Properties

Label 5850.2.a
Level $5850$
Weight $2$
Character orbit 5850.a
Rep. character $\chi_{5850}(1,\cdot)$
Character field $\Q$
Dimension $95$
Newform subspaces $72$
Sturm bound $2520$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 72 \)
Sturm bound: \(2520\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\), \(23\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5850))\).

Total New Old
Modular forms 1308 95 1213
Cusp forms 1213 95 1118
Eisenstein series 95 0 95

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

\(2\)\(3\)\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(43\)
Minus space\(-\)\(52\)

Trace form

\( 95q + q^{2} + 95q^{4} + q^{8} + O(q^{10}) \) \( 95q + q^{2} + 95q^{4} + q^{8} + 4q^{11} - q^{13} - 6q^{14} + 95q^{16} - 12q^{17} - 8q^{19} - 20q^{22} + 12q^{23} - 3q^{26} + 2q^{29} + 24q^{31} + q^{32} + 6q^{34} + 14q^{37} + 4q^{38} + 2q^{41} + 14q^{43} + 4q^{44} + 4q^{46} - 8q^{47} + 105q^{49} - q^{52} - 2q^{53} - 6q^{56} + 22q^{58} + 36q^{59} + 2q^{61} - 8q^{62} + 95q^{64} - 16q^{67} - 12q^{68} + 80q^{71} - 10q^{73} - 12q^{74} - 8q^{76} - 24q^{77} - 4q^{79} + 14q^{82} + 24q^{83} + 24q^{86} - 20q^{88} + 34q^{89} - 6q^{91} + 12q^{92} + 26q^{94} + 2q^{97} - 23q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5850))\) 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 13
5850.2.a.a \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{11}+q^{13}+\cdots\)
5850.2.a.b \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}-q^{11}-q^{13}+\cdots\)
5850.2.a.c \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}+q^{13}+4q^{14}+\cdots\)
5850.2.a.d \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}+4q^{11}-q^{13}+\cdots\)
5850.2.a.e \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}+6q^{11}+q^{13}+\cdots\)
5850.2.a.f \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{11}+q^{13}+\cdots\)
5850.2.a.g \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-q^{13}+2q^{14}+\cdots\)
5850.2.a.h \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+4q^{11}+q^{13}+\cdots\)
5850.2.a.i \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+6q^{11}-q^{13}+\cdots\)
5850.2.a.j \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{7}-q^{8}+3q^{11}+q^{13}+\cdots\)
5850.2.a.k \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{7}-q^{8}+5q^{11}-q^{13}+\cdots\)
5850.2.a.l \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{8}-4q^{11}+q^{13}+q^{16}+\cdots\)
5850.2.a.m \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{8}+q^{13}+q^{16}-6q^{17}+\cdots\)
5850.2.a.n \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{8}+3q^{11}+q^{13}+q^{16}+\cdots\)
5850.2.a.o \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{8}+6q^{11}+q^{13}+q^{16}+\cdots\)
5850.2.a.p \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}-6q^{11}-q^{13}+\cdots\)
5850.2.a.q \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{11}-q^{13}+\cdots\)
5850.2.a.r \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}-6q^{11}+q^{13}+\cdots\)
5850.2.a.s \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}-4q^{11}+q^{13}+\cdots\)
5850.2.a.t \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}-2q^{11}-q^{13}+\cdots\)
5850.2.a.u \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}-q^{13}+\cdots\)
5850.2.a.v \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}+4q^{11}+q^{13}+\cdots\)
5850.2.a.w \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(3\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}+3q^{7}-q^{8}+3q^{11}+q^{13}+\cdots\)
5850.2.a.x \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-2q^{11}+q^{13}+\cdots\)
5850.2.a.y \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-q^{13}-4q^{14}+\cdots\)
5850.2.a.z \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-q^{13}-4q^{14}+\cdots\)
5850.2.a.ba \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}+6q^{11}-q^{13}+\cdots\)
5850.2.a.bb \(1\) \(46.712\) \(\Q\) None \(-1\) \(0\) \(0\) \(5\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+5q^{7}-q^{8}+3q^{11}+q^{13}+\cdots\)
5850.2.a.bc \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-5\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-5q^{7}+q^{8}+3q^{11}-q^{13}+\cdots\)
5850.2.a.bd \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-4q^{7}+q^{8}-2q^{11}-q^{13}+\cdots\)
5850.2.a.be \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-4q^{7}+q^{8}+q^{13}-4q^{14}+\cdots\)
5850.2.a.bf \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-3q^{7}+q^{8}+3q^{11}-q^{13}+\cdots\)
5850.2.a.bg \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-6q^{11}-q^{13}+\cdots\)
