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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7 11
8470.2.a.a \(1\) \(67.633\) \(\Q\) None \(-1\) \(-3\) \(1\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
8470.2.a.b \(1\) \(67.633\) \(\Q\) None \(-1\) \(-2\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.c \(1\) \(67.633\) \(\Q\) None \(-1\) \(-2\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.d \(1\) \(67.633\) \(\Q\) None \(-1\) \(-2\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+q^{7}+\cdots\)
8470.2.a.e \(1\) \(67.633\) \(\Q\) None \(-1\) \(-2\) \(1\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.f \(1\) \(67.633\) \(\Q\) None \(-1\) \(-2\) \(1\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
8470.2.a.g \(1\) \(67.633\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.h \(1\) \(67.633\) \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
8470.2.a.i \(1\) \(67.633\) \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
8470.2.a.j \(1\) \(67.633\) \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
8470.2.a.k \(1\) \(67.633\) \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.l \(1\) \(67.633\) \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.m \(1\) \(67.633\) \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
8470.2.a.n \(1\) \(67.633\) \(\Q\) None \(-1\) \(1\) \(1\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.o \(1\) \(67.633\) \(\Q\) None \(-1\) \(1\) \(1\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
8470.2.a.p \(1\) \(67.633\) \(\Q\) None \(-1\) \(3\) \(1\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+3q^{3}+q^{4}+q^{5}-3q^{6}+q^{7}+\cdots\)
8470.2.a.q \(1\) \(67.633\) \(\Q\) None \(1\) \(-3\) \(1\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}+q^{7}+\cdots\)
8470.2.a.r \(1\) \(67.633\) \(\Q\) None \(1\) \(-2\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.s \(1\) \(67.633\) \(\Q\) None \(1\) \(-2\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.t \(1\) \(67.633\) \(\Q\) None \(1\) \(-2\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
8470.2.a.u \(1\) \(67.633\) \(\Q\) None \(1\) \(-2\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.v \(1\) \(67.633\) \(\Q\) None \(1\) \(-2\) \(1\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\cdots\)
8470.2.a.w \(1\) \(67.633\) \(\Q\) None \(1\) \(-1\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
8470.2.a.x \(1\) \(67.633\) \(\Q\) None \(1\) \(0\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.y \(1\) \(67.633\) \(\Q\) None \(1\) \(0\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.z \(1\) \(67.633\) \(\Q\) None \(1\) \(0\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.ba \(1\) \(67.633\) \(\Q\) None \(1\) \(0\) \(1\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
8470.2.a.bb \(1\) \(67.633\) \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
8470.2.a.bc \(1\) \(67.633\) \(\Q\) None \(1\) \(1\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.bd \(1\) \(67.633\) \(\Q\) None \(1\) \(1\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.be \(1\) \(67.633\) \(\Q\) None \(1\) \(1\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
8470.2.a.bf \(1\) \(67.633\) \(\Q\) None \(1\) \(1\) \(1\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
8470.2.a.bg \(1\) \(67.633\) \(\Q\) None \(1\) \(2\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8470.2.a.bh \(1\) \(67.633\) \(\Q\) None \(1\) \(3\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
8470.2.a.bi \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
8470.2.a.bj \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.bk \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(-2\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bl \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.bm \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.bn \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-2\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bo \(2\) \(67.633\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bp \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(-2\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{7}+\cdots\)
8470.2.a.bq \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(2\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.br \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bs \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bt \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(-2\) \(3\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bu \(2\) \(67.633\) \(\Q(\sqrt{33}) \) None \(-2\) \(4\) \(2\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-q^{7}+\cdots\)
8470.2.a.bv \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(2\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.bw \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
