Properties

Label 8450.2.a
Level $8450$
Weight $2$
Character orbit 8450.a
Rep. character $\chi_{8450}(1,\cdot)$
Character field $\Q$
Dimension $245$
Newform subspaces $79$
Sturm bound $2730$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 79 \)
Sturm bound: \(2730\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(17\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8450))\).

Total New Old
Modular forms 1448 245 1203
Cusp forms 1281 245 1036
Eisenstein series 167 0 167

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

\(2\)\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(24\)
\(+\)\(+\)\(-\)\(-\)\(33\)
\(+\)\(-\)\(+\)\(-\)\(34\)
\(+\)\(-\)\(-\)\(+\)\(31\)
\(-\)\(+\)\(+\)\(-\)\(35\)
\(-\)\(+\)\(-\)\(+\)\(24\)
\(-\)\(-\)\(+\)\(+\)\(27\)
\(-\)\(-\)\(-\)\(-\)\(37\)
Plus space\(+\)\(106\)
Minus space\(-\)\(139\)

Trace form

\( 245 q + q^{2} - 2 q^{3} + 245 q^{4} - 2 q^{6} - 8 q^{7} + q^{8} + 245 q^{9} + O(q^{10}) \) \( 245 q + q^{2} - 2 q^{3} + 245 q^{4} - 2 q^{6} - 8 q^{7} + q^{8} + 245 q^{9} + 2 q^{11} - 2 q^{12} + 245 q^{16} - 10 q^{17} + 5 q^{18} - 2 q^{19} - 16 q^{21} - 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{27} - 8 q^{28} - 8 q^{29} - 20 q^{31} + q^{32} + 20 q^{33} + 4 q^{34} + 245 q^{36} + 2 q^{37} - 2 q^{38} + 8 q^{41} - 12 q^{42} - 10 q^{43} + 2 q^{44} - 8 q^{46} - 2 q^{48} + 225 q^{49} + 70 q^{51} + 28 q^{53} - 2 q^{54} - 8 q^{57} + 2 q^{58} + 36 q^{59} - 16 q^{61} + 245 q^{64} + 14 q^{66} - 10 q^{68} + 72 q^{69} + 24 q^{71} + 5 q^{72} + 2 q^{73} + 8 q^{74} - 2 q^{76} + 36 q^{77} + 317 q^{81} + 22 q^{82} + 24 q^{83} - 16 q^{84} - 8 q^{86} + 28 q^{87} - 6 q^{88} + 28 q^{89} + 8 q^{92} - 32 q^{93} + 16 q^{94} - 2 q^{96} + 6 q^{97} - 7 q^{98} + 40 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\) 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 13
8450.2.a.a 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(-2\) \(0\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-5q^{7}-q^{8}+\cdots\)
8450.2.a.b 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
8450.2.a.c 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
8450.2.a.d 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
8450.2.a.e 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
8450.2.a.f 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-4q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots\)
8450.2.a.g 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(0\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+3q^{7}-q^{8}-3q^{9}-3q^{11}+\cdots\)
8450.2.a.h 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(1\) \(0\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+3q^{7}-q^{8}+\cdots\)
8450.2.a.i 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(2\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-4q^{7}-q^{8}+\cdots\)
8450.2.a.j 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(2\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
8450.2.a.k 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(2\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
8450.2.a.l 8450.a 1.a $1$ $67.474$ \(\Q\) None \(-1\) \(3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
8450.2.a.m 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(-3\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
8450.2.a.n 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(-2\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}-4q^{7}+q^{8}+\cdots\)
8450.2.a.o 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
8450.2.a.p 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(-2\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots\)
8450.2.a.q 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{7}+q^{8}-3q^{9}+3q^{11}+\cdots\)
8450.2.a.r 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}+q^{16}-2q^{17}+\cdots\)
8450.2.a.s 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(0\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+4q^{7}+q^{8}-3q^{9}-4q^{11}+\cdots\)
8450.2.a.t 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
8450.2.a.u 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(1\) \(0\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots\)
8450.2.a.v 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
8450.2.a.w 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(2\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots\)
8450.2.a.x 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(2\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+5q^{7}+q^{8}+\cdots\)
8450.2.a.y 8450.a 1.a $1$ $67.474$ \(\Q\) None \(1\) \(3\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{7}+q^{8}+\cdots\)
8450.2.a.z 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
8450.2.a.ba 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{13}) \) None \(-2\) \(-1\) \(0\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(-3+\beta )q^{7}+\cdots\)
8450.2.a.bb 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{21}) \) None \(-2\) \(-1\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(1-\beta )q^{7}+\cdots\)
8450.2.a.bc 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{10}) \) None \(-2\) \(0\) \(0\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
8450.2.a.bd 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{13}) \) None \(-2\) \(1\) \(0\) \(-5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-3+\beta )q^{7}+\cdots\)
8450.2.a.be 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{21}) \) None \(-2\) \(1\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(1-\beta )q^{7}+\cdots\)
8450.2.a.bf 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(0\) \(6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
8450.2.a.bg 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
8450.2.a.bh 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{21}) \) None \(2\) \(-1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(-1+\beta )q^{7}+\cdots\)
8450.2.a.bi 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{13}) \) None \(2\) \(-1\) \(0\) \(5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(3-\beta )q^{7}+\cdots\)
