Properties

Label 1600.4.a
Level $1600$
Weight $4$
Character orbit 1600.a
Rep. character $\chi_{1600}(1,\cdot)$
Character field $\Q$
Dimension $111$
Newform subspaces $76$
Sturm bound $960$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 76 \)
Sturm bound: \(960\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1600))\).

Total New Old
Modular forms 756 117 639
Cusp forms 684 111 573
Eisenstein series 72 6 66

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

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(27\)
\(+\)\(-\)\(-\)\(28\)
\(-\)\(+\)\(-\)\(26\)
\(-\)\(-\)\(+\)\(30\)
Plus space\(+\)\(57\)
Minus space\(-\)\(54\)

Trace form

\( 111 q + 947 q^{9} + O(q^{10}) \) \( 111 q + 947 q^{9} + 70 q^{13} + 54 q^{17} - 168 q^{21} - 202 q^{29} - 176 q^{33} - 514 q^{37} - 282 q^{41} + 4559 q^{49} - 34 q^{53} + 400 q^{57} - 162 q^{61} + 1432 q^{69} - 146 q^{73} - 1488 q^{77} + 9247 q^{81} + 894 q^{89} - 1568 q^{93} + 582 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
1600.4.a.a \(1\) \(94.403\) \(\Q\) None \(0\) \(-10\) \(0\) \(-18\) \(-\) \(+\) \(q-10q^{3}-18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)
1600.4.a.b \(1\) \(94.403\) \(\Q\) None \(0\) \(-9\) \(0\) \(26\) \(-\) \(+\) \(q-9q^{3}+26q^{7}+54q^{9}-59q^{11}+\cdots\)
1600.4.a.c \(1\) \(94.403\) \(\Q\) None \(0\) \(-9\) \(0\) \(26\) \(+\) \(-\) \(q-9q^{3}+26q^{7}+54q^{9}+59q^{11}+\cdots\)
1600.4.a.d \(1\) \(94.403\) \(\Q\) None \(0\) \(-8\) \(0\) \(4\) \(+\) \(+\) \(q-8q^{3}+4q^{7}+37q^{9}-12q^{11}-58q^{13}+\cdots\)
1600.4.a.e \(1\) \(94.403\) \(\Q\) None \(0\) \(-8\) \(0\) \(16\) \(+\) \(+\) \(q-8q^{3}+2^{4}q^{7}+37q^{9}-40q^{11}+\cdots\)
1600.4.a.f \(1\) \(94.403\) \(\Q\) None \(0\) \(-7\) \(0\) \(-34\) \(+\) \(-\) \(q-7q^{3}-34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
1600.4.a.g \(1\) \(94.403\) \(\Q\) None \(0\) \(-7\) \(0\) \(-34\) \(-\) \(+\) \(q-7q^{3}-34q^{7}+22q^{9}+3^{3}q^{11}+\cdots\)
1600.4.a.h \(1\) \(94.403\) \(\Q\) None \(0\) \(-7\) \(0\) \(6\) \(-\) \(-\) \(q-7q^{3}+6q^{7}+22q^{9}-43q^{11}-28q^{13}+\cdots\)
1600.4.a.i \(1\) \(94.403\) \(\Q\) None \(0\) \(-7\) \(0\) \(6\) \(+\) \(+\) \(q-7q^{3}+6q^{7}+22q^{9}+43q^{11}+28q^{13}+\cdots\)
1600.4.a.j \(1\) \(94.403\) \(\Q\) None \(0\) \(-6\) \(0\) \(34\) \(+\) \(+\) \(q-6q^{3}+34q^{7}+9q^{9}-2^{4}q^{11}+58q^{13}+\cdots\)
1600.4.a.k \(1\) \(94.403\) \(\Q\) None \(0\) \(-5\) \(0\) \(-10\) \(-\) \(+\) \(q-5q^{3}-10q^{7}-2q^{9}-15q^{11}-8q^{13}+\cdots\)
1600.4.a.l \(1\) \(94.403\) \(\Q\) None \(0\) \(-5\) \(0\) \(-10\) \(-\) \(-\) \(q-5q^{3}-10q^{7}-2q^{9}+15q^{11}+8q^{13}+\cdots\)
1600.4.a.m \(1\) \(94.403\) \(\Q\) None \(0\) \(-5\) \(0\) \(2\) \(+\) \(+\) \(q-5q^{3}+2q^{7}-2q^{9}-39q^{11}-84q^{13}+\cdots\)
1600.4.a.n \(1\) \(94.403\) \(\Q\) None \(0\) \(-5\) \(0\) \(2\) \(-\) \(-\) \(q-5q^{3}+2q^{7}-2q^{9}+39q^{11}+84q^{13}+\cdots\)
