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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 11
1089.4.a.a \(1\) \(64.253\) \(\Q\) None \(-5\) \(0\) \(14\) \(32\) \(-\) \(-\) \(q-5q^{2}+17q^{4}+14q^{5}+2^{5}q^{7}-45q^{8}+\cdots\)
1089.4.a.b \(1\) \(64.253\) \(\Q\) None \(-4\) \(0\) \(13\) \(26\) \(-\) \(-\) \(q-4q^{2}+8q^{4}+13q^{5}+26q^{7}-52q^{10}+\cdots\)
1089.4.a.c \(1\) \(64.253\) \(\Q\) None \(-3\) \(0\) \(12\) \(12\) \(-\) \(+\) \(q-3q^{2}+q^{4}+12q^{5}+12q^{7}+21q^{8}+\cdots\)
1089.4.a.d \(1\) \(64.253\) \(\Q\) None \(-1\) \(0\) \(-7\) \(4\) \(-\) \(-\) \(q-q^{2}-7q^{4}-7q^{5}+4q^{7}+15q^{8}+\cdots\)
1089.4.a.e \(1\) \(64.253\) \(\Q\) None \(-1\) \(0\) \(4\) \(26\) \(-\) \(-\) \(q-q^{2}-7q^{4}+4q^{5}+26q^{7}+15q^{8}+\cdots\)
1089.4.a.f \(1\) \(64.253\) \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(0\) \(-18\) \(0\) \(-\) \(+\) \(q-8q^{4}-18q^{5}+2^{6}q^{16}+12^{2}q^{20}+\cdots\)
1089.4.a.g \(1\) \(64.253\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) \(+\) \(-\) \(q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
1089.4.a.h \(1\) \(64.253\) \(\Q\) None \(1\) \(0\) \(-7\) \(-4\) \(-\) \(-\) \(q+q^{2}-7q^{4}-7q^{5}-4q^{7}-15q^{8}+\cdots\)
1089.4.a.i \(1\) \(64.253\) \(\Q\) None \(3\) \(0\) \(12\) \(-12\) \(-\) \(+\) \(q+3q^{2}+q^{4}+12q^{5}-12q^{7}-21q^{8}+\cdots\)
1089.4.a.j \(1\) \(64.253\) \(\Q\) None \(4\) \(0\) \(13\) \(-26\) \(-\) \(-\) \(q+4q^{2}+8q^{4}+13q^{5}-26q^{7}+52q^{10}+\cdots\)
1089.4.a.k \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(10\) \(8\) \(-\) \(-\) \(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(5-\beta )q^{5}+\cdots\)
1089.4.a.l \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(20\) \(16\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(10+\beta )q^{5}+\cdots\)
1089.4.a.m \(2\) \(64.253\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-8q^{4}-\beta q^{7}-2\beta q^{13}+2^{6}q^{16}+\cdots\)
1089.4.a.n \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(6\) \(0\) \(-\) \(+\) \(q+\beta q^{2}-5q^{4}+3q^{5}-2\beta q^{7}-13\beta q^{8}+\cdots\)
1089.4.a.o \(2\) \(64.253\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(4\) \(0\) \(-\) \(+\) \(q-\beta q^{2}-3q^{4}+2q^{5}-10\beta q^{7}+11\beta q^{8}+\cdots\)
1089.4.a.p \(2\) \(64.253\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(13\) \(-44\) \(-\) \(+\) \(q+(2-4\beta )q^{2}+12q^{4}+(1+11\beta )q^{5}+\cdots\)
1089.4.a.q \(2\) \(64.253\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(13\) \(44\) \(-\) \(-\) \(q+(2-4\beta )q^{2}+12q^{4}+(12-11\beta )q^{5}+\cdots\)
1089.4.a.r \(2\) \(64.253\) \(\Q(\sqrt{26}) \) None \(0\) \(0\) \(-10\) \(0\) \(-\) \(+\) \(q+\beta q^{2}+18q^{4}-5q^{5}-4\beta q^{7}+10\beta q^{8}+\cdots\)
1089.4.a.s \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-18\) \(0\) \(-\) \(+\) \(q+3\beta q^{2}+19q^{4}-9q^{5}-14\beta q^{7}+\cdots\)
1089.4.a.t \(2\) \(64.253\) \(\Q(\sqrt{33}) \) None \(1\) \(0\) \(-16\) \(-2\) \(-\) \(-\) \(q+\beta q^{2}+\beta q^{4}+(-10+4\beta )q^{5}+(-2+\cdots)q^{7}+\cdots\)
1089.4.a.u \(2\) \(64.253\) \(\Q(\sqrt{97}) \) None \(1\) \(0\) \(14\) \(-24\) \(-\) \(-\) \(q+\beta q^{2}+(2^{4}+\beta )q^{4}+(6+2\beta )q^{5}+(-14+\cdots)q^{7}+\cdots\)
1089.4.a.v \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(-20\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}+(-1-8\beta )q^{5}+\cdots\)
