Properties

Label 8550.2.a
Level $8550$
Weight $2$
Character orbit 8550.a
Rep. character $\chi_{8550}(1,\cdot)$
Character field $\Q$
Dimension $143$
Newform subspaces $76$
Sturm bound $3600$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 76 \)
Sturm bound: \(3600\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(23\), \(53\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8550))\).

Total New Old
Modular forms 1848 143 1705
Cusp forms 1753 143 1610
Eisenstein series 95 0 95

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

\(2\)\(3\)\(5\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(9\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(10\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(13\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(13\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(64\)
Minus space\(-\)\(79\)

Trace form

\( 143q + q^{2} + 143q^{4} - 2q^{7} + q^{8} + O(q^{10}) \) \( 143q + q^{2} + 143q^{4} - 2q^{7} + q^{8} - 12q^{11} - 2q^{13} - 4q^{14} + 143q^{16} + 4q^{17} + q^{19} - 4q^{22} + 6q^{23} - 12q^{26} - 2q^{28} - 18q^{29} + 12q^{31} + q^{32} - 22q^{34} + 34q^{37} + 3q^{38} + 6q^{41} + 12q^{43} - 12q^{44} + 12q^{46} + 141q^{49} - 2q^{52} - 22q^{53} - 4q^{56} + 28q^{58} - 4q^{59} + 6q^{61} + 4q^{62} + 143q^{64} - 4q^{67} + 4q^{68} + 68q^{71} - 12q^{73} - 22q^{74} + q^{76} + 28q^{77} + 4q^{79} - 10q^{82} - 64q^{83} + 24q^{86} - 4q^{88} + 42q^{89} - 32q^{91} + 6q^{92} + 16q^{94} + 18q^{97} + 17q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\) 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 19
8550.2.a.a \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{11}+4q^{14}+\cdots\)
8550.2.a.b \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}-2q^{13}+4q^{14}+\cdots\)
8550.2.a.c \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}+q^{11}+4q^{14}+\cdots\)
8550.2.a.d \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}-2q^{13}+\cdots\)
8550.2.a.e \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{13}+2q^{14}+\cdots\)
8550.2.a.f \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-6q^{13}+\cdots\)
8550.2.a.g \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+6q^{11}+2q^{14}+\cdots\)
8550.2.a.h \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{8}-q^{11}-4q^{13}+q^{16}+\cdots\)
8550.2.a.i \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{8}+2q^{11}+4q^{13}+\cdots\)
8550.2.a.j \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{8}+4q^{11}-4q^{13}+\cdots\)
8550.2.a.k \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{8}+4q^{11}+2q^{13}+\cdots\)
8550.2.a.l \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{13}-q^{14}+\cdots\)
8550.2.a.m \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}+q^{7}-q^{8}+6q^{11}-5q^{13}+\cdots\)
8550.2.a.n \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}-4q^{11}+6q^{13}+\cdots\)
8550.2.a.o \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}-2q^{14}+\cdots\)
8550.2.a.p \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}+4q^{13}+\cdots\)
8550.2.a.q \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-6q^{11}-4q^{14}+\cdots\)
8550.2.a.r \(1\) \(68.272\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}+4q^{11}+6q^{13}+\cdots\)
8550.2.a.s \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-4q^{7}+q^{8}+6q^{13}-4q^{14}+\cdots\)
8550.2.a.t \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-4q^{7}+q^{8}+4q^{11}+2q^{13}+\cdots\)
8550.2.a.u \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-3q^{7}+q^{8}-2q^{11}+q^{13}+\cdots\)
8550.2.a.v \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-6q^{11}+4q^{13}+\cdots\)
8550.2.a.w \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}+4q^{11}-6q^{13}+\cdots\)
8550.2.a.x \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}-4q^{11}-2q^{13}+\cdots\)
8550.2.a.y \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}-2q^{11}+4q^{13}+\cdots\)
8550.2.a.z \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}-q^{11}+4q^{13}+q^{16}+\cdots\)
8550.2.a.ba \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}+4q^{11}-2q^{13}+\cdots\)
8550.2.a.bb \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{8}+4q^{11}-2q^{13}+\cdots\)
8550.2.a.bc \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{8}+4q^{11}+4q^{13}+\cdots\)
8550.2.a.bd \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+q^{7}+q^{8}+q^{13}+q^{14}+\cdots\)
8550.2.a.be \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{11}+2q^{13}+\cdots\)
8550.2.a.bf \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}-4q^{13}+\cdots\)
8550.2.a.bg \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}+4q^{13}+\cdots\)
8550.2.a.bh \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-6q^{13}+2q^{14}+\cdots\)
8550.2.a.bi \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}+6q^{13}+\cdots\)
8550.2.a.bj \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+4q^{13}+4q^{14}+\cdots\)
8550.2.a.bk \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+q^{11}+4q^{14}+\cdots\)
8550.2.a.bl \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+6q^{11}+4q^{14}+\cdots\)
8550.2.a.bm \(1\) \(68.272\) \(\Q\) None \(1\) \(0\) \(0\) \(5\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+5q^{7}+q^{8}+4q^{11}+q^{13}+\cdots\)
