Properties

Label 1728.4.a
Level $1728$
Weight $4$
Character orbit 1728.a
Rep. character $\chi_{1728}(1,\cdot)$
Character field $\Q$
Dimension $96$
Newform subspaces $58$
Sturm bound $1152$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 58 \)
Sturm bound: \(1152\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1728))\).

Total New Old
Modular forms 900 96 804
Cusp forms 828 96 732
Eisenstein series 72 0 72

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(25\)
\(+\)\(-\)\(-\)\(23\)
\(-\)\(+\)\(-\)\(23\)
\(-\)\(-\)\(+\)\(25\)
Plus space\(+\)\(50\)
Minus space\(-\)\(46\)

Trace form

\( 96q + O(q^{10}) \) \( 96q + 72q^{13} + 2400q^{25} - 504q^{37} + 4704q^{49} - 1992q^{61} + 240q^{85} - 1488q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
1728.4.a.a \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-19\) \(-13\) \(+\) \(-\) \(q-19q^{5}-13q^{7}-65q^{11}+56q^{13}+\cdots\)
1728.4.a.b \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-19\) \(13\) \(+\) \(+\) \(q-19q^{5}+13q^{7}+65q^{11}+56q^{13}+\cdots\)
1728.4.a.c \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-15\) \(-25\) \(+\) \(-\) \(q-15q^{5}-5^{2}q^{7}+15q^{11}-20q^{13}+\cdots\)
1728.4.a.d \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-15\) \(25\) \(-\) \(+\) \(q-15q^{5}+5^{2}q^{7}-15q^{11}-20q^{13}+\cdots\)
1728.4.a.e \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-12\) \(-7\) \(+\) \(+\) \(q-12q^{5}-7q^{7}-60q^{11}+79q^{13}+\cdots\)
1728.4.a.f \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-12\) \(7\) \(-\) \(-\) \(q-12q^{5}+7q^{7}+60q^{11}+79q^{13}+\cdots\)
1728.4.a.g \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-9\) \(-1\) \(+\) \(+\) \(q-9q^{5}-q^{7}-63q^{11}+28q^{13}+72q^{17}+\cdots\)
1728.4.a.h \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-9\) \(1\) \(-\) \(-\) \(q-9q^{5}+q^{7}+63q^{11}+28q^{13}+72q^{17}+\cdots\)
1728.4.a.i \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-4\) \(-3\) \(-\) \(+\) \(q-4q^{5}-3q^{7}+28q^{11}+11q^{13}+\cdots\)
1728.4.a.j \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-4\) \(3\) \(+\) \(-\) \(q-4q^{5}+3q^{7}-28q^{11}+11q^{13}+\cdots\)
1728.4.a.k \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-3\) \(-29\) \(-\) \(-\) \(q-3q^{5}-29q^{7}-57q^{11}-20q^{13}+\cdots\)
1728.4.a.l \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-3\) \(29\) \(+\) \(+\) \(q-3q^{5}+29q^{7}+57q^{11}-20q^{13}+\cdots\)
1728.4.a.m \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-1\) \(-9\) \(+\) \(+\) \(q-q^{5}-9q^{7}+17q^{11}+44q^{13}+56q^{17}+\cdots\)
1728.4.a.n \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(-1\) \(9\) \(-\) \(-\) \(q-q^{5}+9q^{7}-17q^{11}+44q^{13}+56q^{17}+\cdots\)
1728.4.a.o \(1\) \(101.955\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-37\) \(+\) \(-\) \(q-37q^{7}+19q^{13}+163q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.p \(1\) \(101.955\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-17\) \(-\) \(-\) \(q-17q^{7}-89q^{13}+107q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.q \(1\) \(101.955\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(17\) \(+\) \(+\) \(q+17q^{7}-89q^{13}-107q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.r \(1\) \(101.955\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(37\) \(-\) \(+\) \(q+37q^{7}+19q^{13}-163q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.s \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(1\) \(-9\) \(+\) \(-\) \(q+q^{5}-9q^{7}-17q^{11}+44q^{13}-56q^{17}+\cdots\)
