Properties

Label 7728.2.a
Level $7728$
Weight $2$
Character orbit 7728.a
Rep. character $\chi_{7728}(1,\cdot)$
Character field $\Q$
Dimension $132$
Newform subspaces $60$
Sturm bound $3072$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 60 \)
Sturm bound: \(3072\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7728))\).

Total New Old
Modular forms 1560 132 1428
Cusp forms 1513 132 1381
Eisenstein series 47 0 47

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

\(2\)\(3\)\(7\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(10\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(10\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(10\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(8\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(8\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(8\)
Plus space\(+\)\(60\)
Minus space\(-\)\(72\)

Trace form

\( 132 q - 8 q^{5} + 132 q^{9} + O(q^{10}) \) \( 132 q - 8 q^{5} + 132 q^{9} - 8 q^{13} + 8 q^{17} + 140 q^{25} - 8 q^{29} - 8 q^{37} - 16 q^{39} + 8 q^{41} - 32 q^{43} - 8 q^{45} + 132 q^{49} - 8 q^{53} - 32 q^{55} - 8 q^{61} + 48 q^{65} + 40 q^{73} - 32 q^{75} + 132 q^{81} - 16 q^{85} + 40 q^{89} + 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\) 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 7 23
7728.2.a.a 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}-q^{7}+q^{9}+5q^{13}+3q^{15}+\cdots\)
7728.2.a.b 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(-2\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
7728.2.a.c 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-6q^{11}+2q^{13}+\cdots\)
7728.2.a.d 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-3q^{11}+2q^{13}+\cdots\)
7728.2.a.e 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-q^{11}+2q^{13}+4q^{17}+\cdots\)
7728.2.a.f 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}-q^{11}-6q^{13}-7q^{19}+\cdots\)
7728.2.a.g 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
7728.2.a.h 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
7728.2.a.i 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}+q^{9}+6q^{13}-2q^{15}+\cdots\)
7728.2.a.j 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(3\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}+q^{7}+q^{9}-4q^{11}-3q^{13}+\cdots\)
7728.2.a.k 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(4\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}-q^{7}+q^{9}-3q^{11}-2q^{13}+\cdots\)
7728.2.a.l 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(-1\) \(4\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}+q^{7}+q^{9}+5q^{11}-2q^{13}+\cdots\)
7728.2.a.m 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(-4\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
7728.2.a.n 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(-3\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}+q^{7}+q^{9}-q^{13}-3q^{15}+\cdots\)
7728.2.a.o 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(-2\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
7728.2.a.p 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(-2\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{7}+q^{9}+4q^{13}-2q^{15}+\cdots\)
7728.2.a.q 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
7728.2.a.r 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}+5q^{11}-6q^{13}+\cdots\)
7728.2.a.s 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}-q^{13}+q^{15}+\cdots\)
7728.2.a.t 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(2\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}-q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
7728.2.a.u 7728.a 1.a $1$ $61.708$ \(\Q\) None \(0\) \(1\) \(3\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}-q^{7}+q^{9}-4q^{11}+3q^{13}+\cdots\)
7728.2.a.v 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-5\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-\beta )q^{5}-q^{7}+q^{9}+(3+\cdots)q^{11}+\cdots\)
7728.2.a.w 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-5\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-\beta )q^{5}-q^{7}+q^{9}+q^{11}+\cdots\)
7728.2.a.x 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{13}) \) None \(0\) \(-2\) \(-5\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-\beta )q^{5}+q^{7}+q^{9}+5q^{11}+\cdots\)
7728.2.a.y 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-1\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{5}-q^{7}+q^{9}+(3-2\beta )q^{11}+\cdots\)
7728.2.a.z 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{41}) \) None \(0\) \(-2\) \(-1\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{5}+q^{7}+q^{9}+(2+\beta )q^{13}+\cdots\)
7728.2.a.ba 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-1\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{5}+q^{7}+q^{9}+3q^{11}+(-1+\cdots)q^{13}+\cdots\)
7728.2.a.bb 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1+3\beta )q^{5}-q^{7}+q^{9}-3q^{11}+\cdots\)
7728.2.a.bc 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{41}) \) None \(0\) \(-2\) \(1\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}-q^{7}+q^{9}+4q^{11}+(2+\cdots)q^{13}+\cdots\)
7728.2.a.bd 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(4\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}+q^{9}-\beta q^{13}-2q^{15}+\cdots\)
7728.2.a.be 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(5\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(3-\beta )q^{5}-q^{7}+q^{9}-q^{11}+\cdots\)
