Properties

Label 88.2.a
Level $88$
Weight $2$
Character orbit 88.a
Rep. character $\chi_{88}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $24$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 88.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(88))\).

Total New Old
Modular forms 16 3 13
Cusp forms 9 3 6
Eisenstein series 7 0 7

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.
\(+\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\( 3q - 2q^{3} - 4q^{7} + 9q^{9} + O(q^{10}) \) \( 3q - 2q^{3} - 4q^{7} + 9q^{9} - 3q^{11} - 2q^{13} + 2q^{15} - 2q^{17} - 4q^{19} - 12q^{21} + 10q^{23} + 7q^{25} - 2q^{27} - 10q^{29} - 14q^{31} + 2q^{33} + 20q^{35} - 12q^{37} + 16q^{39} + 10q^{41} - 22q^{45} + 8q^{47} + 19q^{49} + 20q^{51} + 10q^{53} - 16q^{57} - 6q^{59} - 2q^{61} - 32q^{63} - 20q^{65} + 10q^{67} + 10q^{69} - 2q^{71} + 18q^{73} - 36q^{75} + 4q^{77} - 12q^{79} - 5q^{81} + 8q^{83} + 24q^{85} + 40q^{87} + 8q^{89} - 32q^{91} + 26q^{93} - 24q^{95} + 20q^{97} - 9q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(88))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
88.2.a.a \(1\) \(0.703\) \(\Q\) None \(0\) \(-3\) \(-3\) \(-2\) \(+\) \(+\) \(q-3q^{3}-3q^{5}-2q^{7}+6q^{9}-q^{11}+\cdots\)
88.2.a.b \(2\) \(0.703\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(3\) \(-2\) \(-\) \(+\) \(q+\beta q^{3}+(2-\beta )q^{5}-2\beta q^{7}+(1+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(88))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 + T \))(\( ( 1 + T )^{2} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 26 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - T + 23 T^{2} \))(\( 1 - 9 T + 62 T^{2} - 207 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 8 T + 29 T^{2} \))(\( 1 + 2 T + 42 T^{2} + 58 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 7 T + 31 T^{2} \))(\( 1 + 7 T + 70 T^{2} + 217 T^{3} + 961 T^{4} \))
$37$ (\( 1 + T + 37 T^{2} \))(\( 1 + 11 T + 100 T^{2} + 407 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 4 T + 41 T^{2} \))(\( 1 - 6 T + 74 T^{2} - 246 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 78 T^{2} + 258 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))
$53$ (\( 1 - 2 T + 53 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + T + 59 T^{2} \))(\( 1 + 5 T + 18 T^{2} + 295 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 4 T + 61 T^{2} \))(\( 1 + 6 T + 114 T^{2} + 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 5 T + 67 T^{2} \))(\( 1 - 15 T + 186 T^{2} - 1005 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 3 T + 71 T^{2} \))(\( 1 + 5 T + 110 T^{2} + 355 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 16 T + 73 T^{2} \))(\( 1 - 2 T + 130 T^{2} - 146 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 2 T + 79 T^{2} \))(\( 1 + 14 T + 190 T^{2} + 1106 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 2 T + 83 T^{2} \))(\( 1 - 10 T + 174 T^{2} - 830 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 15 T + 89 T^{2} \))(\( 1 + 7 T + 152 T^{2} + 623 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 7 T + 97 T^{2} \))(\( 1 - 27 T + 372 T^{2} - 2619 T^{3} + 9409 T^{4} \))
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