Properties

Label 1089.2.a
Level $1089$
Weight $2$
Character orbit 1089.a
Rep. character $\chi_{1089}(1,\cdot)$
Character field $\Q$
Dimension $41$
Newform subspaces $23$
Sturm bound $264$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(264\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1089))\).

Total New Old
Modular forms 156 50 106
Cusp forms 109 41 68
Eisenstein series 47 9 38

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

\(3\)\(11\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(36\)\(6\)\(30\)\(25\)\(6\)\(19\)\(11\)\(0\)\(11\)
\(+\)\(-\)\(-\)\(42\)\(12\)\(30\)\(30\)\(12\)\(18\)\(12\)\(0\)\(12\)
\(-\)\(+\)\(-\)\(42\)\(17\)\(25\)\(30\)\(13\)\(17\)\(12\)\(4\)\(8\)
\(-\)\(-\)\(+\)\(36\)\(15\)\(21\)\(24\)\(10\)\(14\)\(12\)\(5\)\(7\)
Plus space\(+\)\(72\)\(21\)\(51\)\(49\)\(16\)\(33\)\(23\)\(5\)\(18\)
Minus space\(-\)\(84\)\(29\)\(55\)\(60\)\(25\)\(35\)\(24\)\(4\)\(20\)

Trace form

\( 41 q - q^{2} + 41 q^{4} - 2 q^{5} + 2 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{13} + 12 q^{14} + 41 q^{16} - 4 q^{17} + 12 q^{19} + 4 q^{20} + 18 q^{23} + 11 q^{25} + 16 q^{26} + 4 q^{28} - 6 q^{29} - 2 q^{31}+ \cdots + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
1089.2.a.a 1089.a 1.a $1$ $8.696$ \(\Q\) None 363.2.a.a \(-2\) \(0\) \(-4\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-4q^{5}-q^{7}+8q^{10}+\cdots\)
1089.2.a.b 1089.a 1.a $1$ $8.696$ \(\Q\) None 11.2.a.a \(-2\) \(0\) \(-1\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}+2q^{7}+2q^{10}+\cdots\)
1089.2.a.c 1089.a 1.a $1$ $8.696$ \(\Q\) None 121.2.a.a \(-1\) \(0\) \(-1\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}-2q^{7}+3q^{8}+q^{10}+\cdots\)
1089.2.a.d 1089.a 1.a $1$ $8.696$ \(\Q\) None 99.2.a.a \(-1\) \(0\) \(4\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+4q^{5}+2q^{7}+3q^{8}-4q^{10}+\cdots\)
1089.2.a.e 1089.a 1.a $1$ $8.696$ \(\Q\) \(\Q(\sqrt{-3}) \) 1089.2.a.e \(0\) \(0\) \(0\) \(-5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}-5q^{7}-2q^{13}+4q^{16}+7q^{19}+\cdots\)
1089.2.a.f 1089.a 1.a $1$ $8.696$ \(\Q\) \(\Q(\sqrt{-3}) \) 1089.2.a.e \(0\) \(0\) \(0\) \(5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+5q^{7}+2q^{13}+4q^{16}-7q^{19}+\cdots\)
1089.2.a.g 1089.a 1.a $1$ $8.696$ \(\Q\) \(\Q(\sqrt{-11}) \) 121.2.a.b \(0\) \(0\) \(3\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+3q^{5}+4q^{16}-6q^{20}+9q^{23}+\cdots\)
1089.2.a.h 1089.a 1.a $1$ $8.696$ \(\Q\) None 99.2.a.a \(1\) \(0\) \(-4\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-4q^{5}+2q^{7}-3q^{8}-4q^{10}+\cdots\)
1089.2.a.i 1089.a 1.a $1$ $8.696$ \(\Q\) None 121.2.a.a \(1\) \(0\) \(-1\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}+2q^{7}-3q^{8}-q^{10}+\cdots\)
1089.2.a.j 1089.a 1.a $1$ $8.696$ \(\Q\) None 33.2.a.a \(1\) \(0\) \(2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{5}-4q^{7}-3q^{8}+2q^{10}+\cdots\)
1089.2.a.k 1089.a 1.a $1$ $8.696$ \(\Q\) None 363.2.a.a \(2\) \(0\) \(-4\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-4q^{5}+q^{7}-8q^{10}+\cdots\)
1089.2.a.l 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{5}) \) None 33.2.e.b \(-3\) \(0\) \(-1\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3\beta q^{4}+(-1+\beta )q^{5}+\cdots\)
1089.2.a.m 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{5}) \) None 33.2.e.a \(-1\) \(0\) \(3\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(2-\beta )q^{5}+3q^{7}+\cdots\)
1089.2.a.n 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) 1089.2.a.n \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}-\beta q^{7}+4\beta q^{13}+4q^{16}-3\beta q^{19}+\cdots\)
1089.2.a.o 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{3}) \) None 363.2.a.f \(0\) \(0\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+3q^{5}+2\beta q^{7}-\beta q^{8}+\cdots\)
1089.2.a.p 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{5}) \) None 363.2.a.g \(0\) \(0\) \(-4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}-2q^{5}-2\beta q^{7}-\beta q^{8}+\cdots\)
1089.2.a.q 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{7}) \) None 1089.2.a.q \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+5q^{4}+\beta q^{5}-2q^{7}+3\beta q^{8}+\cdots\)
1089.2.a.r 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{7}) \) None 1089.2.a.q \(0\) \(0\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+5q^{4}-\beta q^{5}+2q^{7}+3\beta q^{8}+\cdots\)
1089.2.a.s 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{5}) \) None 33.2.e.a \(1\) \(0\) \(3\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(2-\beta )q^{5}-3q^{7}+\cdots\)
1089.2.a.t 1089.a 1.a $2$ $8.696$ \(\Q(\sqrt{5}) \) None 33.2.e.b \(3\) \(0\) \(-1\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3\beta q^{4}+(-1+\beta )q^{5}+\cdots\)
1089.2.a.u 1089.a 1.a $4$ $8.696$ \(\Q(\sqrt{3}, \sqrt{11})\) None 363.2.a.j \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}+\beta _{3}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1089.2.a.v 1089.a 1.a $4$ $8.696$ 4.4.4400.1 None 99.2.f.c \(0\) \(0\) \(0\) \(-8\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{1}q^{5}+(-3+\cdots)q^{7}+\cdots\)
1089.2.a.w 1089.a 1.a $4$ $8.696$ 4.4.4400.1 None 99.2.f.c \(0\) \(0\) \(0\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{1}q^{5}+(3+2\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1089))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1089)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)