Properties

Label 1089.6.a
Level $1089$
Weight $6$
Character orbit 1089.a
Rep. character $\chi_{1089}(1,\cdot)$
Character field $\Q$
Dimension $222$
Newform subspaces $41$
Sturm bound $792$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 41 \)
Sturm bound: \(792\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1089))\).

Total New Old
Modular forms 684 231 453
Cusp forms 636 222 414
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.
\(+\)\(+\)\(+\)\(42\)
\(+\)\(-\)\(-\)\(48\)
\(-\)\(+\)\(-\)\(67\)
\(-\)\(-\)\(+\)\(65\)
Plus space\(+\)\(107\)
Minus space\(-\)\(115\)

Trace form

\( 222q - 2q^{2} + 3456q^{4} + 72q^{5} + 18q^{7} - 312q^{8} + O(q^{10}) \) \( 222q - 2q^{2} + 3456q^{4} + 72q^{5} + 18q^{7} - 312q^{8} + 158q^{10} + 252q^{13} + 1308q^{14} + 53940q^{16} - 1958q^{17} + 1960q^{19} + 2592q^{20} - 3630q^{23} + 131622q^{25} - 4704q^{26} + 7616q^{28} - 10392q^{29} - 19378q^{31} - 2476q^{32} - 1122q^{34} - 14666q^{35} - 2920q^{37} - 4566q^{38} + 3660q^{40} + 36784q^{41} + 34318q^{43} + 41746q^{46} + 29208q^{47} + 482702q^{49} - 66964q^{50} - 70984q^{52} + 52806q^{53} + 30576q^{56} - 70140q^{58} - 7626q^{59} + 100900q^{61} + 28990q^{62} + 705684q^{64} - 111020q^{65} + 52442q^{67} - 131896q^{68} + 75780q^{70} + 52554q^{71} + 72492q^{73} + 142806q^{74} + 263752q^{76} + 122042q^{79} + 282192q^{80} - 47874q^{82} - 112614q^{83} - 230050q^{85} + 521262q^{86} - 83460q^{89} + 223196q^{91} - 255492q^{92} + 443128q^{94} - 252408q^{95} - 305712q^{97} - 66570q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 11
1089.6.a.a \(1\) \(174.658\) \(\Q\) None \(-9\) \(0\) \(-24\) \(-72\) \(-\) \(+\) \(q-9q^{2}+7^{2}q^{4}-24q^{5}-72q^{7}-153q^{8}+\cdots\)
1089.6.a.b \(1\) \(174.658\) \(\Q\) None \(-6\) \(0\) \(-6\) \(40\) \(-\) \(-\) \(q-6q^{2}+4q^{4}-6q^{5}+40q^{7}+168q^{8}+\cdots\)
1089.6.a.c \(1\) \(174.658\) \(\Q\) None \(-4\) \(0\) \(19\) \(-10\) \(-\) \(-\) \(q-4q^{2}-2^{4}q^{4}+19q^{5}-10q^{7}+192q^{8}+\cdots\)
1089.6.a.d \(1\) \(174.658\) \(\Q\) None \(-2\) \(0\) \(-46\) \(-148\) \(-\) \(-\) \(q-2q^{2}-28q^{4}-46q^{5}-148q^{7}+\cdots\)
1089.6.a.e \(1\) \(174.658\) \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(0\) \(-57\) \(0\) \(-\) \(+\) \(q-2^{5}q^{4}-57q^{5}+2^{10}q^{16}+1824q^{20}+\cdots\)
1089.6.a.f \(1\) \(174.658\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-25\) \(+\) \(-\) \(q-2^{5}q^{4}-5^{2}q^{7}+1202q^{13}+2^{10}q^{16}+\cdots\)
1089.6.a.g \(1\) \(174.658\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(25\) \(+\) \(-\) \(q-2^{5}q^{4}+5^{2}q^{7}-1202q^{13}+2^{10}q^{16}+\cdots\)
1089.6.a.h \(1\) \(174.658\) \(\Q\) None \(1\) \(0\) \(92\) \(26\) \(-\) \(-\) \(q+q^{2}-31q^{4}+92q^{5}+26q^{7}-63q^{8}+\cdots\)
1089.6.a.i \(1\) \(174.658\) \(\Q\) None \(9\) \(0\) \(-24\) \(72\) \(-\) \(+\) \(q+9q^{2}+7^{2}q^{4}-24q^{5}+72q^{7}+153q^{8}+\cdots\)
1089.6.a.j \(2\) \(174.658\) \(\Q(\sqrt{177}) \) None \(-5\) \(0\) \(-58\) \(286\) \(-\) \(-\) \(q+(-2-\beta )q^{2}+(2^{4}+5\beta )q^{4}+(-24+\cdots)q^{5}+\cdots\)
1089.6.a.k \(2\) \(174.658\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-2^{5}q^{4}-149\beta q^{7}+116\beta q^{13}+2^{10}q^{16}+\cdots\)
1089.6.a.l \(2\) \(174.658\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(196\) \(0\) \(-\) \(+\) \(q-\beta q^{2}-12q^{4}+98q^{5}+53\beta q^{7}+\cdots\)
1089.6.a.m \(2\) \(174.658\) \(\Q(\sqrt{38}) \) None \(0\) \(0\) \(38\) \(0\) \(-\) \(+\) \(q+\beta q^{2}+6q^{4}+19q^{5}+8\beta q^{7}-26\beta q^{8}+\cdots\)
1089.6.a.n \(2\) \(174.658\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-48\) \(0\) \(-\) \(+\) \(q+4\beta q^{2}+2^{4}q^{4}-24q^{5}-\beta q^{7}-2^{6}\beta q^{8}+\cdots\)
1089.6.a.o \(2\) \(174.658\) \(\Q(\sqrt{313}) \) None \(1\) \(0\) \(38\) \(18\) \(-\) \(-\) \(q+\beta q^{2}+(46+\beta )q^{4}+(24-10\beta )q^{5}+\cdots\)
1089.6.a.p \(2\) \(174.658\) \(\Q(\sqrt{33}) \) None \(13\) \(0\) \(-58\) \(-146\) \(-\) \(-\) \(q+(7-\beta )q^{2}+(5^{2}-13\beta )q^{4}+(-24+\cdots)q^{5}+\cdots\)
1089.6.a.q \(3\) \(174.658\) 3.3.193425.1 None \(-1\) \(0\) \(58\) \(117\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(6+\beta _{1}+2\beta _{2})q^{4}+(17+6\beta _{1}+\cdots)q^{5}+\cdots\)
1089.6.a.r \(3\) \(174.658\) 3.3.54492.1 None \(0\) \(0\) \(-24\) \(-84\) \(-\) \(-\) \(q+\beta _{2}q^{2}+(28-2\beta _{1}-4\beta _{2})q^{4}+(-8+\cdots)q^{5}+\cdots\)
1089.6.a.s \(3\) \(174.658\) 3.3.193425.1 None \(1\) \(0\) \(58\) \(-117\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(6+\beta _{1}+2\beta _{2})q^{4}+(17+6\beta _{1}+\cdots)q^{5}+\cdots\)
