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Properties

Label 90.10.a
Level $90$
Weight $10$
Character orbit 90.a
Rep. character $\chi_{90}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $13$
Sturm bound $180$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 90.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(90))\).

Total New Old
Modular forms 170 15 155
Cusp forms 154 15 139
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
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(9\)

Trace form

\( 15 q + 16 q^{2} + 3840 q^{4} + 625 q^{5} + 2964 q^{7} + 4096 q^{8} + O(q^{10}) \) \( 15 q + 16 q^{2} + 3840 q^{4} + 625 q^{5} + 2964 q^{7} + 4096 q^{8} - 10000 q^{10} - 64992 q^{11} + 48138 q^{13} + 25280 q^{14} + 983040 q^{16} - 24078 q^{17} + 806244 q^{19} + 160000 q^{20} - 629952 q^{22} - 4102332 q^{23} + 5859375 q^{25} + 5813792 q^{26} + 758784 q^{28} - 17722962 q^{29} + 21736152 q^{31} + 1048576 q^{32} - 15037344 q^{34} + 12935000 q^{35} - 17949774 q^{37} + 25287104 q^{38} - 2560000 q^{40} + 37442526 q^{41} + 4256880 q^{43} - 16637952 q^{44} + 7152384 q^{46} + 48303996 q^{47} + 243290151 q^{49} + 6250000 q^{50} + 12323328 q^{52} + 31135782 q^{53} - 31567500 q^{55} + 6471680 q^{56} + 180494112 q^{58} + 78698280 q^{59} - 257625846 q^{61} + 22123136 q^{62} + 251658240 q^{64} + 52951250 q^{65} + 1261574976 q^{67} - 6163968 q^{68} - 204080000 q^{70} - 307140552 q^{71} + 345397110 q^{73} + 170620448 q^{74} + 206398464 q^{76} - 816249408 q^{77} - 242998320 q^{79} + 40960000 q^{80} + 765195360 q^{82} + 954869040 q^{83} + 636138750 q^{85} + 182538944 q^{86} - 161267712 q^{88} + 1402781166 q^{89} + 3436645440 q^{91} - 1050196992 q^{92} - 744367872 q^{94} + 365697500 q^{95} - 1913802402 q^{97} + 1512508560 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(90))\) 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
90.10.a.a 90.a 1.a $1$ $46.353$ \(\Q\) None \(-16\) \(0\) \(-625\) \(-10336\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}-5^{4}q^{5}-10336q^{7}+\cdots\)
90.10.a.b 90.a 1.a $1$ $46.353$ \(\Q\) None \(-16\) \(0\) \(-625\) \(-2002\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}-5^{4}q^{5}-2002q^{7}+\cdots\)
90.10.a.c 90.a 1.a $1$ $46.353$ \(\Q\) None \(-16\) \(0\) \(-625\) \(2408\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}-5^{4}q^{5}+2408q^{7}+\cdots\)
90.10.a.d 90.a 1.a $1$ $46.353$ \(\Q\) None \(-16\) \(0\) \(625\) \(3164\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}+3164q^{7}+\cdots\)
90.10.a.e 90.a 1.a $1$ $46.353$ \(\Q\) None \(-16\) \(0\) \(625\) \(4658\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}+4658q^{7}+\cdots\)
90.10.a.f 90.a 1.a $1$ $46.353$ \(\Q\) None \(16\) \(0\) \(-625\) \(-7168\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}-5^{4}q^{5}-7168q^{7}+\cdots\)
90.10.a.g 90.a 1.a $1$ $46.353$ \(\Q\) None \(16\) \(0\) \(-625\) \(5432\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}-5^{4}q^{5}+5432q^{7}+\cdots\)
90.10.a.h 90.a 1.a $1$ $46.353$ \(\Q\) None \(16\) \(0\) \(625\) \(-10318\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}-10318q^{7}+\cdots\)
90.10.a.i 90.a 1.a $1$ $46.353$ \(\Q\) None \(16\) \(0\) \(625\) \(-2002\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}-2002q^{7}+\cdots\)
90.10.a.j 90.a 1.a $1$ $46.353$ \(\Q\) None \(16\) \(0\) \(625\) \(6332\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}+6332q^{7}+\cdots\)
90.10.a.k 90.a 1.a $1$ $46.353$ \(\Q\) None \(16\) \(0\) \(625\) \(7196\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}+7196q^{7}+\cdots\)
90.10.a.l 90.a 1.a $2$ $46.353$ \(\Q(\sqrt{315721}) \) None \(-32\) \(0\) \(1250\) \(2800\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+2^{8}q^{4}+5^{4}q^{5}+(1400-\beta )q^{7}+\cdots\)
90.10.a.m 90.a 1.a $2$ $46.353$ \(\Q(\sqrt{315721}) \) None \(32\) \(0\) \(-1250\) \(2800\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+2^{8}q^{4}-5^{4}q^{5}+(1400-\beta )q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)