Properties

Label 21294.2
Level 21294
Weight 2
Dimension 3065228
Nonzero newspaces 168
Sturm bound 49061376

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Defining parameters

Level: \( N \) = \( 21294 = 2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 168 \)
Sturm bound: \(49061376\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(21294))\).

Total New Old
Modular forms 12309120 3065228 9243892
Cusp forms 12221569 3065228 9156341
Eisenstein series 87551 0 87551

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(21294))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
21294.2.a \(\chi_{21294}(1, \cdot)\) 21294.2.a.a 1 1
21294.2.a.b 1
21294.2.a.c 1
21294.2.a.d 1
21294.2.a.e 1
21294.2.a.f 1
21294.2.a.g 1
21294.2.a.h 1
21294.2.a.i 1
21294.2.a.j 1
21294.2.a.k 1
21294.2.a.l 1
21294.2.a.m 1
21294.2.a.n 1
21294.2.a.o 1
21294.2.a.p 1
21294.2.a.q 1
21294.2.a.r 1
21294.2.a.s 1
21294.2.a.t 1
21294.2.a.u 1
21294.2.a.v 1
21294.2.a.w 1
21294.2.a.x 1
21294.2.a.y 1
21294.2.a.z 1
21294.2.a.ba 1
21294.2.a.bb 1
21294.2.a.bc 1
21294.2.a.bd 1
21294.2.a.be 1
21294.2.a.bf 1
21294.2.a.bg 1
21294.2.a.bh 1
21294.2.a.bi 1
21294.2.a.bj 1
21294.2.a.bk 1
21294.2.a.bl 1
21294.2.a.bm 1
21294.2.a.bn 1
21294.2.a.bo 1
21294.2.a.bp 1
21294.2.a.bq 1
21294.2.a.br 1
21294.2.a.bs 1
21294.2.a.bt 1
21294.2.a.bu 1
21294.2.a.bv 1
21294.2.a.bw 1
21294.2.a.bx 1
21294.2.a.by 1
21294.2.a.bz 1
21294.2.a.ca 1
21294.2.a.cb 1
21294.2.a.cc 1
21294.2.a.cd 1
21294.2.a.ce 1
21294.2.a.cf 1
21294.2.a.cg 1
21294.2.a.ch 1
21294.2.a.ci 1
21294.2.a.cj 1
21294.2.a.ck 1
21294.2.a.cl 1
21294.2.a.cm 1
21294.2.a.cn 1
21294.2.a.co 1
21294.2.a.cp 1
21294.2.a.cq 1
21294.2.a.cr 1
21294.2.a.cs 1
21294.2.a.ct 1
21294.2.a.cu 1
21294.2.a.cv 1
21294.2.a.cw 1
21294.2.a.cx 1
21294.2.a.cy 2
21294.2.a.cz 2
21294.2.a.da 2
21294.2.a.db 2
21294.2.a.dc 2
21294.2.a.dd 2
21294.2.a.de 2
21294.2.a.df 2
21294.2.a.dg 2
21294.2.a.dh 2
21294.2.a.di 2
21294.2.a.dj 2
21294.2.a.dk 2
21294.2.a.dl 2
21294.2.a.dm 2
21294.2.a.dn 2
21294.2.a.do 2
21294.2.a.dp 2
21294.2.a.dq 2
21294.2.a.dr 2
21294.2.a.ds 2
21294.2.a.dt 2
21294.2.a.du 2
21294.2.a.dv 2
21294.2.a.dw 2
21294.2.a.dx 2
21294.2.a.dy 2
21294.2.a.dz 2
21294.2.a.ea 2
21294.2.a.eb 2
21294.2.a.ec 2
21294.2.a.ed 3
21294.2.a.ee 3
21294.2.a.ef 3
21294.2.a.eg 3
21294.2.a.eh 3
21294.2.a.ei 3
21294.2.a.ej 3
21294.2.a.ek 3
21294.2.a.el 3
21294.2.a.em 3
21294.2.a.en 3
21294.2.a.eo 3
21294.2.a.ep 3
21294.2.a.eq 3
21294.2.a.er 3
21294.2.a.es 3
21294.2.a.et 3
21294.2.a.eu 3
21294.2.a.ev 3
21294.2.a.ew 3
21294.2.a.ex 3
21294.2.a.ey 3
21294.2.a.ez 3
21294.2.a.fa 3
21294.2.a.fb 3
21294.2.a.fc 3