1600.4.a.o \(1\) \(94.403\) \(\Q\) None \(0\) \(-4\) \(0\) \(-24\) \(+\) \(+\) \(q-4q^{3}-24q^{7}-11q^{9}+44q^{11}+\cdots\)
1600.4.a.p \(1\) \(94.403\) \(\Q\) None \(0\) \(-4\) \(0\) \(-16\) \(-\) \(+\) \(q-4q^{3}-2^{4}q^{7}-11q^{9}-60q^{11}+\cdots\)
1600.4.a.q \(1\) \(94.403\) \(\Q\) None \(0\) \(-4\) \(0\) \(16\) \(-\) \(+\) \(q-4q^{3}+2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
1600.4.a.r \(1\) \(94.403\) \(\Q\) None \(0\) \(-2\) \(0\) \(-6\) \(+\) \(+\) \(q-2q^{3}-6q^{7}-23q^{9}-60q^{11}+50q^{13}+\cdots\)
1600.4.a.s \(1\) \(94.403\) \(\Q\) None \(0\) \(-2\) \(0\) \(6\) \(-\) \(+\) \(q-2q^{3}+6q^{7}-23q^{9}+2^{5}q^{11}-38q^{13}+\cdots\)
1600.4.a.t \(1\) \(94.403\) \(\Q\) None \(0\) \(-2\) \(0\) \(26\) \(-\) \(-\) \(q-2q^{3}+26q^{7}-23q^{9}-28q^{11}+\cdots\)
1600.4.a.u \(1\) \(94.403\) \(\Q\) None \(0\) \(-2\) \(0\) \(26\) \(+\) \(-\) \(q-2q^{3}+26q^{7}-23q^{9}+28q^{11}+\cdots\)
1600.4.a.v \(1\) \(94.403\) \(\Q\) None \(0\) \(-1\) \(0\) \(-6\) \(-\) \(+\) \(q-q^{3}-6q^{7}-26q^{9}-19q^{11}+12q^{13}+\cdots\)
1600.4.a.w \(1\) \(94.403\) \(\Q\) None \(0\) \(-1\) \(0\) \(-6\) \(+\) \(-\) \(q-q^{3}-6q^{7}-26q^{9}+19q^{11}-12q^{13}+\cdots\)
1600.4.a.x \(1\) \(94.403\) \(\Q\) None \(0\) \(-1\) \(0\) \(26\) \(+\) \(-\) \(q-q^{3}+26q^{7}-26q^{9}-45q^{11}-44q^{13}+\cdots\)
1600.4.a.y \(1\) \(94.403\) \(\Q\) None \(0\) \(-1\) \(0\) \(26\) \(-\) \(+\) \(q-q^{3}+26q^{7}-26q^{9}+45q^{11}+44q^{13}+\cdots\)
1600.4.a.z \(1\) \(94.403\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-3^{3}q^{9}-92q^{13}-104q^{17}-130q^{29}+\cdots\)
1600.4.a.ba \(1\) \(94.403\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-3^{3}q^{9}-18q^{13}+94q^{17}+130q^{29}+\cdots\)
1600.4.a.bb \(1\) \(94.403\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-3^{3}q^{9}+92q^{13}+104q^{17}-130q^{29}+\cdots\)
1600.4.a.bc \(1\) \(94.403\) \(\Q\) None \(0\) \(1\) \(0\) \(-26\) \(+\) \(+\) \(q+q^{3}-26q^{7}-26q^{9}-45q^{11}+44q^{13}+\cdots\)
1600.4.a.bd \(1\) \(94.403\) \(\Q\) None \(0\) \(1\) \(0\) \(-26\) \(-\) \(-\) \(q+q^{3}-26q^{7}-26q^{9}+45q^{11}-44q^{13}+\cdots\)
1600.4.a.be \(1\) \(94.403\) \(\Q\) None \(0\) \(1\) \(0\) \(6\) \(-\) \(-\) \(q+q^{3}+6q^{7}-26q^{9}-19q^{11}-12q^{13}+\cdots\)
1600.4.a.bf \(1\) \(94.403\) \(\Q\) None \(0\) \(1\) \(0\) \(6\) \(+\) \(+\) \(q+q^{3}+6q^{7}-26q^{9}+19q^{11}+12q^{13}+\cdots\)
1600.4.a.bg \(1\) \(94.403\) \(\Q\) None \(0\) \(2\) \(0\) \(-26\) \(-\) \(-\) \(q+2q^{3}-26q^{7}-23q^{9}-28q^{11}+\cdots\)
1600.4.a.bh \(1\) \(94.403\) \(\Q\) None \(0\) \(2\) \(0\) \(-26\) \(+\) \(-\) \(q+2q^{3}-26q^{7}-23q^{9}+28q^{11}+\cdots\)
1600.4.a.bi \(1\) \(94.403\) \(\Q\) None \(0\) \(2\) \(0\) \(-6\) \(+\) \(+\) \(q+2q^{3}-6q^{7}-23q^{9}-2^{5}q^{11}-38q^{13}+\cdots\)
1600.4.a.bj \(1\) \(94.403\) \(\Q\) None \(0\) \(2\) \(0\) \(6\) \(+\) \(+\) \(q+2q^{3}+6q^{7}-23q^{9}+60q^{11}+50q^{13}+\cdots\)
1600.4.a.bk \(1\) \(94.403\) \(\Q\) None \(0\) \(4\) \(0\) \(-16\) \(+\) \(+\) \(q+4q^{3}-2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots\)