8470.2.a.bx \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.by \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{7}+\cdots\)
8470.2.a.bz \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.ca \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.cb \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(2\) \(1\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}-q^{7}+\cdots\)
8470.2.a.cc \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(2\) \(1\) \(2\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+q^{7}+\cdots\)
8470.2.a.cd \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(1+\beta )q^{6}+\cdots\)
8470.2.a.ce \(2\) \(67.633\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(2\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(1+\beta )q^{6}+\cdots\)
8470.2.a.cf \(2\) \(67.633\) \(\Q(\sqrt{5}) \) None \(2\) \(3\) \(-2\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(1+\beta )q^{6}+\cdots\)
8470.2.a.cg \(3\) \(67.633\) 3.3.316.1 None \(-3\) \(0\) \(-3\) \(-3\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.ch \(3\) \(67.633\) 3.3.316.1 None \(-3\) \(2\) \(3\) \(-3\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.ci \(3\) \(67.633\) 3.3.316.1 None \(-3\) \(2\) \(3\) \(3\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.cj \(3\) \(67.633\) 3.3.733.1 None \(-3\) \(2\) \(3\) \(3\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.ck \(3\) \(67.633\) 3.3.316.1 None \(3\) \(0\) \(-3\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cl \(3\) \(67.633\) 3.3.892.1 None \(3\) \(0\) \(-3\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cm \(3\) \(67.633\) 3.3.733.1 None \(3\) \(2\) \(3\) \(-3\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
8470.2.a.cn \(3\) \(67.633\) 3.3.316.1 None \(3\) \(2\) \(3\) \(3\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\beta _{2})q^{6}+\cdots\)
8470.2.a.co \(4\) \(67.633\) 4.4.4400.1 None \(-4\) \(-4\) \(4\) \(-4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.cp \(4\) \(67.633\) \(\Q(\zeta_{24})^+\) None \(-4\) \(-4\) \(4\) \(4\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+(-1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.cq \(4\) \(67.633\) 4.4.5225.1 None \(-4\) \(2\) \(-4\) \(-4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta _{3}q^{3}+q^{4}-q^{5}-\beta _{3}q^{6}+\cdots\)
8470.2.a.cr \(4\) \(67.633\) \(\Q(\zeta_{24})^+\) None \(4\) \(-4\) \(4\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+(-1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.cs \(4\) \(67.633\) 4.4.4400.1 None \(4\) \(-4\) \(4\) \(4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.ct \(4\) \(67.633\) 4.4.5225.1 None \(4\) \(2\) \(-4\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+\beta _{3}q^{3}+q^{4}-q^{5}+\beta _{3}q^{6}+\cdots\)
8470.2.a.cu \(6\) \(67.633\) 6.6.38498000.1 None \(-6\) \(-5\) \(-6\) \(-6\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)
8470.2.a.cv \(6\) \(67.633\) 6.6.4642000.1 None \(-6\) \(-1\) \(6\) \(6\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cw \(6\) \(67.633\) 6.6.19898000.1 None \(-6\) \(-1\) \(6\) \(6\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cx \(6\) \(67.633\) 6.6.745749504.1 None \(-6\) \(0\) \(-6\) \(6\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
8470.2.a.cy \(6\) \(67.633\) 6.6.13298000.1 None \(-6\) \(1\) \(6\) \(-6\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+(-\beta _{2}-\beta _{5})q^{3}+q^{4}+q^{5}+\cdots\)
8470.2.a.cz \(6\) \(67.633\) 6.6.10784448.1 None \(-6\) \(4\) \(6\) \(-6\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+(1+\beta _{3})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.da \(6\) \(67.633\) 6.6.38498000.1 None \(6\) \(-5\) \(-6\) \(6\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
8470.2.a.db \(6\) \(67.633\) 6.6.4642000.1 None \(6\) \(-1\) \(6\) \(-6\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.dc \(6\) \(67.633\) 6.6.19898000.1 None \(6\) \(-1\) \(6\) \(-6\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.dd \(6\) \(67.633\) 6.6.745749504.1 None \(6\) \(0\) \(-6\) \(-6\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.de \(6\) \(67.633\) 6.6.13298000.1 None \(6\) \(1\) \(6\) \(6\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+(-\beta _{2}-\beta _{5})q^{3}+q^{4}+q^{5}+\cdots\)
8470.2.a.df \(6\) \(67.633\) 6.6.10784448.1 None \(6\) \(4\) \(6\) \(6\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+(1+\beta _{3})q^{3}+q^{4}+q^{5}+(1+\beta _{3}+\cdots)q^{6}+\cdots\)
8470.2.a.dg \(8\) \(67.633\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(-8\) \(8\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
8470.2.a.dh \(8\) \(67.633\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(-8\) \(-8\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

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

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