1089.4.a.w \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-20\) \(16\) \(+\) \(-\) \(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(-10+\beta )q^{5}+\cdots\)
1089.4.a.x \(2\) \(64.253\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(10\) \(-8\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(5+\beta )q^{5}+\cdots\)
1089.4.a.y \(4\) \(64.253\) \(\Q(\sqrt{5}, \sqrt{37})\) None \(-4\) \(0\) \(11\) \(25\) \(-\) \(+\) \(q+(-1+\beta _{1}+\beta _{2})q^{2}+(3-3\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
1089.4.a.z \(4\) \(64.253\) 4.4.5225.1 None \(-3\) \(0\) \(-12\) \(11\) \(-\) \(-\) \(q+(-1-\beta _{1}-\beta _{2})q^{2}+(-2+3\beta _{1}+\cdots)q^{4}+\cdots\)
1089.4.a.ba \(4\) \(64.253\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-1\) \(0\) \(-14\) \(-20\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(6+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bb \(4\) \(64.253\) \(\Q(\sqrt{3}, \sqrt{5})\) None \(0\) \(0\) \(-16\) \(0\) \(-\) \(+\) \(q+(-\beta _{1}+\beta _{3})q^{2}+2\beta _{2}q^{4}+(-4+4\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bc \(4\) \(64.253\) 4.4.14221152.1 None \(0\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+\beta _{2}q^{7}+\cdots\)
1089.4.a.bd \(4\) \(64.253\) 4.4.14221152.1 None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}-\beta _{2}q^{7}+\cdots\)
1089.4.a.be \(4\) \(64.253\) \(\Q(\sqrt{5}, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+9q^{4}-\beta _{3}q^{5}+\beta _{2}q^{7}+\beta _{1}q^{8}+\cdots\)
1089.4.a.bf \(4\) \(64.253\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(1\) \(0\) \(-14\) \(20\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(6+\beta _{1}+\beta _{3})q^{4}+(-3+\beta _{2}+\cdots)q^{5}+\cdots\)
1089.4.a.bg \(4\) \(64.253\) 4.4.5225.1 None \(3\) \(0\) \(-12\) \(-11\) \(-\) \(+\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(-2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
1089.4.a.bh \(4\) \(64.253\) \(\Q(\sqrt{5}, \sqrt{37})\) None \(4\) \(0\) \(11\) \(-25\) \(-\) \(-\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(3+\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
1089.4.a.bi \(6\) \(64.253\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-5\) \(0\) \(-9\) \(1\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.4.a.bj \(6\) \(64.253\) 6.6.\(\cdots\).1 None \(0\) \(0\) \(-14\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(3+\beta _{3}+\beta _{4})q^{4}+(-2-2\beta _{3}+\cdots)q^{5}+\cdots\)
1089.4.a.bk \(6\) \(64.253\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(5\) \(0\) \(-9\) \(-1\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1089.4.a.bl \(12\) \(64.253\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-66\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
1089.4.a.bm \(12\) \(64.253\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(66\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(\beta _{1}+\beta _{5})q^{5}+\cdots\)
1089.4.a.bn \(12\) \(64.253\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(6+\beta _{7})q^{4}+\beta _{4}q^{5}+(-2\beta _{2}+\cdots)q^{7}+\cdots\)

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

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