1728.4.a.t \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(1\) \(9\) \(-\) \(+\) \(q+q^{5}+9q^{7}+17q^{11}+44q^{13}-56q^{17}+\cdots\)
1728.4.a.u \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(3\) \(-29\) \(-\) \(+\) \(q+3q^{5}-29q^{7}+57q^{11}-20q^{13}+\cdots\)
1728.4.a.v \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(3\) \(29\) \(+\) \(-\) \(q+3q^{5}+29q^{7}-57q^{11}-20q^{13}+\cdots\)
1728.4.a.w \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(4\) \(-3\) \(-\) \(+\) \(q+4q^{5}-3q^{7}-28q^{11}+11q^{13}+\cdots\)
1728.4.a.x \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(4\) \(3\) \(+\) \(-\) \(q+4q^{5}+3q^{7}+28q^{11}+11q^{13}+\cdots\)
1728.4.a.y \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(9\) \(-1\) \(+\) \(-\) \(q+9q^{5}-q^{7}+63q^{11}+28q^{13}-72q^{17}+\cdots\)
1728.4.a.z \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(9\) \(1\) \(-\) \(+\) \(q+9q^{5}+q^{7}-63q^{11}+28q^{13}-72q^{17}+\cdots\)
1728.4.a.ba \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(12\) \(-7\) \(+\) \(+\) \(q+12q^{5}-7q^{7}+60q^{11}+79q^{13}+\cdots\)
1728.4.a.bb \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(12\) \(7\) \(-\) \(-\) \(q+12q^{5}+7q^{7}-60q^{11}+79q^{13}+\cdots\)
1728.4.a.bc \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(15\) \(-25\) \(+\) \(+\) \(q+15q^{5}-5^{2}q^{7}-15q^{11}-20q^{13}+\cdots\)
1728.4.a.bd \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(15\) \(25\) \(-\) \(-\) \(q+15q^{5}+5^{2}q^{7}+15q^{11}-20q^{13}+\cdots\)
1728.4.a.be \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(19\) \(-13\) \(+\) \(+\) \(q+19q^{5}-13q^{7}+65q^{11}+56q^{13}+\cdots\)
1728.4.a.bf \(1\) \(101.955\) \(\Q\) None \(0\) \(0\) \(19\) \(13\) \(+\) \(-\) \(q+19q^{5}+13q^{7}-65q^{11}+56q^{13}+\cdots\)
1728.4.a.bg \(2\) \(101.955\) \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-8\) \(-24\) \(-\) \(+\) \(q+(-4-\beta )q^{5}+(-12-\beta )q^{7}+q^{11}+\cdots\)
1728.4.a.bh \(2\) \(101.955\) \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-8\) \(24\) \(+\) \(-\) \(q+(-4-\beta )q^{5}+(12+\beta )q^{7}-q^{11}+\cdots\)
1728.4.a.bi \(2\) \(101.955\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-4\) \(-6\) \(+\) \(+\) \(q+(-2-\beta )q^{5}+(-3+2\beta )q^{7}+(-26+\cdots)q^{11}+\cdots\)
1728.4.a.bj \(2\) \(101.955\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-4\) \(6\) \(-\) \(-\) \(q+(-2-\beta )q^{5}+(3-2\beta )q^{7}+(26-3\beta )q^{11}+\cdots\)
1728.4.a.bk \(2\) \(101.955\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-22\) \(-\) \(+\) \(q+\beta q^{5}-11q^{7}+\beta q^{11}-29q^{13}+\cdots\)
1728.4.a.bl \(2\) \(101.955\) \(\Q(\sqrt{37}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-\beta q^{5}-5\beta q^{7}-43q^{11}-52q^{13}+\cdots\)
1728.4.a.bm \(2\) \(101.955\) \(\Q(\sqrt{73}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta q^{5}+\beta q^{7}-q^{11}+8q^{13}-12\beta q^{17}+\cdots\)
