Properties

Label 6840.2
Level 6840
Weight 2
Dimension 503212
Nonzero newspaces 144
Sturm bound 4976640

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Defining parameters

Level: \( N \) = \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 144 \)
Sturm bound: \(4976640\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(6840))\).

Total New Old
Modular forms 1257984 506884 751100
Cusp forms 1230337 503212 727125
Eisenstein series 27647 3672 23975

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(6840))\)

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
6840.2.a \(\chi_{6840}(1, \cdot)\) 6840.2.a.a 1 1
6840.2.a.b 1
6840.2.a.c 1
6840.2.a.d 1
6840.2.a.e 1
6840.2.a.f 1
6840.2.a.g 1
6840.2.a.h 1
6840.2.a.i 1
6840.2.a.j 1
6840.2.a.k 1
6840.2.a.l 1
6840.2.a.m 1
6840.2.a.n 1
6840.2.a.o 1
6840.2.a.p 1
6840.2.a.q 1
6840.2.a.r 1
6840.2.a.s 1
6840.2.a.t 1
6840.2.a.u 1
6840.2.a.v 2
6840.2.a.w 2
6840.2.a.x 2
6840.2.a.y 2
6840.2.a.z 2
6840.2.a.ba 2
6840.2.a.bb 2
6840.2.a.bc 2
6840.2.a.bd 2
6840.2.a.be 3
6840.2.a.bf 3
6840.2.a.bg 3
6840.2.a.bh 3
6840.2.a.bi 3
6840.2.a.bj 3
6840.2.a.bk 3
6840.2.a.bl 3
6840.2.a.bm 3
6840.2.a.bn 3
6840.2.a.bo 3
6840.2.a.bp 4
6840.2.a.bq 4
6840.2.a.br 5
6840.2.a.bs 5
6840.2.c \(\chi_{6840}(1331, \cdot)\) n/a 288 1
6840.2.d \(\chi_{6840}(2431, \cdot)\) None 0 1
6840.2.f \(\chi_{6840}(4789, \cdot)\) n/a 540 1
6840.2.i \(\chi_{6840}(5129, \cdot)\) n/a 120 1
6840.2.j \(\chi_{6840}(1369, \cdot)\) n/a 134 1
6840.2.m \(\chi_{6840}(1709, \cdot)\) n/a 480 1
6840.2.o \(\chi_{6840}(4751, \cdot)\) None 0 1
6840.2.p \(\chi_{6840}(5851, \cdot)\) n/a 400 1
6840.2.r \(\chi_{6840}(3761, \cdot)\) 6840.2.r.a 40 1
6840.2.r.b 40
6840.2.u \(\chi_{6840}(3421, \cdot)\) n/a 360 1
6840.2.w \(\chi_{6840}(3799, \cdot)\) None 0 1
6840.2.x \(\chi_{6840}(2699, \cdot)\) n/a 432 1
6840.2.ba \(\chi_{6840}(379, \cdot)\) n/a 596 1
6840.2.bb \(\chi_{6840}(6119, \cdot)\) None 0 1
6840.2.bd \(\chi_{6840}(341, \cdot)\) n/a 320 1
6840.2.bg \(\chi_{6840}(121, \cdot)\) n/a 480 2
6840.2.bh \(\chi_{6840}(2281, \cdot)\) n/a 432 2
6840.2.bi \(\chi_{6840}(2401, \cdot)\) n/a 480 2
6840.2.bj \(\chi_{6840}(3241, \cdot)\) n/a 200 2
6840.2.bk \(\chi_{6840}(37, \cdot)\) n/a 1192 2
6840.2.bn \(\chi_{6840}(343, \cdot)\) None 0 2
6840.2.bp \(\chi_{6840}(1673, \cdot)\) n/a 216 2
6840.2.bq \(\chi_{6840}(683, \cdot)\) n/a 960 2
