Properties

Label 8470.2.a
Level $8470$
Weight $2$
Character orbit 8470.a
Rep. character $\chi_{8470}(1,\cdot)$
Character field $\Q$
Dimension $218$
Newform subspaces $86$
Sturm bound $3168$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 86 \)
Sturm bound: \(3168\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8470))\).

Total New Old
Modular forms 1632 218 1414
Cusp forms 1537 218 1319
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.

\(2\)\(5\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(15\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(15\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(12\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(15\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(17\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(10\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(12\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(15\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(12\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(15\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(17\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(10\)
Plus space\(+\)\(98\)
Minus space\(-\)\(120\)

Trace form

\( 218 q - 2 q^{2} - 4 q^{3} + 218 q^{4} - 4 q^{6} - 2 q^{8} + 210 q^{9} + O(q^{10}) \) \( 218 q - 2 q^{2} - 4 q^{3} + 218 q^{4} - 4 q^{6} - 2 q^{8} + 210 q^{9} - 4 q^{12} - 8 q^{13} - 4 q^{15} + 218 q^{16} - 4 q^{17} - 10 q^{18} - 4 q^{19} - 4 q^{21} - 8 q^{23} - 4 q^{24} + 218 q^{25} - 8 q^{26} - 40 q^{27} - 20 q^{29} - 4 q^{30} - 40 q^{31} - 2 q^{32} - 20 q^{34} - 2 q^{35} + 210 q^{36} - 4 q^{37} - 4 q^{38} - 24 q^{39} + 20 q^{41} + 4 q^{42} - 16 q^{43} + 16 q^{45} + 8 q^{46} - 24 q^{47} - 4 q^{48} + 218 q^{49} - 2 q^{50} + 8 q^{51} - 8 q^{52} + 36 q^{53} + 8 q^{54} - 12 q^{58} + 28 q^{59} - 4 q^{60} - 16 q^{61} + 40 q^{62} + 16 q^{63} + 218 q^{64} - 4 q^{65} + 16 q^{67} - 4 q^{68} + 16 q^{69} + 2 q^{70} + 32 q^{71} - 10 q^{72} - 12 q^{73} - 4 q^{74} - 4 q^{75} - 4 q^{76} + 24 q^{78} + 16 q^{79} + 218 q^{81} - 12 q^{82} + 20 q^{83} - 4 q^{84} - 16 q^{85} - 24 q^{86} + 8 q^{87} - 20 q^{89} - 16 q^{90} - 4 q^{91} - 8 q^{92} - 16 q^{93} + 24 q^{94} + 20 q^{95} - 4 q^{96} - 20 q^{97} - 2 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\) 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 5 7 11
8470.2.a.a 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-3\) \(1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
8470.2.a.b 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.c 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.d 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+q^{7}+\cdots\)
8470.2.a.e 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-2\) \(1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.f 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-2\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
8470.2.a.g 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.h 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
8470.2.a.i 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
8470.2.a.j 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
8470.2.a.k 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.l 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.m 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
8470.2.a.n 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(1\) \(1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.o 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
8470.2.a.p 8470.a 1.a $1$ $67.633$ \(\Q\) None \(-1\) \(3\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}+q^{5}-3q^{6}+q^{7}+\cdots\)
8470.2.a.q 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-3\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}+q^{7}+\cdots\)
8470.2.a.r 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-2\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.s 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-2\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.t 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-2\) \(-1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
8470.2.a.u 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-2\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.v 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-2\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\cdots\)
8470.2.a.w 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(-1\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.x 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(0\) \(-1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.y 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(0\) \(-1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.z 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.ba 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(0\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.bb 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
8470.2.a.bc 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.bd 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.be 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
8470.2.a.bf 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(1\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.bg 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(2\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.bh 8470.a 1.a $1$ $67.633$ \(\Q\) None \(1\) \(3\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
8470.2.a.bi 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