21294.2.a.fd 4
21294.2.a.fe 4
21294.2.a.ff 6
21294.2.a.fg 6
21294.2.a.fh 6
21294.2.a.fi 6
21294.2.a.fj 6
21294.2.a.fk 6
21294.2.a.fl 6
21294.2.a.fm 6
21294.2.a.fn 6
21294.2.a.fo 6
21294.2.a.fp 8
21294.2.a.fq 8
21294.2.a.fr 8
21294.2.a.fs 8
21294.2.a.ft 9
21294.2.a.fu 9
21294.2.a.fv 9
21294.2.a.fw 9
21294.2.a.fx 9
21294.2.a.fy 9
21294.2.a.fz 9
21294.2.a.ga 9
21294.2.c \(\chi_{21294}(7435, \cdot)\) n/a 384 1
21294.2.e \(\chi_{21294}(21293, \cdot)\) n/a 416 1
21294.2.g \(\chi_{21294}(13859, \cdot)\) n/a 416 1
21294.2.i \(\chi_{21294}(4057, \cdot)\) n/a 2480 2
21294.2.j \(\chi_{21294}(6085, \cdot)\) n/a 1032 2
21294.2.k \(\chi_{21294}(3571, \cdot)\) n/a 1848 2
21294.2.l \(\chi_{21294}(8089, \cdot)\) n/a 2464 2
21294.2.m \(\chi_{21294}(10117, \cdot)\) n/a 1028 2
21294.2.n \(\chi_{21294}(7099, \cdot)\) n/a 1860 2
21294.2.o \(\chi_{21294}(10669, \cdot)\) n/a 1848 2
21294.2.p \(\chi_{21294}(991, \cdot)\) n/a 1028 2
21294.2.q \(\chi_{21294}(529, \cdot)\) n/a 2464 2
21294.2.r \(\chi_{21294}(4033, \cdot)\) n/a 772 2
21294.2.s \(\chi_{21294}(9655, \cdot)\) n/a 2464 2
21294.2.t \(\chi_{21294}(7627, \cdot)\) n/a 2464 2
21294.2.u \(\chi_{21294}(11155, \cdot)\) n/a 2480 2
21294.2.x \(\chi_{21294}(6859, \cdot)\) n/a 1032 2
21294.2.y \(\chi_{21294}(2465, \cdot)\) n/a 616 2
21294.2.z \(\chi_{21294}(18589, \cdot)\) n/a 2464 2
21294.2.bc \(\chi_{21294}(3527, \cdot)\) n/a 816 2
21294.2.bd \(\chi_{21294}(6107, \cdot)\) n/a 2464 2
21294.2.bg \(\chi_{21294}(4079, \cdot)\) n/a 2464 2
21294.2.bi \(\chi_{21294}(2851, \cdot)\) n/a 2464 2
21294.2.bj \(\chi_{21294}(3403, \cdot)\) n/a 768 2
21294.2.bm \(\chi_{21294}(823, \cdot)\) n/a 2464 2
21294.2.bn \(\chi_{21294}(8111, \cdot)\) n/a 2464 2
21294.2.bq \(\chi_{21294}(4247, \cdot)\) n/a 824 2
21294.2.br \(\chi_{21294}(10793, \cdot)\) n/a 2464 2
21294.2.bu \(\chi_{21294}(6275, \cdot)\) n/a 2464 2
21294.2.cd \(\chi_{21294}(11345, \cdot)\) n/a 2464 2
21294.2.ce \(\chi_{21294}(3233, \cdot)\) n/a 2464 2
21294.2.cf \(\chi_{21294}(7775, \cdot)\) n/a 824 2
21294.2.cg \(\chi_{21294}(9803, \cdot)\) n/a 2480 2
21294.2.cl \(\chi_{21294}(6761, \cdot)\) n/a 2480 2
21294.2.cm \(\chi_{21294}(11807, \cdot)\) n/a 824 2
21294.2.cr \(\chi_{21294}(361, \cdot)\) n/a 1028 2
21294.2.cs \(\chi_{21294}(3865, \cdot)\) n/a 1848 2
21294.2.cv \(\chi_{21294}(2389, \cdot)\) n/a 2464 2
21294.2.cy \(\chi_{21294}(3065, \cdot)\) n/a 2464 2
21294.2.cz \(\chi_{21294}(3041, \cdot)\) n/a 816 2
21294.2.da \(\chi_{21294}(1013, \cdot)\) n/a 2464 2
21294.2.db \(\chi_{21294}(4541, \cdot)\) n/a 2464 2
21294.2.dg \(\chi_{21294}(6569, \cdot)\) n/a 824 2
21294.2.dh \(\chi_{21294}(7097, \cdot)\) n/a 2464 2