1600.4.a.bl \(1\) \(94.403\) \(\Q\) None \(0\) \(4\) \(0\) \(16\) \(+\) \(+\) \(q+4q^{3}+2^{4}q^{7}-11q^{9}+60q^{11}+\cdots\)
1600.4.a.bm \(1\) \(94.403\) \(\Q\) None \(0\) \(4\) \(0\) \(24\) \(-\) \(+\) \(q+4q^{3}+24q^{7}-11q^{9}-44q^{11}+\cdots\)
1600.4.a.bn \(1\) \(94.403\) \(\Q\) None \(0\) \(5\) \(0\) \(-2\) \(+\) \(-\) \(q+5q^{3}-2q^{7}-2q^{9}-39q^{11}+84q^{13}+\cdots\)
1600.4.a.bo \(1\) \(94.403\) \(\Q\) None \(0\) \(5\) \(0\) \(-2\) \(-\) \(+\) \(q+5q^{3}-2q^{7}-2q^{9}+39q^{11}-84q^{13}+\cdots\)
1600.4.a.bp \(1\) \(94.403\) \(\Q\) None \(0\) \(5\) \(0\) \(10\) \(-\) \(-\) \(q+5q^{3}+10q^{7}-2q^{9}-15q^{11}+8q^{13}+\cdots\)
1600.4.a.bq \(1\) \(94.403\) \(\Q\) None \(0\) \(5\) \(0\) \(10\) \(-\) \(+\) \(q+5q^{3}+10q^{7}-2q^{9}+15q^{11}-8q^{13}+\cdots\)
1600.4.a.br \(1\) \(94.403\) \(\Q\) None \(0\) \(6\) \(0\) \(-34\) \(-\) \(+\) \(q+6q^{3}-34q^{7}+9q^{9}+2^{4}q^{11}+58q^{13}+\cdots\)
1600.4.a.bs \(1\) \(94.403\) \(\Q\) None \(0\) \(7\) \(0\) \(-6\) \(-\) \(+\) \(q+7q^{3}-6q^{7}+22q^{9}-43q^{11}+28q^{13}+\cdots\)
1600.4.a.bt \(1\) \(94.403\) \(\Q\) None \(0\) \(7\) \(0\) \(-6\) \(+\) \(-\) \(q+7q^{3}-6q^{7}+22q^{9}+43q^{11}-28q^{13}+\cdots\)
1600.4.a.bu \(1\) \(94.403\) \(\Q\) None \(0\) \(7\) \(0\) \(34\) \(+\) \(+\) \(q+7q^{3}+34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
1600.4.a.bv \(1\) \(94.403\) \(\Q\) None \(0\) \(7\) \(0\) \(34\) \(-\) \(-\) \(q+7q^{3}+34q^{7}+22q^{9}+3^{3}q^{11}+\cdots\)
1600.4.a.bw \(1\) \(94.403\) \(\Q\) None \(0\) \(8\) \(0\) \(-16\) \(+\) \(+\) \(q+8q^{3}-2^{4}q^{7}+37q^{9}+40q^{11}+\cdots\)
1600.4.a.bx \(1\) \(94.403\) \(\Q\) None \(0\) \(8\) \(0\) \(-4\) \(-\) \(+\) \(q+8q^{3}-4q^{7}+37q^{9}+12q^{11}-58q^{13}+\cdots\)
1600.4.a.by \(1\) \(94.403\) \(\Q\) None \(0\) \(9\) \(0\) \(-26\) \(-\) \(-\) \(q+9q^{3}-26q^{7}+54q^{9}-59q^{11}+\cdots\)
1600.4.a.bz \(1\) \(94.403\) \(\Q\) None \(0\) \(9\) \(0\) \(-26\) \(+\) \(+\) \(q+9q^{3}-26q^{7}+54q^{9}+59q^{11}+\cdots\)
1600.4.a.ca \(1\) \(94.403\) \(\Q\) None \(0\) \(10\) \(0\) \(18\) \(+\) \(+\) \(q+10q^{3}+18q^{7}+73q^{9}+2^{4}q^{11}+\cdots\)
1600.4.a.cb \(2\) \(94.403\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(-14\) \(0\) \(18\) \(+\) \(-\) \(q+(-7-\beta )q^{3}+(9+11\beta )q^{7}+(3^{3}+14\beta )q^{9}+\cdots\)
1600.4.a.cc \(2\) \(94.403\) \(\Q(\sqrt{29}) \) None \(0\) \(-8\) \(0\) \(-8\) \(+\) \(-\) \(q-4q^{3}-4q^{7}-11q^{9}-2\beta q^{11}+\beta q^{13}+\cdots\)
1600.4.a.cd \(2\) \(94.403\) \(\Q(\sqrt{6}) \) None \(0\) \(-8\) \(0\) \(8\) \(+\) \(+\) \(q+(-4+\beta )q^{3}+(4+5\beta )q^{7}+(13-8\beta )q^{9}+\cdots\)
1600.4.a.ce \(2\) \(94.403\) \(\Q(\sqrt{6}) \) None \(0\) \(-4\) \(0\) \(4\) \(+\) \(-\) \(q+(-2+\beta )q^{3}+(2+3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
1600.4.a.cf \(2\) \(94.403\) \(\Q(\sqrt{6}) \) None \(0\) \(-4\) \(0\) \(4\) \(-\) \(-\) \(q+(-2+\beta )q^{3}+(2+3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
