Properties

Label 6864.2.a
Level $6864$
Weight $2$
Character orbit 6864.a
Rep. character $\chi_{6864}(1,\cdot)$
Character field $\Q$
Dimension $120$
Newform subspaces $58$
Sturm bound $2688$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 58 \)
Sturm bound: \(2688\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(7\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6864))\).

Total New Old
Modular forms 1368 120 1248
Cusp forms 1321 120 1201
Eisenstein series 47 0 47

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

\(2\)\(3\)\(11\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(52\)
Minus space\(-\)\(68\)

Trace form

\( 120q + 120q^{9} + O(q^{10}) \) \( 120q + 120q^{9} - 8q^{15} - 16q^{19} + 120q^{25} - 40q^{31} - 48q^{35} - 48q^{47} + 120q^{49} - 16q^{51} + 32q^{61} - 24q^{67} + 32q^{69} + 48q^{71} - 16q^{79} + 120q^{81} + 48q^{83} + 32q^{85} + 24q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 11 13
6864.2.a.a \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-3\) \(-5\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-3q^{5}-5q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
6864.2.a.b \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-3\) \(-3\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-3q^{5}-3q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
6864.2.a.c \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-2\) \(-4\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{5}-4q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
6864.2.a.d \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-2q^{5}+q^{9}-q^{11}-q^{13}+2q^{15}+\cdots\)
6864.2.a.e \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{5}+q^{9}+q^{11}+q^{13}+2q^{15}+\cdots\)
6864.2.a.f \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-2\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{5}+2q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
6864.2.a.g \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}-q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
6864.2.a.h \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{7}+q^{9}+q^{11}+q^{13}+6q^{17}+\cdots\)
6864.2.a.i \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-q^{11}-q^{13}-4q^{17}+\cdots\)
6864.2.a.j \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+4q^{7}+q^{9}+q^{11}+q^{13}+4q^{19}+\cdots\)
6864.2.a.k \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(2\) \(-4\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+2q^{5}-4q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
6864.2.a.l \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(2\) \(-4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}+2q^{5}-4q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
6864.2.a.m \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(2\) \(-4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+2q^{5}-4q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
6864.2.a.n \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(2\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+2q^{5}+q^{9}+q^{11}+q^{13}-2q^{15}+\cdots\)
6864.2.a.o \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(3\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+3q^{5}+q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
6864.2.a.p \(1\) \(54.809\) \(\Q\) None \(0\) \(-1\) \(4\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+4q^{5}+4q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
6864.2.a.q \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(-3\) \(-1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}-3q^{5}-q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
6864.2.a.r \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-q^{11}+q^{13}-2q^{15}+\cdots\)
6864.2.a.s \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-q^{11}+q^{13}-2q^{15}+\cdots\)
6864.2.a.t \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(-1\) \(-3\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}-3q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
6864.2.a.u \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(-1\) \(3\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}+3q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
6864.2.a.v \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-4q^{7}+q^{9}+q^{11}-q^{13}+8q^{17}+\cdots\)
6864.2.a.w \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{9}-q^{11}+q^{13}-4q^{17}+\cdots\)
6864.2.a.x \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{7}+q^{9}+q^{11}+q^{13}-2q^{17}+\cdots\)
6864.2.a.y \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(2\) \(-4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+2q^{5}-4q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
6864.2.a.z \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(2\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+2q^{5}+q^{9}-q^{11}-q^{13}+2q^{15}+\cdots\)
6864.2.a.ba \(1\) \(54.809\) \(\Q\) None \(0\) \(1\) \(4\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+4q^{5}+q^{9}-q^{11}+q^{13}+4q^{15}+\cdots\)
6864.2.a.bb \(2\) \(54.809\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-4\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+(-2+\beta )q^{5}+2\beta q^{7}+q^{9}+\cdots\)
6864.2.a.bc \(2\) \(54.809\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-4\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+(-2+\beta )q^{5}+(2+2\beta )q^{7}+q^{9}+\cdots\)
