Properties

Label 88.4.a
Level $88$
Weight $4$
Character orbit 88.a
Rep. character $\chi_{88}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $4$
Sturm bound $48$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 88.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(88))\).

Total New Old
Modular forms 40 7 33
Cusp forms 32 7 25
Eisenstein series 8 0 8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(11\)FrickeDim
\(+\)\(+\)$+$\(1\)
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(1\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(3\)

Trace form

\( 7 q + 6 q^{3} + 4 q^{5} - 36 q^{7} + 73 q^{9} + O(q^{10}) \) \( 7 q + 6 q^{3} + 4 q^{5} - 36 q^{7} + 73 q^{9} + 33 q^{11} - 18 q^{13} - 94 q^{15} + 86 q^{17} - 180 q^{19} + 316 q^{21} + 66 q^{23} + 287 q^{25} - 282 q^{27} - 362 q^{29} + 250 q^{31} - 66 q^{33} - 236 q^{35} + 192 q^{37} - 352 q^{39} + 250 q^{41} - 704 q^{43} - 54 q^{45} - 608 q^{47} - 9 q^{49} + 468 q^{51} - 934 q^{53} + 864 q^{57} + 1170 q^{59} - 674 q^{61} + 1904 q^{63} + 668 q^{65} - 718 q^{67} - 1274 q^{69} + 374 q^{71} + 290 q^{73} + 1500 q^{75} - 308 q^{77} + 820 q^{79} + 623 q^{81} + 568 q^{83} - 3360 q^{85} - 760 q^{87} + 1540 q^{89} - 320 q^{91} - 3994 q^{93} + 824 q^{95} - 1672 q^{97} + 891 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(88))\) 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 11
88.4.a.a 88.a 1.a $1$ $5.192$ \(\Q\) None \(0\) \(-1\) \(-7\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-7q^{5}-6q^{7}-26q^{9}-11q^{11}+\cdots\)
88.4.a.b 88.a 1.a $1$ $5.192$ \(\Q\) None \(0\) \(7\) \(9\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+7q^{3}+9q^{5}+2q^{7}+22q^{9}-11q^{11}+\cdots\)
88.4.a.c 88.a 1.a $2$ $5.192$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-6\) \(-56\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(-3+4\beta )q^{5}+(-28+\cdots)q^{7}+\cdots\)
88.4.a.d 88.a 1.a $3$ $5.192$ 3.3.11109.1 None \(0\) \(2\) \(8\) \(24\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(8-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(88))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(88)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)