Properties

Label 1089.4.a
Level $1089$
Weight $4$
Character orbit 1089.a
Rep. character $\chi_{1089}(1,\cdot)$
Character field $\Q$
Dimension $132$
Newform subspaces $40$
Sturm bound $528$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(528\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1089))\).

Total New Old
Modular forms 420 141 279
Cusp forms 372 132 240
Eisenstein series 48 9 39

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

\(3\)\(11\)FrickeDim
\(+\)\(+\)$+$\(30\)
\(+\)\(-\)$-$\(25\)
\(-\)\(+\)$-$\(37\)
\(-\)\(-\)$+$\(40\)
Plus space\(+\)\(70\)
Minus space\(-\)\(62\)

Trace form

\( 132 q - 2 q^{2} + 512 q^{4} - 18 q^{5} + 24 q^{7} + 24 q^{8} + O(q^{10}) \) \( 132 q - 2 q^{2} + 512 q^{4} - 18 q^{5} + 24 q^{7} + 24 q^{8} + 46 q^{10} - 54 q^{13} - 84 q^{14} + 1908 q^{16} + 28 q^{17} - 64 q^{19} - 552 q^{20} + 60 q^{23} + 2948 q^{25} + 540 q^{26} + 160 q^{28} + 24 q^{29} + 454 q^{31} + 260 q^{32} + 518 q^{34} + 748 q^{35} + 8 q^{37} + 78 q^{38} - 228 q^{40} - 224 q^{41} + 68 q^{43} + 2 q^{46} - 1290 q^{47} + 4970 q^{49} + 380 q^{50} - 8 q^{52} - 1074 q^{53} + 144 q^{56} + 2512 q^{58} + 144 q^{59} + 350 q^{61} + 3166 q^{62} + 7028 q^{64} - 1760 q^{65} + 1402 q^{67} + 1640 q^{68} + 1012 q^{70} + 582 q^{71} - 2370 q^{73} + 2262 q^{74} + 1208 q^{76} - 1904 q^{79} - 7296 q^{80} + 390 q^{82} + 732 q^{83} - 2180 q^{85} + 498 q^{86} - 2322 q^{89} - 528 q^{91} + 1164 q^{92} - 3064 q^{94} + 2784 q^{95} - 1838 q^{97} - 9450 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
1089.4.a.a 1089.a 1.a $1$ $64.253$ \(\Q\) None \(-5\) \(0\) \(14\) \(32\) $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}+14q^{5}+2^{5}q^{7}-45q^{8}+\cdots\)
1089.4.a.b 1089.a 1.a $1$ $64.253$ \(\Q\) None \(-4\) \(0\) \(13\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}+13q^{5}+26q^{7}-52q^{10}+\cdots\)
1089.4.a.c 1089.a 1.a $1$ $64.253$ \(\Q\) None \(-3\) \(0\) \(12\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+12q^{5}+12q^{7}+21q^{8}+\cdots\)
1089.4.a.d 1089.a 1.a $1$ $64.253$ \(\Q\) None \(-1\) \(0\) \(-7\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}-7q^{5}+4q^{7}+15q^{8}+\cdots\)
1089.4.a.e 1089.a 1.a $1$ $64.253$ \(\Q\) None \(-1\) \(0\) \(4\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+4q^{5}+26q^{7}+15q^{8}+\cdots\)
1089.4.a.f 1089.a 1.a $1$ $64.253$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(0\) \(-18\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}-18q^{5}+2^{6}q^{16}+12^{2}q^{20}+\cdots\)
1089.4.a.g 1089.a 1.a $1$ $64.253$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
1089.4.a.h 1089.a 1.a $1$ $64.253$ \(\Q\) None \(1\) \(0\) \(-7\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}-7q^{5}-4q^{7}-15q^{8}+\cdots\)
1089.4.a.i 1089.a 1.a $1$ $64.253$ \(\Q\) None \(3\) \(0\) \(12\) \(-12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+12q^{5}-12q^{7}-21q^{8}+\cdots\)
1089.4.a.j 1089.a 1.a $1$ $64.253$ \(\Q\) None \(4\) \(0\) \(13\) \(-26\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+8q^{4}+13q^{5}-26q^{7}+52q^{10}+\cdots\)
1089.4.a.k 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(10\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(5-\beta )q^{5}+\cdots\)
1089.4.a.l 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(20\) \(16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(10+\beta )q^{5}+\cdots\)
1089.4.a.m 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}-\beta q^{7}-2\beta q^{13}+2^{6}q^{16}+\cdots\)
1089.4.a.n 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-5q^{4}+3q^{5}-2\beta q^{7}-13\beta q^{8}+\cdots\)
1089.4.a.o 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-3q^{4}+2q^{5}-10\beta q^{7}+11\beta q^{8}+\cdots\)
1089.4.a.p 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(13\) \(-44\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-4\beta )q^{2}+12q^{4}+(1+11\beta )q^{5}+\cdots\)
1089.4.a.q 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(13\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-4\beta )q^{2}+12q^{4}+(12-11\beta )q^{5}+\cdots\)
