Properties

Label 9800.2.a
Level $9800$
Weight $2$
Character orbit 9800.a
Rep. character $\chi_{9800}(1,\cdot)$
Character field $\Q$
Dimension $195$
Newform subspaces $81$
Sturm bound $3360$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 81 \)
Sturm bound: \(3360\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9800))\).

Total New Old
Modular forms 1776 195 1581
Cusp forms 1585 195 1390
Eisenstein series 191 0 191

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

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(19\)
\(+\)\(+\)\(-\)\(-\)\(26\)
\(+\)\(-\)\(+\)\(-\)\(28\)
\(+\)\(-\)\(-\)\(+\)\(24\)
\(-\)\(+\)\(+\)\(-\)\(25\)
\(-\)\(+\)\(-\)\(+\)\(22\)
\(-\)\(-\)\(+\)\(+\)\(24\)
\(-\)\(-\)\(-\)\(-\)\(27\)
Plus space\(+\)\(89\)
Minus space\(-\)\(106\)

Trace form

\( 195 q - 2 q^{3} + 197 q^{9} + O(q^{10}) \) \( 195 q - 2 q^{3} + 197 q^{9} - 6 q^{11} + 6 q^{17} - 4 q^{19} - 20 q^{27} + 2 q^{29} - 8 q^{31} + 24 q^{33} - 10 q^{37} + 16 q^{39} - 8 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{51} + 2 q^{53} - 4 q^{57} - 6 q^{59} - 36 q^{61} - 20 q^{67} - 12 q^{69} + 24 q^{71} + 2 q^{73} - 32 q^{79} + 243 q^{81} + 10 q^{83} - 12 q^{87} + 8 q^{89} - 12 q^{93} + 22 q^{97} - 64 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\) 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
9800.2.a.a 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+6q^{9}-5q^{11}-5q^{13}-7q^{17}+\cdots\)
9800.2.a.b 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-3\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+6q^{9}-q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.c 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+6q^{9}+q^{11}+4q^{13}+5q^{17}+\cdots\)
9800.2.a.d 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-4q^{11}-4q^{13}+4q^{19}+\cdots\)
9800.2.a.e 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-3q^{11}+2q^{13}-4q^{17}+\cdots\)
9800.2.a.f 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-3q^{11}+2q^{13}-4q^{17}+\cdots\)
9800.2.a.g 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-q^{11}-3q^{13}-2q^{17}+\cdots\)
9800.2.a.h 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+q^{11}-4q^{13}-6q^{19}+\cdots\)
9800.2.a.i 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{11}-2q^{13}+2q^{19}+\cdots\)
9800.2.a.j 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.k 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.l 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+5q^{11}-8q^{17}+2q^{19}+\cdots\)
9800.2.a.m 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-5q^{11}-7q^{13}+3q^{17}+\cdots\)
9800.2.a.n 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-5q^{11}+q^{13}+3q^{17}+\cdots\)
9800.2.a.o 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-2q^{11}+4q^{13}-6q^{19}+\cdots\)
9800.2.a.p 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots\)
9800.2.a.q 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}-q^{11}+6q^{13}-7q^{17}+\cdots\)
9800.2.a.r 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}+2q^{11}-4q^{17}-2q^{19}+\cdots\)
9800.2.a.s 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}+3q^{11}-6q^{13}-5q^{17}+\cdots\)
9800.2.a.t 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{9}+3q^{11}+q^{13}-5q^{17}+\cdots\)
9800.2.a.u 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{9}-4q^{11}+2q^{13}-6q^{17}-8q^{19}+\cdots\)
9800.2.a.v 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{9}+q^{11}-2q^{13}-4q^{17}+2q^{19}+\cdots\)
9800.2.a.w 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{9}+q^{11}+2q^{13}+4q^{17}+2q^{19}+\cdots\)
9800.2.a.x 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{9}+4q^{11}-2q^{13}+2q^{17}-4q^{19}+\cdots\)
9800.2.a.y 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}-5q^{11}+7q^{13}-3q^{17}+\cdots\)
9800.2.a.z 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}-2q^{11}-4q^{13}+6q^{19}+\cdots\)
9800.2.a.ba 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}-q^{11}-6q^{13}+7q^{17}+\cdots\)
9800.2.a.bb 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}-q^{11}-q^{13}-3q^{17}+\cdots\)
9800.2.a.bc 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}+2q^{11}+4q^{17}+2q^{19}+\cdots\)
9800.2.a.bd 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}+3q^{11}-q^{13}+5q^{17}+\cdots\)
9800.2.a.be 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{9}+3q^{11}+6q^{13}+5q^{17}+\cdots\)
9800.2.a.bf 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-4q^{11}+4q^{13}+4q^{19}+\cdots\)
9800.2.a.bg 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-3q^{11}-2q^{13}+4q^{17}+\cdots\)
9800.2.a.bh 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-3q^{11}-2q^{13}+4q^{17}+\cdots\)
9800.2.a.bi 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-q^{11}+3q^{13}+2q^{17}+\cdots\)
9800.2.a.bj 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-2q^{17}+2q^{19}-8q^{23}+\cdots\)
9800.2.a.bk 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}+q^{11}+4q^{13}-6q^{19}+\cdots\)
9800.2.a.bl 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}+4q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.bm 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}+4q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.bn 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{19}+\cdots\)