8550.2.a.bn \(2\) \(68.272\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(-6\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+(-3+\beta )q^{7}-q^{8}-\beta q^{11}+\cdots\)
8550.2.a.bo \(2\) \(68.272\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
8550.2.a.bp \(2\) \(68.272\) \(\Q(\sqrt{17}) \) None \(-2\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}+2q^{11}+\cdots\)
8550.2.a.bq \(2\) \(68.272\) \(\Q(\sqrt{10}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+\beta q^{7}-q^{8}+(-1-\beta )q^{11}+\cdots\)
8550.2.a.br \(2\) \(68.272\) \(\Q(\sqrt{17}) \) None \(-2\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+\beta q^{7}-q^{8}-4q^{11}+(2+\cdots)q^{13}+\cdots\)
8550.2.a.bs \(2\) \(68.272\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots\)
8550.2.a.bt \(2\) \(68.272\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}-3\beta q^{11}+\cdots\)
8550.2.a.bu \(2\) \(68.272\) \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.bv \(2\) \(68.272\) \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.bw \(2\) \(68.272\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
8550.2.a.bx \(2\) \(68.272\) \(\Q(\sqrt{17}) \) None \(2\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}-2q^{11}+\cdots\)
8550.2.a.by \(2\) \(68.272\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+(-2+\cdots)q^{11}+\cdots\)
8550.2.a.bz \(2\) \(68.272\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+3\beta q^{11}+\cdots\)
8550.2.a.ca \(2\) \(68.272\) \(\Q(\sqrt{10}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+\beta q^{7}+q^{8}+(-1+\beta )q^{11}+\cdots\)
8550.2.a.cb \(2\) \(68.272\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(6\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+(3+\beta )q^{7}+q^{8}+\beta q^{11}+\cdots\)
8550.2.a.cc \(3\) \(68.272\) 3.3.788.1 None \(-3\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+(-1+\beta _{1})q^{7}-q^{8}+(1+\cdots)q^{11}+\cdots\)
8550.2.a.cd \(3\) \(68.272\) 3.3.564.1 None \(-3\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+(-\beta _{1}+\beta _{2})q^{7}-q^{8}+\cdots\)
8550.2.a.ce \(3\) \(68.272\) 3.3.148.1 None \(-3\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(-3-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cf \(3\) \(68.272\) 3.3.148.1 None \(-3\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cg \(3\) \(68.272\) 3.3.564.1 None \(-3\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots\)
8550.2.a.ch \(3\) \(68.272\) 3.3.788.1 None \(-3\) \(0\) \(0\) \(2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+(1-\beta _{1})q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.ci \(3\) \(68.272\) 3.3.993.1 None \(-3\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+(1-\beta _{2})q^{7}-q^{8}-2\beta _{1}q^{11}+\cdots\)
8550.2.a.cj \(3\) \(68.272\) 3.3.257.1 None \(-3\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+(1-\beta _{1}+2\beta _{2})q^{7}-q^{8}+\cdots\)
8550.2.a.ck \(3\) \(68.272\) 3.3.568.1 None \(-3\) \(0\) \(0\) \(4\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+(1+\beta _{1})q^{7}-q^{8}+(\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cl \(3\) \(68.272\) 3.3.568.1 None \(3\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+(-1-\beta _{1})q^{7}+q^{8}+(\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cm \(3\) \(68.272\) 3.3.788.1 None \(3\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+(-1+\beta _{1})q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.cn \(3\) \(68.272\) 3.3.564.1 None \(3\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{4}+(-\beta _{1}+\beta _{2})q^{7}+q^{8}+\cdots\)
8550.2.a.co \(3\) \(68.272\) 3.3.257.1 None \(3\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+(-1+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
8550.2.a.cp \(3\) \(68.272\) 3.3.993.1 None \(3\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+(-1+\beta _{2})q^{7}+q^{8}-2\beta _{1}q^{11}+\cdots\)
8550.2.a.cq \(3\) \(68.272\) 3.3.148.1 None \(3\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}+(-3-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cr \(3\) \(68.272\) 3.3.148.1 None \(3\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cs \(3\) \(68.272\) 3.3.564.1 None \(3\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}+q^{8}+(2+\cdots)q^{11}+\cdots\)
8550.2.a.ct \(3\) \(68.272\) 3.3.788.1 None \(3\) \(0\) \(0\) \(2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+(1-\beta _{1})q^{7}+q^{8}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cu \(4\) \(68.272\) \(\Q(\zeta_{24})^+\) None \(-4\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}-\beta _{3}q^{11}+\cdots\)
8550.2.a.cv \(4\) \(68.272\) \(\Q(\zeta_{24})^+\) None \(4\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+\beta _{3}q^{11}+\cdots\)
8550.2.a.cw \(6\) \(68.272\) 6.6.3356224.1 None \(-6\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}+(-\beta _{1}+\beta _{4}+\cdots)q^{11}+\cdots\)
8550.2.a.cx \(6\) \(68.272\) 6.6.3356224.1 None \(6\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+(\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8550))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8550)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(855))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\)\(^{\oplus 2}\)