1089.4.a.r 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{26}) \) None \(0\) \(0\) \(-10\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+18q^{4}-5q^{5}-4\beta q^{7}+10\beta q^{8}+\cdots\)
1089.4.a.s 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-18\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3\beta q^{2}+19q^{4}-9q^{5}-14\beta q^{7}+\cdots\)
1089.4.a.t 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{33}) \) None \(1\) \(0\) \(-16\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+\beta q^{4}+(-10+4\beta )q^{5}+(-2+\cdots)q^{7}+\cdots\)
1089.4.a.u 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{97}) \) None \(1\) \(0\) \(14\) \(-24\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2^{4}+\beta )q^{4}+(6+2\beta )q^{5}+(-14+\cdots)q^{7}+\cdots\)
1089.4.a.v 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(-20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}+(-1-8\beta )q^{5}+\cdots\)
1089.4.a.w 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-20\) \(16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(-10+\beta )q^{5}+\cdots\)
1089.4.a.x 1089.a 1.a $2$ $64.253$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(10\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(5+\beta )q^{5}+\cdots\)
1089.4.a.y 1089.a 1.a $4$ $64.253$ \(\Q(\sqrt{5}, \sqrt{37})\) None \(-4\) \(0\) \(11\) \(25\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}+\beta _{2})q^{2}+(3-3\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
1089.4.a.z 1089.a 1.a $4$ $64.253$ 4.4.5225.1 None \(-3\) \(0\) \(-12\) \(11\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}-\beta _{2})q^{2}+(-2+3\beta _{1}+\cdots)q^{4}+\cdots\)
1089.4.a.ba 1089.a 1.a $4$ $64.253$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-1\) \(0\) \(-14\) \(-20\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(6+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bb 1089.a 1.a $4$ $64.253$ \(\Q(\sqrt{3}, \sqrt{5})\) None \(0\) \(0\) \(-16\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{3})q^{2}+2\beta _{2}q^{4}+(-4+4\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bc 1089.a 1.a $4$ $64.253$ 4.4.14221152.1 None \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+\beta _{2}q^{7}+\cdots\)
1089.4.a.bd 1089.a 1.a $4$ $64.253$ 4.4.14221152.1 None \(0\) \(0\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}-\beta _{2}q^{7}+\cdots\)
1089.4.a.be 1089.a 1.a $4$ $64.253$ \(\Q(\sqrt{5}, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+9q^{4}-\beta _{3}q^{5}+\beta _{2}q^{7}+\beta _{1}q^{8}+\cdots\)
1089.4.a.bf 1089.a 1.a $4$ $64.253$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(1\) \(0\) \(-14\) \(20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bg 1089.a 1.a $4$ $64.253$ 4.4.5225.1 None \(3\) \(0\) \(-12\) \(-11\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(-2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
1089.4.a.bh 1089.a 1.a $4$ $64.253$ \(\Q(\sqrt{5}, \sqrt{37})\) None \(4\) \(0\) \(11\) \(-25\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(3+\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
1089.4.a.bi 1089.a 1.a $6$ $64.253$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-5\) \(0\) \(-9\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.4.a.bj 1089.a 1.a $6$ $64.253$ 6.6.\(\cdots\).1 None \(0\) \(0\) \(-14\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{3}+\beta _{4})q^{4}+(-2-2\beta _{3}+\cdots)q^{5}+\cdots\)
1089.4.a.bk 1089.a 1.a $6$ $64.253$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(5\) \(0\) \(-9\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1089.4.a.bl 1089.a 1.a $12$ $64.253$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-66\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
1089.4.a.bm 1089.a 1.a $12$ $64.253$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(66\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(\beta _{1}+\beta _{5})q^{5}+\cdots\)
1089.4.a.bn 1089.a 1.a $12$ $64.253$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6+\beta _{7})q^{4}+\beta _{4}q^{5}+(-2\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1089))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1089)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)