Properties

Label 90.4.a
Level $90$
Weight $4$
Character orbit 90.a
Rep. character $\chi_{90}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $72$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(72\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(90))\).

Total New Old
Modular forms 62 5 57
Cusp forms 46 5 41
Eisenstein series 16 0 16

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
Plus space\(+\)\(4\)
Minus space\(-\)\(1\)

Trace form

\( 5 q - 2 q^{2} + 20 q^{4} - 5 q^{5} + 52 q^{7} - 8 q^{8} + O(q^{10}) \) \( 5 q - 2 q^{2} + 20 q^{4} - 5 q^{5} + 52 q^{7} - 8 q^{8} + 10 q^{10} + 96 q^{11} + 46 q^{13} + 80 q^{14} + 80 q^{16} + 6 q^{17} + 52 q^{19} - 20 q^{20} + 24 q^{22} - 204 q^{23} + 125 q^{25} + 44 q^{26} + 208 q^{28} - 126 q^{29} - 296 q^{31} - 32 q^{32} - 492 q^{34} - 160 q^{35} - 842 q^{37} - 232 q^{38} + 40 q^{40} + 402 q^{41} - 440 q^{43} + 384 q^{44} - 72 q^{46} + 348 q^{47} - 267 q^{49} - 50 q^{50} + 184 q^{52} - 366 q^{53} - 60 q^{55} + 320 q^{56} - 276 q^{58} - 1320 q^{59} + 1438 q^{61} - 1312 q^{62} + 320 q^{64} + 470 q^{65} - 152 q^{67} + 24 q^{68} - 40 q^{70} + 216 q^{71} + 1570 q^{73} + 1004 q^{74} + 208 q^{76} + 1776 q^{77} + 1840 q^{79} - 80 q^{80} + 180 q^{82} + 1080 q^{83} + 330 q^{85} - 352 q^{86} + 96 q^{88} - 342 q^{89} + 1040 q^{91} - 816 q^{92} - 1656 q^{94} + 1580 q^{95} - 326 q^{97} + 2670 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(90))\) 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 5
90.4.a.a 90.a 1.a $1$ $5.310$ \(\Q\) None \(-2\) \(0\) \(-5\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-4q^{7}-8q^{8}+\cdots\)
90.4.a.b 90.a 1.a $1$ $5.310$ \(\Q\) None \(-2\) \(0\) \(-5\) \(14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+14q^{7}-8q^{8}+\cdots\)
90.4.a.c 90.a 1.a $1$ $5.310$ \(\Q\) None \(-2\) \(0\) \(5\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-4q^{7}-8q^{8}+\cdots\)
90.4.a.d 90.a 1.a $1$ $5.310$ \(\Q\) None \(2\) \(0\) \(-5\) \(32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+2^{5}q^{7}+8q^{8}+\cdots\)
90.4.a.e 90.a 1.a $1$ $5.310$ \(\Q\) None \(2\) \(0\) \(5\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+14q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)