9800.2.a.bo 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}+5q^{11}+8q^{17}+2q^{19}+\cdots\)
9800.2.a.bp 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(3\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+6q^{9}-q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.bq 9800.a 1.a $1$ $78.253$ \(\Q\) None \(0\) \(3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+6q^{9}+q^{11}-4q^{13}-5q^{17}+\cdots\)
9800.2.a.br 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-2\beta q^{9}+(-2+2\beta )q^{11}+\cdots\)
9800.2.a.bs 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-2\beta q^{9}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
9800.2.a.bt 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
9800.2.a.bu 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(5+\beta )q^{9}+(4-\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
9800.2.a.bv 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{9}+6q^{11}-4\beta q^{13}+\beta q^{17}+\cdots\)
9800.2.a.bw 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+5q^{9}-4q^{11}-\beta q^{13}-2\beta q^{17}+\cdots\)
9800.2.a.bx 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
9800.2.a.by 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1+\beta )q^{9}-\beta q^{11}+(2-3\beta )q^{13}+\cdots\)
9800.2.a.bz 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+2\beta q^{9}+(-2-2\beta )q^{11}+\cdots\)
9800.2.a.ca 9800.a 1.a $2$ $78.253$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+2\beta q^{9}-q^{11}+(1-\beta )q^{13}+\cdots\)
9800.2.a.cb 9800.a 1.a $3$ $78.253$ \(\Q(\zeta_{18})^+\) None \(0\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cc 9800.a 1.a $3$ $78.253$ \(\Q(\zeta_{18})^+\) None \(0\) \(-3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cd 9800.a 1.a $3$ $78.253$ 3.3.568.1 None \(0\) \(-1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{2})q^{9}+(2-\beta _{2})q^{11}+\cdots\)
9800.2.a.ce 9800.a 1.a $3$ $78.253$ 3.3.1944.1 None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(3+\beta _{1}+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.cf 9800.a 1.a $3$ $78.253$ 3.3.1944.1 None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(3+\beta _{1}+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.cg 9800.a 1.a $3$ $78.253$ 3.3.568.1 None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{2})q^{9}+(2-\beta _{2})q^{11}+\cdots\)
9800.2.a.ch 9800.a 1.a $3$ $78.253$ \(\Q(\zeta_{18})^+\) None \(0\) \(3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots\)
9800.2.a.ci 9800.a 1.a $3$ $78.253$ \(\Q(\zeta_{18})^+\) None \(0\) \(3\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots\)
9800.2.a.cj 9800.a 1.a $4$ $78.253$ 4.4.43449.1 None \(0\) \(-3\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.ck 9800.a 1.a $4$ $78.253$ 4.4.43449.1 None \(0\) \(-3\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cl 9800.a 1.a $4$ $78.253$ 4.4.16448.2 None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cm 9800.a 1.a $4$ $78.253$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{3}q^{9}-q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cn 9800.a 1.a $4$ $78.253$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{3}q^{9}-q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.co 9800.a 1.a $4$ $78.253$ 4.4.13448.1 None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(2+\beta _{2})q^{11}+\cdots\)
9800.2.a.cp 9800.a 1.a $4$ $78.253$ 4.4.13448.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(2+\beta _{2})q^{11}+\cdots\)
9800.2.a.cq 9800.a 1.a $4$ $78.253$ \(\Q(\sqrt{7}, \sqrt{15})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cr 9800.a 1.a $4$ $78.253$ \(\Q(\sqrt{7}, \sqrt{15})\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cs 9800.a 1.a $4$ $78.253$ 4.4.16448.2 None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.ct 9800.a 1.a $4$ $78.253$ 4.4.43449.1 None \(0\) \(3\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cu 9800.a 1.a $4$ $78.253$ 4.4.43449.1 None \(0\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cv 9800.a 1.a $6$ $78.253$ 6.6.239575536.1 None \(0\) \(-1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cw 9800.a 1.a $6$ $78.253$ 6.6.239575536.1 None \(0\) \(-1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cx 9800.a 1.a $6$ $78.253$ 6.6.239575536.1 None \(0\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cy 9800.a 1.a $6$ $78.253$ 6.6.239575536.1 None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cz 9800.a 1.a $8$ $78.253$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
9800.2.a.da 9800.a 1.a $8$ $78.253$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
9800.2.a.db 9800.a 1.a $10$ $78.253$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{7}q^{3}+(1-\beta _{4}-\beta _{8})q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.dc 9800.a 1.a $10$ $78.253$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{7}q^{3}+(1-\beta _{4}-\beta _{8})q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(980))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\)\(^{\oplus 2}\)