8470.2.a.bj 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.bk 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(-2\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bl 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(-2\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.bm 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(-2\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.bn 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-2\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bo 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bp 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(-2\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bq 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(2\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.br 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bs 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bt 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(-2\) \(3\) \(-2\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bu 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{33}) \) None \(-2\) \(4\) \(2\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.bv 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(2\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bw 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
8470.2.a.bx 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(-2\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.by 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-2\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.bz 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.ca 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.cb 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(2\) \(1\) \(-2\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}-q^{7}+\cdots\)
8470.2.a.cc 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(2\) \(1\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.cd 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(-2\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(1+\beta )q^{6}+\cdots\)
8470.2.a.ce 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(2\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(1+\beta )q^{6}+\cdots\)
8470.2.a.cf 8470.a 1.a $2$ $67.633$ \(\Q(\sqrt{5}) \) None \(2\) \(3\) \(-2\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(1+\beta )q^{6}+\cdots\)
8470.2.a.cg 8470.a 1.a $3$ $67.633$ 3.3.316.1 None \(-3\) \(0\) \(-3\) \(-3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.ch 8470.a 1.a $3$ $67.633$ 3.3.316.1 None \(-3\) \(2\) \(3\) \(-3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.ci 8470.a 1.a $3$ $67.633$ 3.3.316.1 None \(-3\) \(2\) \(3\) \(3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.cj 8470.a 1.a $3$ $67.633$ 3.3.733.1 None \(-3\) \(2\) \(3\) \(3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.ck 8470.a 1.a $3$ $67.633$ 3.3.316.1 None \(3\) \(0\) \(-3\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cl 8470.a 1.a $3$ $67.633$ 3.3.892.1 None \(3\) \(0\) \(-3\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cm 8470.a 1.a $3$ $67.633$ 3.3.733.1 None \(3\) \(2\) \(3\) \(-3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
8470.2.a.cn 8470.a 1.a $3$ $67.633$ 3.3.316.1 None \(3\) \(2\) \(3\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\beta _{2})q^{6}+\cdots\)
8470.2.a.co 8470.a 1.a $4$ $67.633$ 4.4.4400.1 None \(-4\) \(-4\) \(4\) \(-4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.cp 8470.a 1.a $4$ $67.633$ \(\Q(\zeta_{24})^+\) None \(-4\) \(-4\) \(4\) \(4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.cq 8470.a 1.a $4$ $67.633$ 4.4.5225.1 None \(-4\) \(2\) \(-4\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{3}q^{3}+q^{4}-q^{5}-\beta _{3}q^{6}+\cdots\)
8470.2.a.cr 8470.a 1.a $4$ $67.633$ \(\Q(\zeta_{24})^+\) None \(4\) \(-4\) \(4\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.cs 8470.a 1.a $4$ $67.633$ 4.4.4400.1 None \(4\) \(-4\) \(4\) \(4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.ct 8470.a 1.a $4$ $67.633$ 4.4.5225.1 None \(4\) \(2\) \(-4\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{3}q^{3}+q^{4}-q^{5}+\beta _{3}q^{6}+\cdots\)
8470.2.a.cu 8470.a 1.a $6$ $67.633$ 6.6.38498000.1 None \(-6\) \(-5\) \(-6\) \(-6\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.cv 8470.a 1.a $6$ $67.633$ 6.6.4642000.1 None \(-6\) \(-1\) \(6\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cw 8470.a 1.a $6$ $67.633$ 6.6.19898000.1 None \(-6\) \(-1\) \(6\) \(6\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cx 8470.a 1.a $6$ $67.633$ 6.6.745749504.1 None \(-6\) \(0\) \(-6\) \(6\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cy 8470.a 1.a $6$ $67.633$ 6.6.13298000.1 None \(-6\) \(1\) \(6\) \(-6\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-\beta _{2}-\beta _{5})q^{3}+q^{4}+q^{5}+\cdots\)
8470.2.a.cz 8470.a 1.a $6$ $67.633$ 6.6.10784448.1 None \(-6\) \(4\) \(6\) \(-6\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{3})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.da 8470.a 1.a $6$ $67.633$ 6.6.38498000.1 None \(6\) \(-5\) \(-6\) \(6\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.db 8470.a 1.a $6$ $67.633$ 6.6.4642000.1 None \(6\) \(-1\) \(6\) \(-6\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.dc 8470.a 1.a $6$ $67.633$ 6.6.19898000.1 None \(6\) \(-1\) \(6\) \(-6\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.dd 8470.a 1.a $6$ $67.633$ 6.6.745749504.1 None \(6\) \(0\) \(-6\) \(-6\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.de 8470.a 1.a $6$ $67.633$ 6.6.13298000.1 None \(6\) \(1\) \(6\) \(6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-\beta _{2}-\beta _{5})q^{3}+q^{4}+q^{5}+\cdots\)
8470.2.a.df 8470.a 1.a $6$ $67.633$ 6.6.10784448.1 None \(6\) \(4\) \(6\) \(6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{3})q^{3}+q^{4}+q^{5}+(1+\beta _{3}+\cdots)q^{6}+\cdots\)
8470.2.a.dg 8470.a 1.a $8$ $67.633$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(-8\) \(8\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.dh 8470.a 1.a $8$ $67.633$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(-8\) \(-8\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8470))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8470)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(385))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(605))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(770))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(847))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1694))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\)\(^{\oplus 2}\)