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

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

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

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

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