8450.2.a.bj 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{10}) \) None \(2\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots\)
8450.2.a.bk 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{21}) \) None \(2\) \(1\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(-1+\beta )q^{7}+\cdots\)
8450.2.a.bl 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{13}) \) None \(2\) \(1\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(3-\beta )q^{7}+\cdots\)
8450.2.a.bm 8450.a 1.a $2$ $67.474$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(-6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
8450.2.a.bn 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-3\) \(0\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(1+\cdots)q^{6}+\cdots\)
8450.2.a.bo 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-1\) \(0\) \(-5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
8450.2.a.bp 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-1\) \(0\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.bq 8450.a 1.a $3$ $67.474$ 3.3.2808.1 None \(-3\) \(0\) \(0\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
8450.2.a.br 8450.a 1.a $3$ $67.474$ 3.3.2808.1 None \(-3\) \(0\) \(0\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
8450.2.a.bs 8450.a 1.a $3$ $67.474$ 3.3.940.1 None \(-3\) \(0\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.bt 8450.a 1.a $3$ $67.474$ 3.3.148.1 None \(-3\) \(0\) \(0\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.bu 8450.a 1.a $3$ $67.474$ 3.3.148.1 None \(-3\) \(0\) \(0\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.bv 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(-3\) \(1\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-2\beta _{1}q^{7}+\cdots\)
8450.2.a.bw 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(-3\) \(1\) \(0\) \(4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.bx 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(3\) \(-3\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots\)
8450.2.a.by 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(3\) \(-1\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots\)
8450.2.a.bz 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(3\) \(-1\) \(0\) \(5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.ca 8450.a 1.a $3$ $67.474$ 3.3.148.1 None \(3\) \(0\) \(0\) \(-5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(-2+\cdots)q^{7}+\cdots\)
8450.2.a.cb 8450.a 1.a $3$ $67.474$ 3.3.148.1 None \(3\) \(0\) \(0\) \(-5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(-2+\cdots)q^{7}+\cdots\)
8450.2.a.cc 8450.a 1.a $3$ $67.474$ 3.3.940.1 None \(3\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
8450.2.a.cd 8450.a 1.a $3$ $67.474$ 3.3.2808.1 None \(3\) \(0\) \(0\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.ce 8450.a 1.a $3$ $67.474$ 3.3.2808.1 None \(3\) \(0\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.cf 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(3\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots\)
8450.2.a.cg 8450.a 1.a $3$ $67.474$ \(\Q(\zeta_{14})^+\) None \(3\) \(1\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+2\beta _{1}q^{7}+\cdots\)
8450.2.a.ch 8450.a 1.a $4$ $67.474$ 4.4.4752.1 None \(-4\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
8450.2.a.ci 8450.a 1.a $4$ $67.474$ 4.4.4752.1 None \(-4\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(1+\cdots)q^{6}+\cdots\)
8450.2.a.cj 8450.a 1.a $4$ $67.474$ \(\Q(\sqrt{3}, \sqrt{11})\) None \(-4\) \(0\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+\beta _{2}q^{7}+\cdots\)
8450.2.a.ck 8450.a 1.a $4$ $67.474$ 4.4.4752.1 None \(-4\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
8450.2.a.cl 8450.a 1.a $4$ $67.474$ 4.4.4752.1 None \(4\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.cm 8450.a 1.a $4$ $67.474$ 4.4.4752.1 None \(4\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots\)
8450.2.a.cn 8450.a 1.a $4$ $67.474$ \(\Q(\sqrt{3}, \sqrt{11})\) None \(4\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-\beta _{2}q^{7}+\cdots\)
8450.2.a.co 8450.a 1.a $4$ $67.474$ 4.4.4752.1 None \(4\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
8450.2.a.cp 8450.a 1.a $6$ $67.474$ 6.6.20439713.1 None \(-6\) \(2\) \(0\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
8450.2.a.cq 8450.a 1.a $6$ $67.474$ 6.6.20439713.1 None \(6\) \(2\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
8450.2.a.cr 8450.a 1.a $8$ $67.474$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(0\) \(-10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
8450.2.a.cs 8450.a 1.a $8$ $67.474$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(0\) \(10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{7})q^{7}+\cdots\)
8450.2.a.ct 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(-7\) \(0\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{4})q^{3}+q^{4}+(1+\beta _{4}+\cdots)q^{6}+\cdots\)
8450.2.a.cu 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(0\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-\beta _{8}q^{7}+\cdots\)
8450.2.a.cv 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(0\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-\beta _{8}q^{7}+\cdots\)
8450.2.a.cw 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(7\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{4})q^{3}+q^{4}+(-1-\beta _{4}+\cdots)q^{6}+\cdots\)
8450.2.a.cx 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(-7\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{4})q^{3}+q^{4}+(-1-\beta _{4}+\cdots)q^{6}+\cdots\)
8450.2.a.cy 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(0\) \(0\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+\beta _{8}q^{7}+\cdots\)
8450.2.a.cz 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+\beta _{8}q^{7}+\cdots\)
8450.2.a.da 8450.a 1.a $9$ $67.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(7\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{4})q^{3}+q^{4}+(1+\beta _{4})q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8450))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8450)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\)\(^{\oplus 2}\)