Properties

Label 288.6.a
Level $288$
Weight $6$
Character orbit 288.a
Rep. character $\chi_{288}(1,\cdot)$
Character field $\Q$
Dimension $25$
Newform subspaces $18$
Sturm bound $288$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(288))\).

Total New Old
Modular forms 256 25 231
Cusp forms 224 25 199
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.

\(2\)\(3\)FrickeDim
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(5\)
\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(12\)
Minus space\(-\)\(13\)

Trace form

\( 25 q + 38 q^{5} + O(q^{10}) \) \( 25 q + 38 q^{5} - 354 q^{13} - 398 q^{17} + 13679 q^{25} - 12242 q^{29} - 4034 q^{37} + 4058 q^{41} + 39761 q^{49} + 1222 q^{53} - 28426 q^{61} + 65508 q^{65} - 2046 q^{73} + 160416 q^{77} + 33236 q^{85} + 55370 q^{89} + 110490 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(288))\) 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
288.6.a.a 288.a 1.a $1$ $46.191$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-76\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-76q^{5}+1194q^{13}+808q^{17}+2651q^{25}+\cdots\)
288.6.a.b 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(-26\) \(-36\) $-$ $-$ $\mathrm{SU}(2)$ \(q-26q^{5}-6^{2}q^{7}+180q^{11}+318q^{13}+\cdots\)
288.6.a.c 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(-26\) \(36\) $-$ $-$ $\mathrm{SU}(2)$ \(q-26q^{5}+6^{2}q^{7}-180q^{11}+318q^{13}+\cdots\)
288.6.a.d 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(-14\) \(-208\) $-$ $-$ $\mathrm{SU}(2)$ \(q-14q^{5}-208q^{7}+536q^{11}+694q^{13}+\cdots\)
288.6.a.e 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(-14\) \(208\) $+$ $-$ $\mathrm{SU}(2)$ \(q-14q^{5}+208q^{7}-536q^{11}+694q^{13}+\cdots\)
288.6.a.f 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(14\) \(-100\) $+$ $-$ $\mathrm{SU}(2)$ \(q+14q^{5}-10^{2}q^{7}+220q^{11}-818q^{13}+\cdots\)
288.6.a.g 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(14\) \(100\) $-$ $-$ $\mathrm{SU}(2)$ \(q+14q^{5}+10^{2}q^{7}-220q^{11}-818q^{13}+\cdots\)
288.6.a.h 288.a 1.a $1$ $46.191$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(76\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+76q^{5}+1194q^{13}-808q^{17}+2651q^{25}+\cdots\)
288.6.a.i 288.a 1.a $1$ $46.191$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(82\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+82q^{5}-1194q^{13}-2242q^{17}+\cdots\)
288.6.a.j 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(86\) \(-180\) $+$ $-$ $\mathrm{SU}(2)$ \(q+86q^{5}-180q^{7}-684q^{11}+222q^{13}+\cdots\)
288.6.a.k 288.a 1.a $1$ $46.191$ \(\Q\) None \(0\) \(0\) \(86\) \(180\) $+$ $-$ $\mathrm{SU}(2)$ \(q+86q^{5}+180q^{7}+684q^{11}+222q^{13}+\cdots\)
288.6.a.l 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-92\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-46q^{5}+2\beta q^{7}+\beta q^{11}-42q^{13}+\cdots\)
288.6.a.m 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{15}) \) None \(0\) \(0\) \(-88\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-44q^{5}-\beta q^{7}-4\beta q^{11}-726q^{13}+\cdots\)
288.6.a.n 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{31}) \) None \(0\) \(0\) \(-36\) \(-120\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-18+\beta )q^{5}+(-60+\beta )q^{7}+(10^{2}+\cdots)q^{11}+\cdots\)
288.6.a.o 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{31}) \) None \(0\) \(0\) \(-36\) \(120\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-18+\beta )q^{5}+(60-\beta )q^{7}+(-10^{2}+\cdots)q^{11}+\cdots\)
288.6.a.p 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{181}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+2\beta q^{7}-2^{5}q^{11}+10q^{13}+\cdots\)
288.6.a.q 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{181}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{5}-2\beta q^{7}+2^{5}q^{11}+10q^{13}+\cdots\)
288.6.a.r 288.a 1.a $2$ $46.191$ \(\Q(\sqrt{15}) \) None \(0\) \(0\) \(88\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+44q^{5}-\beta q^{7}+4\beta q^{11}-726q^{13}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(288)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)