Properties

Label 6480.2.a
Level $6480$
Weight $2$
Character orbit 6480.a
Rep. character $\chi_{6480}(1,\cdot)$
Character field $\Q$
Dimension $96$
Newform subspaces $54$
Sturm bound $2592$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 54 \)
Sturm bound: \(2592\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6480))\).

Total New Old
Modular forms 1368 96 1272
Cusp forms 1225 96 1129
Eisenstein series 143 0 143

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.
\(+\)\(+\)\(+\)\(+\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(13\)
\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(13\)
Plus space\(+\)\(44\)
Minus space\(-\)\(52\)

Trace form

\( 96 q + O(q^{10}) \) \( 96 q + 96 q^{25} - 24 q^{31} - 24 q^{43} + 96 q^{49} - 48 q^{67} + 24 q^{73} - 48 q^{79} - 24 q^{91} + 24 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6480))\) 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
6480.2.a.a 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-5q^{7}-q^{13}-6q^{17}-5q^{19}+\cdots\)
6480.2.a.b 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-3q^{11}-4q^{13}+6q^{17}+\cdots\)
6480.2.a.c 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+3q^{11}-4q^{13}-6q^{17}+\cdots\)
6480.2.a.d 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{11}+7q^{19}-6q^{23}+q^{25}+\cdots\)
6480.2.a.e 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+5q^{11}+3q^{17}-5q^{19}-6q^{23}+\cdots\)
6480.2.a.f 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+5q^{11}+4q^{13}+4q^{17}+5q^{19}+\cdots\)
6480.2.a.g 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}-6q^{11}+2q^{13}+4q^{19}+\cdots\)
6480.2.a.h 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}-4q^{13}+6q^{17}-2q^{19}+\cdots\)
6480.2.a.i 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}+5q^{13}+6q^{17}-5q^{19}+\cdots\)
6480.2.a.j 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-3q^{11}-2q^{17}-q^{19}+\cdots\)
6480.2.a.k 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}+2q^{11}-2q^{13}+4q^{17}+\cdots\)
6480.2.a.l 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+3q^{11}-4q^{13}-3q^{17}+\cdots\)
6480.2.a.m 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+3q^{11}-4q^{13}-5q^{19}+\cdots\)
6480.2.a.n 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(-5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-5q^{7}-q^{13}+6q^{17}-5q^{19}+\cdots\)
6480.2.a.o 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-3q^{11}-4q^{13}+6q^{17}+\cdots\)
6480.2.a.p 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+3q^{11}-4q^{13}-6q^{17}+\cdots\)
6480.2.a.q 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-5q^{11}-3q^{17}-5q^{19}+6q^{23}+\cdots\)
6480.2.a.r 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-5q^{11}+4q^{13}-4q^{17}+5q^{19}+\cdots\)
6480.2.a.s 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{11}+7q^{19}+6q^{23}+q^{25}+\cdots\)
6480.2.a.t 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{7}-4q^{13}-6q^{17}-2q^{19}+\cdots\)
6480.2.a.u 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{7}+5q^{13}-6q^{17}-5q^{19}+\cdots\)
6480.2.a.v 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{7}+6q^{11}+2q^{13}+4q^{19}+\cdots\)
6480.2.a.w 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}+3q^{11}+2q^{17}-q^{19}+\cdots\)
6480.2.a.x 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+3q^{7}-2q^{11}-2q^{13}-4q^{17}+\cdots\)
6480.2.a.y 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{7}-3q^{11}-4q^{13}-5q^{19}+\cdots\)
6480.2.a.z 6480.a 1.a $1$ $51.743$ \(\Q\) None \(0\) \(0\) \(1\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{7}-3q^{11}-4q^{13}+3q^{17}+\cdots\)
6480.2.a.ba 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-2+\beta )q^{7}+(-2+2\beta )q^{11}+\cdots\)
6480.2.a.bb 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-2+\beta )q^{7}+2\beta q^{11}+(2-\beta )q^{13}+\cdots\)
6480.2.a.bc 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(-2\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta )q^{7}-2q^{17}+(2-2\beta )q^{19}+\cdots\)
6480.2.a.bd 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-2\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-\beta q^{7}+(-1+\beta )q^{11}-\beta q^{13}+\cdots\)
