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

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

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

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

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