21294.2.dk \(\chi_{21294}(7459, \cdot)\) n/a 2464 2
21294.2.dl \(\chi_{21294}(6421, \cdot)\) n/a 2464 2
21294.2.dm \(\chi_{21294}(4393, \cdot)\) n/a 1024 2
21294.2.dn \(\chi_{21294}(17599, \cdot)\) n/a 1848 2
21294.2.ds \(\chi_{21294}(337, \cdot)\) n/a 1848 2
21294.2.dt \(\chi_{21294}(7921, \cdot)\) n/a 1028 2
21294.2.dv \(\chi_{21294}(10163, \cdot)\) n/a 2464 2
21294.2.dw \(\chi_{21294}(18737, \cdot)\) n/a 824 2
21294.2.dz \(\chi_{21294}(18275, \cdot)\) n/a 2464 2
21294.2.ea \(\chi_{21294}(677, \cdot)\) n/a 2480 2
21294.2.eg \(\chi_{21294}(6737, \cdot)\) n/a 2464 2
21294.2.eh \(\chi_{21294}(10331, \cdot)\) n/a 816 2
21294.2.ek \(\chi_{21294}(4709, \cdot)\) n/a 2464 2
21294.2.eu \(\chi_{21294}(2803, \cdot)\) n/a 4928 4
21294.2.ev \(\chi_{21294}(5389, \cdot)\) n/a 4928 4
21294.2.ew \(\chi_{21294}(6403, \cdot)\) n/a 4928 4
21294.2.ex \(\chi_{21294}(695, \cdot)\) n/a 4928 4
21294.2.ey \(\chi_{21294}(1709, \cdot)\) n/a 1232 4
21294.2.ez \(\chi_{21294}(8549, \cdot)\) n/a 1632 4
21294.2.fa \(\chi_{21294}(995, \cdot)\) n/a 3696 4
21294.2.fb \(\chi_{21294}(7793, \cdot)\) n/a 1648 4
21294.2.fc \(\chi_{21294}(4295, \cdot)\) n/a 4928 4
21294.2.fd \(\chi_{21294}(7117, \cdot)\) n/a 4928 4
21294.2.fe \(\chi_{21294}(1441, \cdot)\) n/a 2048 4
21294.2.ff \(\chi_{21294}(5647, \cdot)\) n/a 4928 4
21294.2.fg \(\chi_{21294}(19, \cdot)\) n/a 2056 4
21294.2.fh \(\chi_{21294}(577, \cdot)\) n/a 2048 4
21294.2.fi \(\chi_{21294}(8539, \cdot)\) n/a 4928 4
21294.2.fj \(\chi_{21294}(1451, \cdot)\) n/a 4928 4
21294.2.fk \(\chi_{21294}(7079, \cdot)\) n/a 4928 4
21294.2.fl \(\chi_{21294}(4145, \cdot)\) n/a 3696 4
21294.2.ge \(\chi_{21294}(1333, \cdot)\) n/a 2056 4
21294.2.gf \(\chi_{21294}(5765, \cdot)\) n/a 4928 4
21294.2.gg \(\chi_{21294}(5051, \cdot)\) n/a 4928 4
21294.2.gh \(\chi_{21294}(239, \cdot)\) n/a 3696 4
21294.2.gi \(\chi_{21294}(2047, \cdot)\) n/a 4928 4
21294.2.gj \(\chi_{21294}(8431, \cdot)\) n/a 4928 4
21294.2.gk \(\chi_{21294}(4633, \cdot)\) n/a 4928 4
21294.2.gl \(\chi_{21294}(3131, \cdot)\) n/a 1648 4
21294.2.gm \(\chi_{21294}(1639, \cdot)\) n/a 5448 12
21294.2.go \(\chi_{21294}(755, \cdot)\) n/a 5760 12
21294.2.gq \(\chi_{21294}(1637, \cdot)\) n/a 5760 12
21294.2.gs \(\chi_{21294}(883, \cdot)\) n/a 5472 12
21294.2.gu \(\chi_{21294}(79, \cdot)\) n/a 34944 24
21294.2.gv \(\chi_{21294}(835, \cdot)\) n/a 34944 24
21294.2.gw \(\chi_{21294}(757, \cdot)\) n/a 10896 24
21294.2.gx \(\chi_{21294}(445, \cdot)\) n/a 34944 24
21294.2.gy \(\chi_{21294}(1381, \cdot)\) n/a 34944 24
21294.2.gz \(\chi_{21294}(841, \cdot)\) n/a 26208 24
21294.2.ha \(\chi_{21294}(919, \cdot)\) n/a 14544 24