5850.2.a.bh \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-4q^{11}+q^{13}+\cdots\)
5850.2.a.bi \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-4q^{11}+q^{13}+\cdots\)
5850.2.a.bj \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-2q^{11}+q^{13}+\cdots\)
5850.2.a.bk \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-q^{13}-2q^{14}+\cdots\)
5850.2.a.bl \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}+2q^{11}+q^{13}+\cdots\)
5850.2.a.bm \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-q^{7}+q^{8}-3q^{11}+q^{13}+\cdots\)
5850.2.a.bn \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-q^{7}+q^{8}+2q^{11}+q^{13}+\cdots\)
5850.2.a.bo \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}-4q^{11}-q^{13}+q^{16}+\cdots\)
5850.2.a.bp \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}-4q^{11}-q^{13}+q^{16}+\cdots\)
5850.2.a.bq \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}-q^{13}+q^{16}+2q^{17}+\cdots\)
5850.2.a.br \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{8}+3q^{11}-q^{13}+q^{16}+\cdots\)
5850.2.a.bs \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{8}+6q^{11}-q^{13}+q^{16}+\cdots\)
5850.2.a.bt \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}+3q^{11}-q^{13}+\cdots\)
5850.2.a.bu \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{11}+q^{13}+\cdots\)
5850.2.a.bv \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-4q^{11}+q^{13}+\cdots\)
5850.2.a.bw \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}-q^{13}+\cdots\)
5850.2.a.bx \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}+6q^{11}+q^{13}+\cdots\)
5850.2.a.by \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}-4q^{11}-q^{13}+\cdots\)
5850.2.a.bz \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}-q^{11}+q^{13}+\cdots\)
5850.2.a.ca \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}-q^{13}+4q^{14}+\cdots\)
5850.2.a.cb \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+2q^{11}+q^{13}+\cdots\)
5850.2.a.cc \(1\) \(46.712\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+6q^{11}-q^{13}+\cdots\)
5850.2.a.cd \(2\) \(46.712\) \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+(-2+\beta )q^{7}-q^{8}+3q^{11}+\cdots\)
5850.2.a.ce \(2\) \(46.712\) \(\Q(\sqrt{13}) \) None \(-2\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}-q^{13}+\cdots\)
5850.2.a.cf \(2\) \(46.712\) \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}-2\beta q^{11}+\cdots\)
5850.2.a.cg \(2\) \(46.712\) \(\Q(\sqrt{41}) \) None \(-2\) \(0\) \(0\) \(3\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
5850.2.a.ch \(2\) \(46.712\) \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(0\) \(4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}-3q^{11}+\cdots\)
5850.2.a.ci \(2\) \(46.712\) \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}-3q^{11}+\cdots\)
5850.2.a.cj \(2\) \(46.712\) \(\Q(\sqrt{41}) \) None \(2\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots\)
5850.2.a.ck \(2\) \(46.712\) \(\Q(\sqrt{13}) \) None \(2\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}-q^{13}+\cdots\)
5850.2.a.cl \(2\) \(46.712\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+\beta q^{7}+q^{8}-2\beta q^{11}+\cdots\)
5850.2.a.cm \(2\) \(46.712\) \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+(1+\beta )q^{7}+q^{8}-2\beta q^{11}+\cdots\)
5850.2.a.cn \(2\) \(46.712\) \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(0\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}+(2+\beta )q^{7}+q^{8}+3q^{11}+\cdots\)
5850.2.a.co \(3\) \(46.712\) 3.3.148.1 None \(-3\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+(-1-\beta _{1})q^{7}-q^{8}+2q^{11}+\cdots\)
5850.2.a.cp \(3\) \(46.712\) 3.3.940.1 None \(-3\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{7}-q^{8}+\cdots\)
5850.2.a.cq \(3\) \(46.712\) 3.3.148.1 None \(-3\) \(0\) \(0\) \(4\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+(1+\beta _{1})q^{7}-q^{8}-2q^{11}+\cdots\)
5850.2.a.cr \(3\) \(46.712\) 3.3.148.1 None \(3\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+(-1-\beta _{1})q^{7}+q^{8}-2q^{11}+\cdots\)
5850.2.a.cs \(3\) \(46.712\) 3.3.940.1 None \(3\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+(1+\beta _{1}+\beta _{2})q^{7}+q^{8}+\cdots\)
5850.2.a.ct \(3\) \(46.712\) 3.3.148.1 None \(3\) \(0\) \(0\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{4}+(1+\beta _{1})q^{7}+q^{8}+2q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5850)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(585))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(975))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1170))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2925))\)\(^{\oplus 2}\)