1089.6.a.t \(4\) \(174.658\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-9\) \(0\) \(-42\) \(-14\) \(-\) \(-\) \(q+(-2-\beta _{1})q^{2}+(12+6\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.u \(4\) \(174.658\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(9\) \(0\) \(-42\) \(14\) \(-\) \(-\) \(q+(2+\beta _{1})q^{2}+(12+6\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.v \(5\) \(174.658\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(0\) \(-29\) \(-102\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(21-\beta _{1}+\beta _{3})q^{4}+\cdots\)
1089.6.a.w \(5\) \(174.658\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(0\) \(100\) \(18\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
1089.6.a.x \(5\) \(174.658\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(0\) \(-100\) \(18\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
1089.6.a.y \(5\) \(174.658\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(0\) \(-29\) \(102\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(21-\beta _{1}+\beta _{3})q^{4}+(-6+\cdots)q^{5}+\cdots\)
1089.6.a.z \(6\) \(174.658\) 6.6.\(\cdots\).1 None \(0\) \(0\) \(50\) \(0\) \(-\) \(+\) \(q+(-\beta _{1}+3\beta _{2})q^{2}+(14+\beta _{3}-7\beta _{4}+\cdots)q^{4}+\cdots\)
1089.6.a.ba \(6\) \(174.658\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(48\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(28+\beta _{2})q^{4}+(8+\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)
1089.6.a.bb \(8\) \(174.658\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(-70\) \(292\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(11-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.bc \(8\) \(174.658\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-\beta _{4}q^{2}+(19-\beta _{1})q^{4}-\beta _{2}q^{5}+\beta _{7}q^{7}+\cdots\)
1089.6.a.bd \(8\) \(174.658\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(256\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(19-2\beta _{2}+\beta _{6})q^{4}+(33+\cdots)q^{5}+\cdots\)
1089.6.a.be \(8\) \(174.658\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-106\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(5^{2}+\beta _{2})q^{4}+(-3\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1089.6.a.bf \(8\) \(174.658\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(106\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(5^{2}+\beta _{2})q^{4}+(3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
1089.6.a.bg \(8\) \(174.658\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(-70\) \(-292\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+(11-\beta _{1}+\beta _{2})q^{4}+(-9+\cdots)q^{5}+\cdots\)
1089.6.a.bh \(10\) \(174.658\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-9\) \(0\) \(-11\) \(470\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+(19-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.bi \(10\) \(174.658\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-7\) \(0\) \(33\) \(-78\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(15-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.6.a.bj \(10\) \(174.658\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(-198\) \(0\) \(-\) \(+\) \(q+\beta _{6}q^{2}+(18+\beta _{1})q^{4}+(-20-\beta _{1}+\cdots)q^{5}+\cdots\)
1089.6.a.bk \(10\) \(174.658\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(7\) \(0\) \(33\) \(78\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+(15-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.6.a.bl \(10\) \(174.658\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(9\) \(0\) \(-11\) \(-470\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(19-\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1089.6.a.bm \(12\) \(174.658\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{6}q^{2}+(14+\beta _{3})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
1089.6.a.bn \(20\) \(174.658\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(-472\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(15+\beta _{2})q^{4}+(-\beta _{1}-\beta _{8}+\cdots)q^{5}+\cdots\)
1089.6.a.bo \(20\) \(174.658\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(472\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(15+\beta _{2})q^{4}+(\beta _{1}+\beta _{8})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1089))\) into lower level spaces

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