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Properties

Label 2088.4.a
Level $2088$
Weight $4$
Character orbit 2088.a
Rep. character $\chi_{2088}(1,\cdot)$
Character field $\Q$
Dimension $105$
Newform subspaces $17$
Sturm bound $1440$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2088.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(1440\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2088))\).

Total New Old
Modular forms 1096 105 991
Cusp forms 1064 105 959
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.

\(2\)\(3\)\(29\)FrickeDim
\(+\)\(+\)\(+\)$+$\(11\)
\(+\)\(+\)\(-\)$-$\(10\)
\(+\)\(-\)\(+\)$-$\(15\)
\(+\)\(-\)\(-\)$+$\(17\)
\(-\)\(+\)\(+\)$-$\(10\)
\(-\)\(+\)\(-\)$+$\(11\)
\(-\)\(-\)\(+\)$+$\(15\)
\(-\)\(-\)\(-\)$-$\(16\)
Plus space\(+\)\(54\)
Minus space\(-\)\(51\)

Trace form

\( 105 q + 36 q^{7} + O(q^{10}) \) \( 105 q + 36 q^{7} - 144 q^{11} - 36 q^{13} + 182 q^{17} + 272 q^{19} - 20 q^{23} + 2697 q^{25} + 87 q^{29} + 120 q^{31} - 100 q^{35} - 38 q^{37} + 642 q^{41} + 432 q^{43} + 1192 q^{47} + 5121 q^{49} - 888 q^{53} - 1220 q^{55} + 700 q^{59} + 174 q^{61} - 2178 q^{65} + 2476 q^{67} - 304 q^{71} + 2254 q^{73} - 1208 q^{77} - 2888 q^{79} - 2524 q^{83} - 2728 q^{85} + 410 q^{89} - 1636 q^{91} - 984 q^{95} + 518 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2088))\) 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 29
2088.4.a.a 2088.a 1.a $3$ $123.196$ 3.3.4481.1 None \(0\) \(0\) \(-11\) \(-38\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1}-\beta _{2})q^{5}+(-13+\beta _{1}+\cdots)q^{7}+\cdots\)
2088.4.a.b 2088.a 1.a $3$ $123.196$ 3.3.229.1 None \(0\) \(0\) \(-4\) \(16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-3\beta _{1}+4\beta _{2})q^{5}+(2+2\beta _{1}+\cdots)q^{7}+\cdots\)
2088.4.a.c 2088.a 1.a $4$ $123.196$ 4.4.6020961.1 None \(0\) \(0\) \(-1\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{5}+(-5-\beta _{1})q^{7}+(9+\cdots)q^{11}+\cdots\)
2088.4.a.d 2088.a 1.a $4$ $123.196$ 4.4.518057.1 None \(0\) \(0\) \(11\) \(-6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{2}-\beta _{3})q^{5}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
2088.4.a.e 2088.a 1.a $4$ $123.196$ 4.4.225792.1 None \(0\) \(0\) \(20\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(5-\beta _{2})q^{5}+(-2-2\beta _{1}-\beta _{2}+2\beta _{3})q^{7}+\cdots\)
2088.4.a.f 2088.a 1.a $5$ $123.196$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-10\) \(32\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{3})q^{5}+(6+\beta _{2}+\beta _{4})q^{7}+\cdots\)
2088.4.a.g 2088.a 1.a $5$ $123.196$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-9\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{2})q^{5}+(-1+\beta _{3})q^{7}+(12+\cdots)q^{11}+\cdots\)
2088.4.a.h 2088.a 1.a $5$ $123.196$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(19\) \(10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(4-\beta _{2})q^{5}+(2-\beta _{3}+\beta _{4})q^{7}+(-1+\cdots)q^{11}+\cdots\)
2088.4.a.i 2088.a 1.a $6$ $123.196$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-16\) \(-11\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{2})q^{5}+(-2+\beta _{4})q^{7}+(-8+\cdots)q^{11}+\cdots\)
2088.4.a.j 2088.a 1.a $6$ $123.196$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-6\) \(41\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{5}+(7-\beta _{2})q^{7}+(-9+\cdots)q^{11}+\cdots\)
2088.4.a.k 2088.a 1.a $6$ $123.196$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-4\) \(29\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{5}+(5+\beta _{3})q^{7}+(-5+\cdots)q^{11}+\cdots\)
2088.4.a.l 2088.a 1.a $6$ $123.196$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(5\) \(-38\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{3}+\beta _{4})q^{5}+(-6-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
2088.4.a.m 2088.a 1.a $6$ $123.196$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(6\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{4})q^{5}+(\beta _{1}+\beta _{3}+\beta _{4})q^{7}+(-5+\cdots)q^{11}+\cdots\)
2088.4.a.n 2088.a 1.a $10$ $123.196$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(-15\) \(-19\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{5}+(-2-\beta _{5})q^{7}+(-1+\cdots)q^{11}+\cdots\)
2088.4.a.o 2088.a 1.a $10$ $123.196$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(15\) \(-19\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{5}+(-2-\beta _{5})q^{7}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
2088.4.a.p 2088.a 1.a $11$ $123.196$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(0\) \(-5\) \(37\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{5}+(3-\beta _{3})q^{7}+(2+\beta _{2})q^{11}+\cdots\)
2088.4.a.q 2088.a 1.a $11$ $123.196$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(0\) \(5\) \(37\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+(3-\beta _{3})q^{7}+(-2-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2088))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2088)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(522))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(696))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1044))\)\(^{\oplus 2}\)