21294.2.hb \(\chi_{21294}(289, \cdot)\) n/a 14544 24
21294.2.hc \(\chi_{21294}(547, \cdot)\) n/a 26208 24
21294.2.hd \(\chi_{21294}(235, \cdot)\) n/a 14592 24
21294.2.he \(\chi_{21294}(625, \cdot)\) n/a 34944 24
21294.2.hf \(\chi_{21294}(211, \cdot)\) n/a 26208 24
21294.2.hg \(\chi_{21294}(373, \cdot)\) n/a 34944 24
21294.2.hh \(\chi_{21294}(307, \cdot)\) n/a 14496 24
21294.2.hi \(\chi_{21294}(827, \cdot)\) n/a 8736 24
21294.2.hm \(\chi_{21294}(815, \cdot)\) n/a 34944 24
21294.2.hp \(\chi_{21294}(185, \cdot)\) n/a 34944 24
21294.2.hq \(\chi_{21294}(503, \cdot)\) n/a 11712 24
21294.2.hw \(\chi_{21294}(1067, \cdot)\) n/a 34944 24
21294.2.hx \(\chi_{21294}(257, \cdot)\) n/a 34944 24
21294.2.ia \(\chi_{21294}(335, \cdot)\) n/a 34944 24
21294.2.ib \(\chi_{21294}(647, \cdot)\) n/a 11616 24
21294.2.id \(\chi_{21294}(1429, \cdot)\) n/a 26208 24
21294.2.ie \(\chi_{21294}(1297, \cdot)\) n/a 14544 24
21294.2.ij \(\chi_{21294}(121, \cdot)\) n/a 34944 24
21294.2.ik \(\chi_{21294}(415, \cdot)\) n/a 14592 24
21294.2.il \(\chi_{21294}(25, \cdot)\) n/a 34944 24
21294.2.im \(\chi_{21294}(43, \cdot)\) n/a 26208 24
21294.2.ip \(\chi_{21294}(17, \cdot)\) n/a 11616 24
21294.2.iq \(\chi_{21294}(545, \cdot)\) n/a 34944 24
21294.2.iv \(\chi_{21294}(1343, \cdot)\) n/a 34944 24
21294.2.iw \(\chi_{21294}(857, \cdot)\) n/a 34944 24
21294.2.ix \(\chi_{21294}(467, \cdot)\) n/a 11712 24
21294.2.iy \(\chi_{21294}(101, \cdot)\) n/a 34944 24
21294.2.jb \(\chi_{21294}(277, \cdot)\) n/a 34944 24
21294.2.je \(\chi_{21294}(667, \cdot)\) n/a 14544 24
21294.2.jf \(\chi_{21294}(589, \cdot)\) n/a 26208 24
21294.2.jk \(\chi_{21294}(209, \cdot)\) n/a 34944 24
21294.2.jl \(\chi_{21294}(269, \cdot)\) n/a 11616 24
21294.2.jq \(\chi_{21294}(731, \cdot)\) n/a 34944 24
21294.2.jr \(\chi_{21294}(419, \cdot)\) n/a 34944 24
21294.2.js \(\chi_{21294}(131, \cdot)\) n/a 34944 24
21294.2.jt \(\chi_{21294}(521, \cdot)\) n/a 11712 24
21294.2.kc \(\chi_{21294}(887, \cdot)\) n/a 34944 24
21294.2.kf \(\chi_{21294}(971, \cdot)\) n/a 11616 24
21294.2.kg \(\chi_{21294}(965, \cdot)\) n/a 34944 24
21294.2.kj \(\chi_{21294}(311, \cdot)\) n/a 34944 24
21294.2.kk \(\chi_{21294}(205, \cdot)\) n/a 34944 24
21294.2.kn \(\chi_{21294}(1213, \cdot)\) n/a 34944 24
21294.2.ko \(\chi_{21294}(127, \cdot)\) n/a 10944 24
21294.2.kq \(\chi_{21294}(563, \cdot)\) n/a 34944 24
21294.2.kt \(\chi_{21294}(251, \cdot)\) n/a 11712 24
21294.2.ku \(\chi_{21294}(173, \cdot)\) n/a 34944 24
21294.2.kx \(\chi_{21294}(571, \cdot)\) n/a 34944 24
21294.2.ky \(\chi_{21294}(145, \cdot)\) n/a 29088 48
21294.2.kz \(\chi_{21294}(281, \cdot)\) n/a 52416 48
