $\Q$
Dimension $25$
Newform subspaces $16$
Sturm bound $432$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(432\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(360))\).

Total New Old
Modular forms 376 25 351
Cusp forms 344 25 319
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
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(11\)
Minus space\(-\)\(14\)

Trace form

\( 25 q - 25 q^{5} - 124 q^{7} + O(q^{10}) \) \( 25 q - 25 q^{5} - 124 q^{7} - 448 q^{11} + 454 q^{13} - 3746 q^{17} + 3340 q^{19} + 1076 q^{23} + 15625 q^{25} - 8494 q^{29} - 4616 q^{31} + 8800 q^{35} + 5022 q^{37} - 25470 q^{41} - 9704 q^{43} + 8780 q^{47} + 109905 q^{49} - 61398 q^{53} + 13900 q^{55} + 73768 q^{59} + 12166 q^{61} + 14750 q^{65} + 107480 q^{67} - 36456 q^{71} - 70566 q^{73} + 81664 q^{77} + 57360 q^{79} + 136264 q^{83} + 14450 q^{85} + 202050 q^{89} + 64688 q^{91} - 37900 q^{95} + 128818 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(360))\) 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
360.6.a.a 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(-25\) \(-160\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}-160q^{7}+596q^{11}-122q^{13}+\cdots\)
360.6.a.b 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(-25\) \(-108\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}-108q^{7}+604q^{11}-306q^{13}+\cdots\)
360.6.a.c 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(-25\) \(-80\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}-80q^{7}-684q^{11}-978q^{13}+\cdots\)
360.6.a.d 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(-25\) \(128\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+2^{7}q^{7}+308q^{11}-1058q^{13}+\cdots\)
360.6.a.e 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(25\) \(-100\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}-10^{2}q^{7}+136q^{11}+82q^{13}+\cdots\)
360.6.a.f 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(25\) \(-62\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}-62q^{7}+12^{2}q^{11}-654q^{13}+\cdots\)
360.6.a.g 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(25\) \(-28\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}-28q^{7}+208q^{11}-422q^{13}+\cdots\)
360.6.a.h 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(25\) \(108\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+108q^{7}+8q^{11}+162q^{13}+\cdots\)
360.6.a.i 360.a 1.a $1$ $57.738$ \(\Q\) None \(0\) \(0\) \(25\) \(242\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+242q^{7}-656q^{11}-206q^{13}+\cdots\)
360.6.a.j 360.a 1.a $2$ $57.738$ \(\Q(\sqrt{3289}) \) None \(0\) \(0\) \(-50\) \(-80\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+(-40-\beta )q^{7}+(140+\beta )q^{11}+\cdots\)
360.6.a.k 360.a 1.a $2$ $57.738$ \(\Q(\sqrt{2161}) \) None \(0\) \(0\) \(-50\) \(-8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+(-4-\beta )q^{7}+(-244-2\beta )q^{11}+\cdots\)
360.6.a.l 360.a 1.a $2$ $57.738$ \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(-50\) \(52\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+(26-\beta )q^{7}+(-280-2\beta )q^{11}+\cdots\)
360.6.a.m 360.a 1.a $2$ $57.738$ \(\Q(\sqrt{3289}) \) None \(0\) \(0\) \(50\) \(-80\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+(-40-\beta )q^{7}+(-140-\beta )q^{11}+\cdots\)
360.6.a.n 360.a 1.a $2$ $57.738$ \(\Q(\sqrt{1489}) \) None \(0\) \(0\) \(50\) \(16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+(8+\beta )q^{7}+(-2^{5}+4\beta )q^{11}+\cdots\)
360.6.a.o 360.a 1.a $3$ $57.738$ 3.3.2521041.1 None \(0\) \(0\) \(-75\) \(18\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+(6+\beta _{1})q^{7}+(-186-\beta _{1}+\cdots)q^{11}+\cdots\)
360.6.a.p 360.a 1.a $3$ $57.738$ 3.3.2521041.1 None \(0\) \(0\) \(75\) \(18\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+(6+\beta _{1})q^{7}+(186+\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(360))\) into lower level spaces

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