Properties

Label 5520.2.a
Level $5520$
Weight $2$
Character orbit 5520.a
Rep. character $\chi_{5520}(1,\cdot)$
Character field $\Q$
Dimension $88$
Newform subspaces $55$
Sturm bound $2304$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 55 \)
Sturm bound: \(2304\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5520))\).

Total New Old
Modular forms 1176 88 1088
Cusp forms 1129 88 1041
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.

\(2\)\(3\)\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(40\)
Minus space\(-\)\(48\)

Trace form

\( 88q + 4q^{3} + 8q^{7} + 88q^{9} + O(q^{10}) \) \( 88q + 4q^{3} + 8q^{7} + 88q^{9} + 8q^{19} + 88q^{25} + 4q^{27} + 8q^{31} + 8q^{39} + 8q^{43} + 104q^{49} - 32q^{53} - 16q^{55} + 16q^{57} - 32q^{59} - 32q^{61} + 8q^{63} + 8q^{67} - 32q^{71} + 16q^{73} + 4q^{75} - 32q^{77} + 8q^{79} + 88q^{81} - 24q^{87} - 64q^{91} + 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 23
5520.2.a.a \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-4\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
5520.2.a.b \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-4\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}-4q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
5520.2.a.c \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-3\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}-3q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
5520.2.a.d \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}+q^{9}-2q^{11}+q^{15}+6q^{17}+\cdots\)
5520.2.a.e \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}+q^{9}-2q^{13}+q^{15}-6q^{17}+\cdots\)
5520.2.a.f \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
5520.2.a.g \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}+q^{7}+q^{9}-4q^{13}+q^{15}+\cdots\)
5520.2.a.h \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(-1\) \(3\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}+3q^{7}+q^{9}+4q^{11}+q^{15}+\cdots\)
5520.2.a.i \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
5520.2.a.j \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
5520.2.a.k \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{9}-2q^{11}-q^{15}+6q^{17}+\cdots\)
5520.2.a.l \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{9}+2q^{11}+4q^{13}+\cdots\)
5520.2.a.m \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(3\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}+3q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
5520.2.a.n \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(3\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}+3q^{7}+q^{9}+4q^{11}-q^{15}+\cdots\)
5520.2.a.o \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+4q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
5520.2.a.p \(1\) \(44.077\) \(\Q\) None \(0\) \(-1\) \(1\) \(5\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+5q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
5520.2.a.q \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(-4\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
5520.2.a.r \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-4q^{7}+q^{9}+2q^{11}-q^{15}+\cdots\)
5520.2.a.s \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(-4\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-4q^{7}+q^{9}+6q^{11}-4q^{13}+\cdots\)
5520.2.a.t \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(-3\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}-3q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
5520.2.a.u \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}-q^{15}+\cdots\)
5520.2.a.v \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
5520.2.a.w \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+q^{9}-6q^{13}-q^{15}+2q^{17}+\cdots\)
5520.2.a.x \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+q^{9}+6q^{13}-q^{15}+2q^{17}+\cdots\)
5520.2.a.y \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+2q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
5520.2.a.z \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}+2q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
5520.2.a.ba \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(-1\) \(5\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+5q^{7}+q^{9}+4q^{13}-q^{15}+\cdots\)
5520.2.a.bb \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(1\) \(-4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}-4q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
5520.2.a.bc \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(1\) \(-3\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
5520.2.a.bd \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(1\) \(-2\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
5520.2.a.be \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(1\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{13}+q^{15}+\cdots\)
5520.2.a.bf \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
5520.2.a.bg \(1\) \(44.077\) \(\Q\) None \(0\) \(1\) \(1\) \(2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
5520.2.a.bh \(2\) \(44.077\) \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(-2\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}+\beta q^{7}+q^{9}+2\beta q^{11}+\cdots\)
5520.2.a.bi \(2\) \(44.077\) \(\Q(\sqrt{6}) \) None \(0\) \(-2\) \(-2\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}+q^{7}+q^{9}-\beta q^{11}+(2+\cdots)q^{13}+\cdots\)
5520.2.a.bj \(2\) \(44.077\) \(\Q(\sqrt{15}) \) None \(0\) \(-2\) \(2\) \(-6\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}-3q^{7}+q^{9}+(-1-\beta )q^{11}+\cdots\)
5520.2.a.bk \(2\) \(44.077\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{7}+q^{9}+(2+3\beta )q^{11}+\cdots\)
5520.2.a.bl \(2\) \(44.077\) \(\Q(\sqrt{6}) \) None \(0\) \(2\) \(-2\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
5520.2.a.bm \(2\) \(44.077\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+(1+2\beta )q^{7}+q^{9}+(4+\beta )q^{11}+\cdots\)
5520.2.a.bn \(2\) \(44.077\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(-2\) \(3\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}-2q^{11}+\cdots\)
5520.2.a.bo \(2\) \(44.077\) \(\Q(\sqrt{11}) \) None \(0\) \(2\) \(-2\) \(6\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}+3q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
5520.2.a.bp \(2\) \(44.077\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(-2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+(-1+2\beta )q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
5520.2.a.bq \(2\) \(44.077\) \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(2\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}-\beta q^{7}+q^{9}-4q^{11}+(-2+\cdots)q^{13}+\cdots\)
5520.2.a.br \(2\) \(44.077\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(2\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}-\beta q^{7}+q^{9}-2\beta q^{11}+\cdots\)
5520.2.a.bs \(2\) \(44.077\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(2\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}+(1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.bt \(2\) \(44.077\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+(1+2\beta )q^{7}+q^{9}-\beta q^{11}+\cdots\)
5520.2.a.bu \(2\) \(44.077\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(6\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}+3q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
5520.2.a.bv \(3\) \(44.077\) 3.3.3144.1 None \(0\) \(-3\) \(-3\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}+(-1+\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.bw \(3\) \(44.077\) 3.3.1436.1 None \(0\) \(-3\) \(-3\) \(2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}+(1-\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.bx \(3\) \(44.077\) 3.3.316.1 None \(0\) \(-3\) \(-3\) \(6\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}+(2-\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.by \(3\) \(44.077\) 3.3.316.1 None \(0\) \(-3\) \(3\) \(-6\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}+(-2-\beta _{2})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
5520.2.a.bz \(3\) \(44.077\) 3.3.568.1 None \(0\) \(-3\) \(3\) \(-2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+(-1+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
5520.2.a.ca \(3\) \(44.077\) 3.3.229.1 None \(0\) \(-3\) \(3\) \(-1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{5}+\beta _{1}q^{7}+q^{9}+(1-\beta _{2})q^{13}+\cdots\)
5520.2.a.cb \(4\) \(44.077\) 4.4.54764.1 None \(0\) \(4\) \(-4\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+\beta _{3}q^{11}+\cdots\)
5520.2.a.cc \(5\) \(44.077\) 5.5.20087896.1 None \(0\) \(5\) \(5\) \(4\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+(1-\beta _{1})q^{7}+q^{9}+(1-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5520)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(920))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2760))\)\(^{\oplus 2}\)