Properties

Label 90.12.a
Level $90$
Weight $12$
Character orbit 90.a
Rep. character $\chi_{90}(1,\cdot)$
Character field $\Q$
Dimension $17$
Newform subspaces $14$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 90.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(90))\).

Total New Old
Modular forms 206 17 189
Cusp forms 190 17 173
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
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(10\)
Minus space\(-\)\(7\)

Trace form

\( 17 q - 32 q^{2} + 17408 q^{4} - 3125 q^{5} - 22880 q^{7} - 32768 q^{8} + 100000 q^{10} - 1453056 q^{11} + 1463038 q^{13} + 310016 q^{14} + 17825792 q^{16} + 8801982 q^{17} - 40388924 q^{19} - 3200000 q^{20}+ \cdots - 274465665312 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(90))\) 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
90.12.a.a 90.a 1.a $1$ $69.151$ \(\Q\) None 30.12.a.e \(-32\) \(0\) \(-3125\) \(-5152\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-5152q^{7}+\cdots\)
90.12.a.b 90.a 1.a $1$ $69.151$ \(\Q\) None 10.12.a.c \(-32\) \(0\) \(3125\) \(-70714\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}-70714q^{7}+\cdots\)
90.12.a.c 90.a 1.a $1$ $69.151$ \(\Q\) None 90.12.a.c \(-32\) \(0\) \(3125\) \(-14014\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}-14014q^{7}+\cdots\)
90.12.a.d 90.a 1.a $1$ $69.151$ \(\Q\) None 30.12.a.f \(-32\) \(0\) \(3125\) \(10556\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+10556q^{7}+\cdots\)
90.12.a.e 90.a 1.a $1$ $69.151$ \(\Q\) None 30.12.a.d \(-32\) \(0\) \(3125\) \(29348\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+29348q^{7}+\cdots\)
90.12.a.f 90.a 1.a $1$ $69.151$ \(\Q\) None 30.12.a.b \(32\) \(0\) \(-3125\) \(-57376\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-57376q^{7}+\cdots\)
90.12.a.g 90.a 1.a $1$ $69.151$ \(\Q\) None 10.12.a.a \(32\) \(0\) \(-3125\) \(-14176\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-14176q^{7}+\cdots\)
90.12.a.h 90.a 1.a $1$ $69.151$ \(\Q\) None 90.12.a.c \(32\) \(0\) \(-3125\) \(-14014\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-14014q^{7}+\cdots\)
90.12.a.i 90.a 1.a $1$ $69.151$ \(\Q\) None 30.12.a.c \(32\) \(0\) \(-3125\) \(56672\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}+56672q^{7}+\cdots\)
90.12.a.j 90.a 1.a $1$ $69.151$ \(\Q\) None 30.12.a.a \(32\) \(0\) \(3125\) \(-22876\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}-22876q^{7}+\cdots\)
90.12.a.k 90.a 1.a $1$ $69.151$ \(\Q\) None 10.12.a.b \(32\) \(0\) \(3125\) \(25574\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+25574q^{7}+\cdots\)
90.12.a.l 90.a 1.a $2$ $69.151$ \(\Q(\sqrt{1969}) \) None 10.12.a.d \(-64\) \(0\) \(-6250\) \(14092\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}+(7046+\cdots)q^{7}+\cdots\)
90.12.a.m 90.a 1.a $2$ $69.151$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 90.12.a.m \(-64\) \(0\) \(-6250\) \(19600\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}+(9800+\cdots)q^{7}+\cdots\)
90.12.a.n 90.a 1.a $2$ $69.151$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 90.12.a.m \(64\) \(0\) \(6250\) \(19600\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+(9800+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

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