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Properties

Label 288.8.a
Level $288$
Weight $8$
Character orbit 288.a
Rep. character $\chi_{288}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $18$
Sturm bound $384$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(384\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(288))\).

Total New Old
Modular forms 352 35 317
Cusp forms 320 35 285
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
\(+\)\(+\)$+$\(7\)
\(+\)\(-\)$-$\(10\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(11\)
Plus space\(+\)\(18\)
Minus space\(-\)\(17\)

Trace form

\( 35 q - 278 q^{5} + O(q^{10}) \) \( 35 q - 278 q^{5} - 510 q^{13} - 19610 q^{17} + 550573 q^{25} + 67602 q^{29} + 675410 q^{37} - 1309394 q^{41} + 2571547 q^{49} - 4188502 q^{53} - 1428854 q^{61} + 1491596 q^{65} - 965658 q^{73} - 3745696 q^{77} - 371252 q^{85} + 11948574 q^{89} + 12237438 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(288))\) 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
288.8.a.a 288.a 1.a $1$ $89.967$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-556\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-556q^{5}+8898q^{13}-5816q^{17}+\cdots\)
288.8.a.b 288.a 1.a $1$ $89.967$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(58\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+58q^{5}-8898q^{13}+40094q^{17}+\cdots\)
288.8.a.c 288.a 1.a $1$ $89.967$ \(\Q\) None \(0\) \(0\) \(70\) \(-92\) $-$ $-$ $\mathrm{SU}(2)$ \(q+70q^{5}-92q^{7}+3124q^{11}+1174q^{13}+\cdots\)
288.8.a.d 288.a 1.a $1$ $89.967$ \(\Q\) None \(0\) \(0\) \(70\) \(92\) $+$ $-$ $\mathrm{SU}(2)$ \(q+70q^{5}+92q^{7}-3124q^{11}+1174q^{13}+\cdots\)
288.8.a.e 288.a 1.a $1$ $89.967$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(556\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+556q^{5}+8898q^{13}+5816q^{17}+\cdots\)
288.8.a.f 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{435}) \) None \(0\) \(0\) \(-280\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-140q^{5}+\beta q^{7}+4\beta q^{11}-2238q^{13}+\cdots\)
288.8.a.g 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(-196\) \(-504\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-98+\beta )q^{5}+(-252+3\beta )q^{7}+\cdots\)
288.8.a.h 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(-196\) \(504\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-98+\beta )q^{5}+(252-3\beta )q^{7}+(-828+\cdots)q^{11}+\cdots\)
288.8.a.i 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{235}) \) None \(0\) \(0\) \(-180\) \(-1032\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-90+\beta )q^{5}+(-516-5\beta )q^{7}+\cdots\)
288.8.a.j 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{235}) \) None \(0\) \(0\) \(-180\) \(1032\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-90+\beta )q^{5}+(516+5\beta )q^{7}+(-1420+\cdots)q^{11}+\cdots\)
288.8.a.k 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{15}) \) None \(0\) \(0\) \(-140\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-70q^{5}+2\beta q^{7}+13\beta q^{11}+13758q^{13}+\cdots\)
288.8.a.l 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(28\) \(-936\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(14+3\beta )q^{5}+(-468+7\beta )q^{7}+(-1692+\cdots)q^{11}+\cdots\)
288.8.a.m 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(28\) \(936\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(14+3\beta )q^{5}+(468-7\beta )q^{7}+(1692+\cdots)q^{11}+\cdots\)
288.8.a.n 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(180\) \(-1248\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(90+8\beta )q^{5}+(-624-14\beta )q^{7}+\cdots\)
288.8.a.o 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(180\) \(1248\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(90+8\beta )q^{5}+(624+14\beta )q^{7}+(-4520+\cdots)q^{11}+\cdots\)
288.8.a.p 288.a 1.a $2$ $89.967$ \(\Q(\sqrt{435}) \) None \(0\) \(0\) \(280\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+140q^{5}+\beta q^{7}-4\beta q^{11}-2238q^{13}+\cdots\)
288.8.a.q 288.a 1.a $4$ $89.967$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{5}+\beta _{3}q^{7}+(-160-\beta _{1})q^{11}+\cdots\)
288.8.a.r 288.a 1.a $4$ $89.967$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{5}-\beta _{3}q^{7}+(160+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(288)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 15}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)