6864.2.a.bd \(2\) \(54.809\) \(\Q(\sqrt{41}) \) None \(0\) \(-2\) \(-1\) \(-3\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-\beta q^{5}+(-2+\beta )q^{7}+q^{9}+q^{11}+\cdots\)
6864.2.a.be \(2\) \(54.809\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(4\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+\beta q^{5}+2q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
6864.2.a.bf \(2\) \(54.809\) \(\Q(\sqrt{41}) \) None \(0\) \(-2\) \(4\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+2q^{5}+q^{9}+q^{11}+q^{13}-2q^{15}+\cdots\)
6864.2.a.bg \(2\) \(54.809\) \(\Q(\sqrt{6}) \) None \(0\) \(2\) \(-4\) \(4\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+(-2+\beta )q^{5}+2q^{7}+q^{9}-q^{11}+\cdots\)
6864.2.a.bh \(2\) \(54.809\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(-3\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+(-1-\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
6864.2.a.bi \(2\) \(54.809\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+(-1+\beta )q^{5}+q^{9}-q^{11}-q^{13}+\cdots\)
6864.2.a.bj \(2\) \(54.809\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1-\beta )q^{5}+q^{9}+q^{11}-q^{13}+\cdots\)
6864.2.a.bk \(2\) \(54.809\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1+\beta )q^{5}+2q^{7}+q^{9}+q^{11}+\cdots\)
6864.2.a.bl \(2\) \(54.809\) \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(1\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
6864.2.a.bm \(2\) \(54.809\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+(1+\beta )q^{5}-2\beta q^{7}+q^{9}-q^{11}+\cdots\)
6864.2.a.bn \(3\) \(54.809\) 3.3.404.1 None \(0\) \(-3\) \(-4\) \(6\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+(-1-\beta _{2})q^{5}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
6864.2.a.bo \(3\) \(54.809\) 3.3.892.1 None \(0\) \(-3\) \(-1\) \(-5\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}-\beta _{1}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
6864.2.a.bp \(3\) \(54.809\) 3.3.148.1 None \(0\) \(-3\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-\beta _{2}q^{5}+(1-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
6864.2.a.bq \(3\) \(54.809\) 3.3.148.1 None \(0\) \(-3\) \(0\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-\beta _{2}q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
6864.2.a.br \(3\) \(54.809\) 3.3.564.1 None \(0\) \(-3\) \(2\) \(-4\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
6864.2.a.bs \(3\) \(54.809\) 3.3.148.1 None \(0\) \(-3\) \(4\) \(2\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+(1+\beta _{1})q^{5}+(1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
6864.2.a.bt \(3\) \(54.809\) 3.3.404.1 None \(0\) \(-3\) \(4\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+(1+2\beta _{1}-\beta _{2})q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
6864.2.a.bu \(3\) \(54.809\) 3.3.564.1 None \(0\) \(3\) \(-2\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
6864.2.a.bv \(3\) \(54.809\) 3.3.1620.1 None \(0\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+\beta _{1}q^{5}-\beta _{2}q^{7}+q^{9}+q^{11}+\cdots\)
6864.2.a.bw \(4\) \(54.809\) 4.4.70164.1 None \(0\) \(-4\) \(1\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}+(1+\beta _{2}-\beta _{3})q^{5}+\beta _{2}q^{7}+q^{9}+\cdots\)
6864.2.a.bx \(4\) \(54.809\) 4.4.83476.1 None \(0\) \(-4\) \(2\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+\beta _{3}q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
6864.2.a.by \(4\) \(54.809\) 4.4.22676.1 None \(0\) \(4\) \(-3\) \(-3\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1+\beta _{1})q^{5}+(-1-\beta _{3})q^{7}+\cdots\)
6864.2.a.bz \(4\) \(54.809\) 4.4.8468.1 None \(0\) \(4\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta _{3}q^{5}+(-1-\beta _{2}-\beta _{3})q^{7}+\cdots\)
6864.2.a.ca \(4\) \(54.809\) 4.4.70164.1 None \(0\) \(4\) \(1\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+\beta _{1}q^{5}+(1+\beta _{2}-\beta _{3})q^{7}+q^{9}+\cdots\)
6864.2.a.cb \(4\) \(54.809\) 4.4.90996.1 None \(0\) \(4\) \(2\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+q^{9}+\cdots\)
6864.2.a.cc \(4\) \(54.809\) 4.4.23252.1 None \(0\) \(4\) \(4\) \(-2\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+(1+\beta _{2})q^{5}-\beta _{1}q^{7}+q^{9}-q^{11}+\cdots\)
6864.2.a.cd \(4\) \(54.809\) 4.4.29268.1 None \(0\) \(4\) \(4\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+(1-\beta _{1})q^{5}+(1+\beta _{2})q^{7}+q^{9}+\cdots\)
6864.2.a.ce \(5\) \(54.809\) 5.5.46437524.1 None \(0\) \(-5\) \(1\) \(3\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+\beta _{1}q^{5}+(1+\beta _{4})q^{7}+q^{9}-q^{11}+\cdots\)
6864.2.a.cf \(5\) \(54.809\) 5.5.2172244.1 None \(0\) \(5\) \(1\) \(5\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta _{1}q^{5}+(1+\beta _{4})q^{7}+q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6864))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6864)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(286))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(429))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(572))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(624))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(858))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1716))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3432))\)\(^{\oplus 2}\)