1600.4.a.cg \(2\) \(94.403\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta q^{3}-7\beta q^{7}-7q^{9}-2\beta q^{11}-62q^{13}+\cdots\)
1600.4.a.ch \(2\) \(94.403\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta q^{3}-3\beta q^{7}+13q^{9}+2\beta q^{11}+\cdots\)
1600.4.a.ci \(2\) \(94.403\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta q^{3}+\beta q^{7}+5^{2}q^{9}+6\beta q^{11}+34q^{13}+\cdots\)
1600.4.a.cj \(2\) \(94.403\) \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{3}-\beta q^{7}+7^{2}q^{9}-20q^{11}-6\beta q^{13}+\cdots\)
1600.4.a.ck \(2\) \(94.403\) \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{3}-\beta q^{7}+7^{2}q^{9}+20q^{11}+6\beta q^{13}+\cdots\)
1600.4.a.cl \(2\) \(94.403\) \(\Q(\sqrt{6}) \) None \(0\) \(4\) \(0\) \(-4\) \(+\) \(-\) \(q+(2+\beta )q^{3}+(-2+3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
1600.4.a.cm \(2\) \(94.403\) \(\Q(\sqrt{6}) \) None \(0\) \(4\) \(0\) \(-4\) \(-\) \(-\) \(q+(2+\beta )q^{3}+(-2+3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
1600.4.a.cn \(2\) \(94.403\) \(\Q(\sqrt{6}) \) None \(0\) \(8\) \(0\) \(-8\) \(+\) \(+\) \(q+(4+\beta )q^{3}+(-4+5\beta )q^{7}+(13+8\beta )q^{9}+\cdots\)
1600.4.a.co \(2\) \(94.403\) \(\Q(\sqrt{29}) \) None \(0\) \(8\) \(0\) \(8\) \(+\) \(-\) \(q+4q^{3}+4q^{7}-11q^{9}+2\beta q^{11}+\beta q^{13}+\cdots\)
1600.4.a.cp \(2\) \(94.403\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(14\) \(0\) \(-18\) \(+\) \(-\) \(q+(7-\beta )q^{3}+(-9+11\beta )q^{7}+(3^{3}-14\beta )q^{9}+\cdots\)
1600.4.a.cq \(3\) \(94.403\) 3.3.5685.1 None \(0\) \(-5\) \(0\) \(-2\) \(+\) \(+\) \(q+(-2-\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1600.4.a.cr \(3\) \(94.403\) 3.3.5685.1 None \(0\) \(-5\) \(0\) \(-2\) \(+\) \(-\) \(q+(-2-\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1600.4.a.cs \(3\) \(94.403\) 3.3.5685.1 None \(0\) \(5\) \(0\) \(2\) \(+\) \(-\) \(q+(2+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(7+\cdots)q^{9}+\cdots\)
1600.4.a.ct \(3\) \(94.403\) 3.3.5685.1 None \(0\) \(5\) \(0\) \(2\) \(+\) \(+\) \(q+(2+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(7+\cdots)q^{9}+\cdots\)
1600.4.a.cu \(4\) \(94.403\) 4.4.37485.2 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(19-3\beta _{2}+\cdots)q^{9}+\cdots\)
1600.4.a.cv \(4\) \(94.403\) 4.4.37485.2 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(19-3\beta _{2}+\cdots)q^{9}+\cdots\)
1600.4.a.cw \(4\) \(94.403\) 4.4.2106005.1 None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(24+\beta _{2}+\cdots)q^{9}+\cdots\)
1600.4.a.cx \(4\) \(94.403\) 4.4.2106005.1 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(24+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1600))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1600)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)