Properties

Label 7200.2.a
Level $7200$
Weight $2$
Character orbit 7200.a
Rep. character $\chi_{7200}(1,\cdot)$
Character field $\Q$
Dimension $95$
Newform subspaces $72$
Sturm bound $2880$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 72 \)
Sturm bound: \(2880\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7200))\).

Total New Old
Modular forms 1536 95 1441
Cusp forms 1345 95 1250
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.

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(10\)
\(+\)\(-\)\(+\)\(-\)\(15\)
\(+\)\(-\)\(-\)\(+\)\(14\)
\(-\)\(+\)\(+\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(+\)\(10\)
\(-\)\(-\)\(+\)\(+\)\(12\)
\(-\)\(-\)\(-\)\(-\)\(16\)
Plus space\(+\)\(45\)
Minus space\(-\)\(50\)

Trace form

\( 95q + O(q^{10}) \) \( 95q - 6q^{13} - 2q^{17} - 6q^{29} - 6q^{37} - 10q^{41} + 87q^{49} + 18q^{53} - 14q^{61} - 34q^{73} - 16q^{77} + 22q^{89} - 10q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7200))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
7200.2.a.a \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q-4q^{7}-4q^{13}+8q^{19}-4q^{23}+6q^{29}+\cdots\)
7200.2.a.b \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q-4q^{7}+2q^{13}-6q^{17}+4q^{23}+2q^{29}+\cdots\)
7200.2.a.c \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q-4q^{7}+4q^{13}-8q^{19}-4q^{23}+6q^{29}+\cdots\)
7200.2.a.d \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(q-4q^{7}+4q^{11}-6q^{13}+2q^{17}-4q^{19}+\cdots\)
7200.2.a.e \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q-4q^{7}+4q^{11}+2q^{13}-6q^{17}+4q^{19}+\cdots\)
7200.2.a.f \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(-\) \(q-3q^{7}-4q^{11}-7q^{13}-4q^{17}-q^{19}+\cdots\)
7200.2.a.g \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q-3q^{7}-5q^{13}+5q^{19}-4q^{23}-4q^{29}+\cdots\)
7200.2.a.h \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(-\) \(q-3q^{7}+5q^{13}-5q^{19}-4q^{23}-4q^{29}+\cdots\)
7200.2.a.i \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q-3q^{7}+4q^{11}+7q^{13}+4q^{17}+q^{19}+\cdots\)
7200.2.a.j \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(q-2q^{7}-6q^{11}-2q^{13}+6q^{17}-4q^{19}+\cdots\)
7200.2.a.k \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(q-2q^{7}-5q^{11}-5q^{17}+5q^{19}-6q^{23}+\cdots\)
7200.2.a.l \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q-2q^{7}-4q^{11}+6q^{13}+2q^{17}-8q^{19}+\cdots\)
7200.2.a.m \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-2q^{7}-2q^{11}-2q^{17}+4q^{19}-2q^{29}+\cdots\)
7200.2.a.n \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-2q^{7}+2q^{11}+2q^{17}+4q^{19}+2q^{29}+\cdots\)
7200.2.a.o \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q-2q^{7}+5q^{11}+5q^{17}-5q^{19}-6q^{23}+\cdots\)
7200.2.a.p \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q-2q^{7}+6q^{11}+2q^{13}-6q^{17}+4q^{19}+\cdots\)
7200.2.a.q \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{7}-4q^{11}-3q^{13}+4q^{17}+q^{19}+\cdots\)
7200.2.a.r \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-q^{13}+3q^{19}+4q^{23}-4q^{29}+\cdots\)
7200.2.a.s \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{7}+q^{13}-3q^{19}+4q^{23}-4q^{29}+\cdots\)
7200.2.a.t \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{7}+4q^{11}+3q^{13}-4q^{17}-q^{19}+\cdots\)
7200.2.a.u \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-4q^{11}-2q^{13}-2q^{17}+8q^{19}+\cdots\)
7200.2.a.v \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q-6q^{13}+2q^{17}+10q^{29}+2q^{37}+\cdots\)
7200.2.a.w \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-4q^{13}-2q^{17}+4q^{29}+12q^{37}+\cdots\)
7200.2.a.x \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-4q^{13}+2q^{17}-4q^{29}+12q^{37}+\cdots\)
7200.2.a.y \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-4q^{13}+8q^{17}-10q^{29}-12q^{37}+\cdots\)
7200.2.a.z \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-2q^{13}+6q^{17}-4q^{19}+8q^{23}+\cdots\)
7200.2.a.ba \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q-2q^{13}+6q^{17}+4q^{19}-8q^{23}+\cdots\)
7200.2.a.bb \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+4q^{13}-8q^{17}-10q^{29}+12q^{37}+\cdots\)
7200.2.a.bc \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+4q^{13}-2q^{17}-4q^{29}-12q^{37}+\cdots\)
7200.2.a.bd \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+4q^{13}+2q^{17}+4q^{29}-12q^{37}+\cdots\)
7200.2.a.be \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+6q^{13}-8q^{17}-4q^{29}+2q^{37}+\cdots\)
7200.2.a.bf \(1\) \(57.492\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+6q^{13}+8q^{17}+4q^{29}+2q^{37}+\cdots\)
7200.2.a.bg \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+4q^{11}-2q^{13}-2q^{17}-8q^{19}+\cdots\)
7200.2.a.bh \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(q+q^{7}-4q^{11}+3q^{13}-4q^{17}+q^{19}+\cdots\)
7200.2.a.bi \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(q+q^{7}-q^{13}-3q^{19}-4q^{23}-4q^{29}+\cdots\)
