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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\) into newform subspaces

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

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

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