6480.2.a.be 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-2\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-\beta q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
6480.2.a.bf 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-2\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-\beta q^{7}+(3-\beta )q^{11}+(2+\beta )q^{13}+\cdots\)
6480.2.a.bg 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta )q^{7}+(-2+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
6480.2.a.bh 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta )q^{7}-\beta q^{11}+(2+2\beta )q^{13}+\cdots\)
6480.2.a.bi 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(3+\beta )q^{7}+(-4-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
6480.2.a.bj 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-2+\beta )q^{7}-2\beta q^{11}+(2-\beta )q^{13}+\cdots\)
6480.2.a.bk 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-2+\beta )q^{7}+(2-2\beta )q^{11}+\cdots\)
6480.2.a.bl 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(2\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+\beta )q^{7}+2q^{17}+(2-2\beta )q^{19}+\cdots\)
6480.2.a.bm 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(2\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-\beta q^{7}+(-3+\beta )q^{11}+(2+\beta )q^{13}+\cdots\)
6480.2.a.bn 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(2\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-\beta q^{7}+(-1-\beta )q^{11}-2\beta q^{13}+\cdots\)
6480.2.a.bo 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(2\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-\beta q^{7}+(1-\beta )q^{11}-\beta q^{13}+\cdots\)
6480.2.a.bp 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta )q^{7}+\beta q^{11}+(2+2\beta )q^{13}+\cdots\)
6480.2.a.bq 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta )q^{7}+(2-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
6480.2.a.br 6480.a 1.a $2$ $51.743$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(3+\beta )q^{7}+(4+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
6480.2.a.bs 6480.a 1.a $3$ $51.743$ 3.3.564.1 None \(0\) \(0\) \(-3\) \(-5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-2+\beta _{1})q^{7}+(-1-\beta _{2})q^{11}+\cdots\)
6480.2.a.bt 6480.a 1.a $3$ $51.743$ 3.3.564.1 None \(0\) \(0\) \(-3\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta _{1})q^{7}+\beta _{2}q^{11}+(2+\cdots)q^{13}+\cdots\)
6480.2.a.bu 6480.a 1.a $3$ $51.743$ 3.3.564.1 None \(0\) \(0\) \(-3\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta _{1}-\beta _{2})q^{7}+(-1-\beta _{2})q^{11}+\cdots\)
6480.2.a.bv 6480.a 1.a $3$ $51.743$ 3.3.564.1 None \(0\) \(0\) \(3\) \(-5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-2+\beta _{1})q^{7}+(1+\beta _{2})q^{11}+\cdots\)
6480.2.a.bw 6480.a 1.a $3$ $51.743$ 3.3.564.1 None \(0\) \(0\) \(3\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+\beta _{1})q^{7}-\beta _{2}q^{11}+(2+\cdots)q^{13}+\cdots\)
6480.2.a.bx 6480.a 1.a $3$ $51.743$ 3.3.564.1 None \(0\) \(0\) \(3\) \(5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta _{1}-\beta _{2})q^{7}+(1+\beta _{2})q^{11}+\cdots\)
6480.2.a.by 6480.a 1.a $4$ $51.743$ 4.4.62352.1 None \(0\) \(0\) \(-4\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2-\beta _{1}+\beta _{3})q^{11}+\cdots\)
6480.2.a.bz 6480.a 1.a $4$ $51.743$ 4.4.29268.1 None \(0\) \(0\) \(-4\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta _{2}q^{7}+(-\beta _{2}-\beta _{3})q^{11}+(1+\cdots)q^{13}+\cdots\)
6480.2.a.ca 6480.a 1.a $4$ $51.743$ 4.4.62352.1 None \(0\) \(0\) \(4\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
6480.2.a.cb 6480.a 1.a $4$ $51.743$ 4.4.29268.1 None \(0\) \(0\) \(4\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta _{2}q^{7}+(\beta _{2}+\beta _{3})q^{11}+(1+\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6480))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6480)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(810))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1080))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1620))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3240))\)\(^{\oplus 2}\)