7200.2.a.bj \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+q^{7}+q^{13}+3q^{19}-4q^{23}-4q^{29}+\cdots\)
7200.2.a.bk \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}+4q^{11}-3q^{13}+4q^{17}-q^{19}+\cdots\)
7200.2.a.bl \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+2q^{7}-6q^{11}+2q^{13}-6q^{17}-4q^{19}+\cdots\)
7200.2.a.bm \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(q+2q^{7}-5q^{11}+5q^{17}+5q^{19}+6q^{23}+\cdots\)
7200.2.a.bn \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(+\) \(+\) \(q+2q^{7}-2q^{11}+2q^{17}-4q^{19}+2q^{29}+\cdots\)
7200.2.a.bo \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(-\) \(+\) \(+\) \(q+2q^{7}+2q^{11}-2q^{17}-4q^{19}-2q^{29}+\cdots\)
7200.2.a.bp \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(q+2q^{7}+4q^{11}+6q^{13}+2q^{17}+8q^{19}+\cdots\)
7200.2.a.bq \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+2q^{7}+5q^{11}-5q^{17}-5q^{19}+6q^{23}+\cdots\)
7200.2.a.br \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+2q^{7}+6q^{11}-2q^{13}+6q^{17}+4q^{19}+\cdots\)
7200.2.a.bs \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(3\) \(+\) \(-\) \(+\) \(q+3q^{7}-4q^{11}+7q^{13}+4q^{17}-q^{19}+\cdots\)
7200.2.a.bt \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(3\) \(-\) \(-\) \(+\) \(q+3q^{7}-5q^{13}-5q^{19}+4q^{23}-4q^{29}+\cdots\)
7200.2.a.bu \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+3q^{7}+5q^{13}+5q^{19}+4q^{23}-4q^{29}+\cdots\)
7200.2.a.bv \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+3q^{7}+4q^{11}-7q^{13}-4q^{17}+q^{19}+\cdots\)
7200.2.a.bw \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(q+4q^{7}-4q^{11}-6q^{13}+2q^{17}+4q^{19}+\cdots\)
7200.2.a.bx \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+4q^{7}-4q^{11}+2q^{13}-6q^{17}-4q^{19}+\cdots\)
7200.2.a.by \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(+\) \(-\) \(-\) \(q+4q^{7}-4q^{13}-8q^{19}+4q^{23}+6q^{29}+\cdots\)
7200.2.a.bz \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(q+4q^{7}+2q^{13}-6q^{17}-4q^{23}+2q^{29}+\cdots\)
7200.2.a.ca \(1\) \(57.492\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+4q^{7}+4q^{13}+8q^{19}+4q^{23}+6q^{29}+\cdots\)
7200.2.a.cb \(2\) \(57.492\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(-6\) \(+\) \(-\) \(-\) \(q+(-3-\beta )q^{7}+(1+3\beta )q^{23}+6q^{29}+\cdots\)
7200.2.a.cc \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q-2q^{7}-\beta q^{11}-\beta q^{13}-\beta q^{17}+4q^{23}+\cdots\)
7200.2.a.cd \(2\) \(57.492\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-\beta q^{7}-2q^{11}-3q^{13}+2\beta q^{17}+\cdots\)
7200.2.a.ce \(2\) \(57.492\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-\beta q^{7}-2q^{11}+3q^{13}+2\beta q^{17}+\cdots\)
7200.2.a.cf \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-2\beta q^{7}-\beta q^{11}-4q^{13}-7q^{17}+\cdots\)
7200.2.a.cg \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta q^{7}-\beta q^{11}-4q^{13}-2q^{17}-2\beta q^{23}+\cdots\)
7200.2.a.ch \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-\beta q^{7}+\beta q^{11}-4q^{13}+2q^{17}+2\beta q^{23}+\cdots\)
7200.2.a.ci \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta q^{7}+2\beta q^{11}-q^{13}-2q^{17}+3\beta q^{19}+\cdots\)
7200.2.a.cj \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-\beta q^{7}-2\beta q^{11}-q^{13}+2q^{17}+3\beta q^{19}+\cdots\)
7200.2.a.ck \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-\beta q^{7}+2\beta q^{11}+q^{13}-2q^{17}-3\beta q^{19}+\cdots\)
7200.2.a.cl \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-\beta q^{7}-2\beta q^{11}+q^{13}+2q^{17}-3\beta q^{19}+\cdots\)
7200.2.a.cm \(2\) \(57.492\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta q^{7}+2\beta q^{11}+2q^{13}+2q^{17}+\cdots\)
7200.2.a.cn \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q-2\beta q^{7}+\beta q^{11}+4q^{13}+7q^{17}+\cdots\)
7200.2.a.co \(2\) \(57.492\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-\beta q^{7}+2q^{11}-3q^{13}-2\beta q^{17}+\cdots\)
7200.2.a.cp \(2\) \(57.492\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta q^{7}+2q^{11}+3q^{13}-2\beta q^{17}+\cdots\)
7200.2.a.cq \(2\) \(57.492\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(4\) \(+\) \(-\) \(-\) \(q+2q^{7}+\beta q^{11}-\beta q^{13}-\beta q^{17}-4q^{23}+\cdots\)
7200.2.a.cr \(2\) \(57.492\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(6\) \(-\) \(-\) \(-\) \(q+(3+\beta )q^{7}+(-1-3\beta )q^{23}+6q^{29}+\cdots\)
7200.2.a.cs \(4\) \(57.492\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{7}-\beta _{2}q^{11}+\beta _{2}q^{13}+\beta _{3}q^{17}+\cdots\)
7200.2.a.ct \(4\) \(57.492\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{7}+\beta _{2}q^{11}+\beta _{2}q^{13}-\beta _{3}q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7200)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1440))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3600))\)\(^{\oplus 2}\)