1728.4.a.bn \(2\) \(101.955\) \(\Q(\sqrt{73}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-\beta q^{5}-\beta q^{7}+q^{11}+8q^{13}-12\beta q^{17}+\cdots\)
1728.4.a.bo \(2\) \(101.955\) \(\Q(\sqrt{37}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta q^{5}+5\beta q^{7}+43q^{11}-52q^{13}+\cdots\)
1728.4.a.bp \(2\) \(101.955\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(22\) \(+\) \(-\) \(q+\beta q^{5}+11q^{7}-\beta q^{11}-29q^{13}+\cdots\)
1728.4.a.bq \(2\) \(101.955\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(4\) \(-6\) \(+\) \(+\) \(q+(2+\beta )q^{5}+(-3+2\beta )q^{7}+(26-3\beta )q^{11}+\cdots\)
1728.4.a.br \(2\) \(101.955\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(4\) \(6\) \(-\) \(-\) \(q+(2+\beta )q^{5}+(3-2\beta )q^{7}+(-26+3\beta )q^{11}+\cdots\)
1728.4.a.bs \(2\) \(101.955\) \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(8\) \(-24\) \(-\) \(-\) \(q+(4+\beta )q^{5}+(-12-\beta )q^{7}-q^{11}+\cdots\)
1728.4.a.bt \(2\) \(101.955\) \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(8\) \(24\) \(+\) \(+\) \(q+(4+\beta )q^{5}+(12+\beta )q^{7}+q^{11}+(-2^{4}+\cdots)q^{13}+\cdots\)
1728.4.a.bu \(3\) \(101.955\) 3.3.2708.1 None \(0\) \(0\) \(-10\) \(-19\) \(+\) \(-\) \(q+(-3-\beta _{1})q^{5}+(-6-\beta _{2})q^{7}+(13+\cdots)q^{11}+\cdots\)
1728.4.a.bv \(3\) \(101.955\) 3.3.2708.1 None \(0\) \(0\) \(-10\) \(19\) \(+\) \(+\) \(q+(-3-\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
1728.4.a.bw \(3\) \(101.955\) 3.3.148.1 None \(0\) \(0\) \(-6\) \(-9\) \(-\) \(+\) \(q+(-2+\beta _{2})q^{5}+(-3-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1728.4.a.bx \(3\) \(101.955\) 3.3.148.1 None \(0\) \(0\) \(-6\) \(9\) \(-\) \(-\) \(q+(-2+\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1728.4.a.by \(3\) \(101.955\) 3.3.229.1 None \(0\) \(0\) \(-3\) \(-3\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-7+\cdots)q^{11}+\cdots\)
1728.4.a.bz \(3\) \(101.955\) 3.3.229.1 None \(0\) \(0\) \(-3\) \(3\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{5}+(1-\beta _{2})q^{7}+(7-3\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.a.ca \(3\) \(101.955\) 3.3.229.1 None \(0\) \(0\) \(3\) \(-3\) \(-\) \(-\) \(q+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(7-3\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.a.cb \(3\) \(101.955\) 3.3.229.1 None \(0\) \(0\) \(3\) \(3\) \(-\) \(+\) \(q+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+(-7+3\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.a.cc \(3\) \(101.955\) 3.3.148.1 None \(0\) \(0\) \(6\) \(-9\) \(-\) \(+\) \(q+(2-\beta _{2})q^{5}+(-3-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1728.4.a.cd \(3\) \(101.955\) 3.3.148.1 None \(0\) \(0\) \(6\) \(9\) \(-\) \(-\) \(q+(2-\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+(6+\cdots)q^{11}+\cdots\)
1728.4.a.ce \(3\) \(101.955\) 3.3.2708.1 None \(0\) \(0\) \(10\) \(-19\) \(+\) \(-\) \(q+(3+\beta _{1})q^{5}+(-6-\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
1728.4.a.cf \(3\) \(101.955\) 3.3.2708.1 None \(0\) \(0\) \(10\) \(19\) \(+\) \(+\) \(q+(3+\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(13+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1728))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1728)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(864))\)\(^{\oplus 2}\)