Properties

Label 90.18.a
Level $90$
Weight $18$
Character orbit 90.a
Rep. character $\chi_{90}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $16$
Sturm bound $324$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 90.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(324\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(90))\).

Total New Old
Modular forms 314 27 287
Cusp forms 298 27 271
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
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(12\)
Minus space\(-\)\(15\)

Trace form

\( 27 q + 256 q^{2} + 1769472 q^{4} + 390625 q^{5} - 2868240 q^{7} + 16777216 q^{8} - 100000000 q^{10} + 1337372256 q^{11} - 9935909766 q^{13} - 773334016 q^{14} + 115964116992 q^{16} + 66106268874 q^{17}+ \cdots + 25\!\cdots\!04 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(90))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
90.18.a.a 90.a 1.a $1$ $164.900$ \(\Q\) None 30.18.a.e \(-256\) \(0\) \(-390625\) \(-12193216\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}-12193216q^{7}+\cdots\)
90.18.a.b 90.a 1.a $1$ $164.900$ \(\Q\) None 10.18.a.a \(-256\) \(0\) \(-390625\) \(14808668\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+14808668q^{7}+\cdots\)
90.18.a.c 90.a 1.a $1$ $164.900$ \(\Q\) None 30.18.a.f \(-256\) \(0\) \(390625\) \(-16972732\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}-16972732q^{7}+\cdots\)
90.18.a.d 90.a 1.a $1$ $164.900$ \(\Q\) None 30.18.a.d \(-256\) \(0\) \(390625\) \(-7079716\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}-7079716q^{7}+\cdots\)
90.18.a.e 90.a 1.a $1$ $164.900$ \(\Q\) None 30.18.a.b \(256\) \(0\) \(-390625\) \(-2533888\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}-2533888q^{7}+\cdots\)
90.18.a.f 90.a 1.a $1$ $164.900$ \(\Q\) None 30.18.a.c \(256\) \(0\) \(-390625\) \(1929536\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+1929536q^{7}+\cdots\)
90.18.a.g 90.a 1.a $1$ $164.900$ \(\Q\) None 30.18.a.a \(256\) \(0\) \(390625\) \(2579612\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+2579612q^{7}+\cdots\)
90.18.a.h 90.a 1.a $2$ $164.900$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 90.18.a.h \(-512\) \(0\) \(-781250\) \(-12328400\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(-6164200+\cdots)q^{7}+\cdots\)
90.18.a.i 90.a 1.a $2$ $164.900$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 30.18.a.h \(-512\) \(0\) \(-781250\) \(6956296\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(3478148+\cdots)q^{7}+\cdots\)
90.18.a.j 90.a 1.a $2$ $164.900$ \(\Q(\sqrt{2941}) \) None 10.18.a.d \(-512\) \(0\) \(781250\) \(27684196\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(13842098+\cdots)q^{7}+\cdots\)
90.18.a.k 90.a 1.a $2$ $164.900$ \(\Q(\sqrt{83281}) \) None 10.18.a.c \(512\) \(0\) \(-781250\) \(603844\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(301922+\cdots)q^{7}+\cdots\)
90.18.a.l 90.a 1.a $2$ $164.900$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 90.18.a.h \(512\) \(0\) \(781250\) \(-12328400\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(-6164200+\cdots)q^{7}+\cdots\)
90.18.a.m 90.a 1.a $2$ $164.900$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 30.18.a.g \(512\) \(0\) \(781250\) \(1059712\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(529856+\cdots)q^{7}+\cdots\)
90.18.a.n 90.a 1.a $2$ $164.900$ \(\Q(\sqrt{36061}) \) None 10.18.a.b \(512\) \(0\) \(781250\) \(6543844\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(3271922+\cdots)q^{7}+\cdots\)
90.18.a.o 90.a 1.a $3$ $164.900$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 90.18.a.o \(-768\) \(0\) \(1171875\) \(-798798\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(-266266+\cdots)q^{7}+\cdots\)
90.18.a.p 90.a 1.a $3$ $164.900$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 90.18.a.o \(768\) \(0\) \(-1171875\) \(-798798\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(-266266+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(90)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)