7728.2.a.bf 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(-5\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-2-\beta )q^{5}+q^{7}+q^{9}+(-3+\cdots)q^{11}+\cdots\)
7728.2.a.bg 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-3\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
7728.2.a.bh 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(-3\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{5}+q^{7}+q^{9}-4q^{11}+\cdots\)
7728.2.a.bi 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{41}) \) None \(0\) \(2\) \(0\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}+\beta q^{11}+2q^{13}+\cdots\)
7728.2.a.bj 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(1\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}-q^{7}+q^{9}+(-1+\beta )q^{11}+\cdots\)
7728.2.a.bk 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(1\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}-q^{7}+q^{9}+(-1+2\beta )q^{11}+\cdots\)
7728.2.a.bl 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{29}) \) None \(0\) \(2\) \(1\) \(2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+q^{7}+q^{9}-5q^{11}+(1+\cdots)q^{13}+\cdots\)
7728.2.a.bm 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+q^{7}+q^{9}+(-2+2\beta )q^{11}+\cdots\)
7728.2.a.bn 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(1\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+q^{7}+q^{9}+(-1+2\beta )q^{11}+\cdots\)
7728.2.a.bo 7728.a 1.a $2$ $61.708$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+q^{7}+q^{9}+(1+2\beta )q^{11}+\cdots\)
7728.2.a.bp 7728.a 1.a $3$ $61.708$ 3.3.229.1 None \(0\) \(-3\) \(-3\) \(-3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1+\beta _{1})q^{5}-q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
7728.2.a.bq 7728.a 1.a $3$ $61.708$ 3.3.733.1 None \(0\) \(-3\) \(-3\) \(3\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{2})q^{5}+q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
7728.2.a.br 7728.a 1.a $3$ $61.708$ 3.3.1101.1 None \(0\) \(-3\) \(1\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
7728.2.a.bs 7728.a 1.a $3$ $61.708$ 3.3.229.1 None \(0\) \(-3\) \(1\) \(3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{2}q^{5}+q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.bt 7728.a 1.a $3$ $61.708$ 3.3.837.1 None \(0\) \(-3\) \(3\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1})q^{5}+q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
7728.2.a.bu 7728.a 1.a $3$ $61.708$ 3.3.621.1 None \(0\) \(3\) \(-3\) \(-3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}-q^{7}+q^{9}+\cdots\)
7728.2.a.bv 7728.a 1.a $3$ $61.708$ 3.3.1509.1 None \(0\) \(3\) \(-1\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta _{1}q^{5}-q^{7}+q^{9}+(-1+2\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.bw 7728.a 1.a $3$ $61.708$ 3.3.229.1 None \(0\) \(3\) \(-1\) \(3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{2}q^{5}+q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.bx 7728.a 1.a $3$ $61.708$ 3.3.837.1 None \(0\) \(3\) \(0\) \(-3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.by 7728.a 1.a $3$ $61.708$ 3.3.1229.1 None \(0\) \(3\) \(1\) \(-3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.bz 7728.a 1.a $4$ $61.708$ 4.4.39605.1 None \(0\) \(-4\) \(-2\) \(4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1+\beta _{1})q^{5}+q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
7728.2.a.ca 7728.a 1.a $4$ $61.708$ 4.4.256549.1 None \(0\) \(-4\) \(2\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+(1+\beta _{2}+\cdots)q^{11}+\cdots\)
7728.2.a.cb 7728.a 1.a $4$ $61.708$ 4.4.2225.1 None \(0\) \(4\) \(-3\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta _{3})q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
7728.2.a.cc 7728.a 1.a $4$ $61.708$ 4.4.75645.1 None \(0\) \(4\) \(1\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.cd 7728.a 1.a $4$ $61.708$ 4.4.24197.1 None \(0\) \(4\) \(5\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1-\beta _{3})q^{5}-q^{7}+q^{9}+(1+\beta _{2}+\cdots)q^{11}+\cdots\)
7728.2.a.ce 7728.a 1.a $4$ $61.708$ 4.4.15317.1 None \(0\) \(4\) \(5\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta _{3})q^{5}+q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.cf 7728.a 1.a $5$ $61.708$ 5.5.17679757.1 None \(0\) \(-5\) \(-6\) \(-5\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{1})q^{5}-q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
7728.2.a.cg 7728.a 1.a $6$ $61.708$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-6\) \(0\) \(6\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{5}q^{5}+q^{7}+q^{9}+(-1-\beta _{4}+\cdots)q^{11}+\cdots\)
7728.2.a.ch 7728.a 1.a $6$ $61.708$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(6\) \(2\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta _{3}q^{5}+q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7728))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7728)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(322))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(483))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(644))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(966))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1932))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\)\(^{\oplus 2}\)