21294.2.la \(\chi_{21294}(137, \cdot)\) n/a 69888 48
21294.2.lb \(\chi_{21294}(821, \cdot)\) n/a 69888 48
21294.2.lc \(\chi_{21294}(265, \cdot)\) n/a 69888 48
21294.2.ld \(\chi_{21294}(241, \cdot)\) n/a 69888 48
21294.2.le \(\chi_{21294}(409, \cdot)\) n/a 69888 48
21294.2.lf \(\chi_{21294}(305, \cdot)\) n/a 23232 48
21294.2.ly \(\chi_{21294}(115, \cdot)\) n/a 69888 48
21294.2.lz \(\chi_{21294}(97, \cdot)\) n/a 69888 48
21294.2.ma \(\chi_{21294}(31, \cdot)\) n/a 69888 48
21294.2.mb \(\chi_{21294}(515, \cdot)\) n/a 69888 48
21294.2.mc \(\chi_{21294}(431, \cdot)\) n/a 23232 48
21294.2.md \(\chi_{21294}(617, \cdot)\) n/a 52416 48
21294.2.me \(\chi_{21294}(359, \cdot)\) n/a 23424 48
21294.2.mf \(\chi_{21294}(71, \cdot)\) n/a 17472 48
21294.2.mg \(\chi_{21294}(977, \cdot)\) n/a 69888 48
21294.2.mh \(\chi_{21294}(223, \cdot)\) n/a 69888 48
21294.2.mi \(\chi_{21294}(73, \cdot)\) n/a 29184 48
21294.2.mj \(\chi_{21294}(397, \cdot)\) n/a 29088 48
21294.2.mk \(\chi_{21294}(229, \cdot)\) n/a 69888 48
21294.2.ml \(\chi_{21294}(1189, \cdot)\) n/a 29184 48
21294.2.mm \(\chi_{21294}(535, \cdot)\) n/a 69888 48
21294.2.mn \(\chi_{21294}(743, \cdot)\) n/a 52416 48
21294.2.mo \(\chi_{21294}(11, \cdot)\) n/a 69888 48
21294.2.mp \(\chi_{21294}(317, \cdot)\) n/a 69888 48

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(21294))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(21294)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 36}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(91))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(117))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(169))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(182))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(234))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(273))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(338))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(507))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(546))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(819))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1014))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1183))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1521))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1638))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2366))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3042))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3549))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7098))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10647))\)\(^{\oplus 2}\)