6840.2.bs \(\chi_{6840}(2357, \cdot)\) n/a 864 2
6840.2.bv \(\chi_{6840}(1367, \cdot)\) None 0 2
6840.2.bx \(\chi_{6840}(3457, \cdot)\) n/a 300 2
6840.2.by \(\chi_{6840}(1027, \cdot)\) n/a 1080 2
6840.2.ca \(\chi_{6840}(449, \cdot)\) n/a 240 2
6840.2.cd \(\chi_{6840}(1189, \cdot)\) n/a 1192 2
6840.2.cf \(\chi_{6840}(4591, \cdot)\) None 0 2
6840.2.cg \(\chi_{6840}(4571, \cdot)\) n/a 640 2
6840.2.cj \(\chi_{6840}(1171, \cdot)\) n/a 800 2
6840.2.ck \(\chi_{6840}(1151, \cdot)\) None 0 2
6840.2.cm \(\chi_{6840}(3869, \cdot)\) n/a 960 2
6840.2.cp \(\chi_{6840}(4609, \cdot)\) n/a 300 2
6840.2.cq \(\chi_{6840}(239, \cdot)\) None 0 2
6840.2.ct \(\chi_{6840}(259, \cdot)\) n/a 2864 2
6840.2.cv \(\chi_{6840}(2621, \cdot)\) n/a 1920 2
6840.2.cy \(\chi_{6840}(221, \cdot)\) n/a 1920 2
6840.2.da \(\chi_{6840}(2659, \cdot)\) n/a 2864 2
6840.2.dd \(\chi_{6840}(1679, \cdot)\) None 0 2
6840.2.de \(\chi_{6840}(1699, \cdot)\) n/a 2864 2
6840.2.dh \(\chi_{6840}(1559, \cdot)\) None 0 2
6840.2.dj \(\chi_{6840}(1661, \cdot)\) n/a 1920 2
6840.2.dm \(\chi_{6840}(2101, \cdot)\) n/a 1920 2
6840.2.dn \(\chi_{6840}(1361, \cdot)\) n/a 480 2
6840.2.dp \(\chi_{6840}(1519, \cdot)\) None 0 2
6840.2.ds \(\chi_{6840}(1379, \cdot)\) n/a 2864 2
6840.2.dt \(\chi_{6840}(1399, \cdot)\) None 0 2
6840.2.dw \(\chi_{6840}(419, \cdot)\) n/a 2592 2
6840.2.dy \(\chi_{6840}(1481, \cdot)\) n/a 480 2
6840.2.dz \(\chi_{6840}(3541, \cdot)\) n/a 1920 2
6840.2.ec \(\chi_{6840}(2801, \cdot)\) n/a 480 2
6840.2.ed \(\chi_{6840}(1141, \cdot)\) n/a 1728 2
6840.2.ef \(\chi_{6840}(2819, \cdot)\) n/a 2864 2
6840.2.ei \(\chi_{6840}(2839, \cdot)\) None 0 2
6840.2.ek \(\chi_{6840}(3029, \cdot)\) n/a 2864 2
6840.2.el \(\chi_{6840}(49, \cdot)\) n/a 720 2
6840.2.en \(\chi_{6840}(191, \cdot)\) None 0 2
6840.2.eq \(\chi_{6840}(331, \cdot)\) n/a 1920 2
6840.2.er \(\chi_{6840}(311, \cdot)\) None 0 2
6840.2.eu \(\chi_{6840}(1291, \cdot)\) n/a 1920 2
6840.2.ew \(\chi_{6840}(3649, \cdot)\) n/a 648 2
6840.2.ex \(\chi_{6840}(749, \cdot)\) n/a 2864 2
6840.2.fa \(\chi_{6840}(1489, \cdot)\) n/a 720 2
6840.2.fb \(\chi_{6840}(3989, \cdot)\) n/a 2864 2
6840.2.fd \(\chi_{6840}(4891, \cdot)\) n/a 1920 2
6840.2.fg \(\chi_{6840}(4871, \cdot)\) None 0 2
6840.2.fh \(\chi_{6840}(1471, \cdot)\) None 0 2
6840.2.fk \(\chi_{6840}(1451, \cdot)\) n/a 1920 2
6840.2.fm \(\chi_{6840}(229, \cdot)\) n/a 2592 2
6840.2.fn \(\chi_{6840}(4169, \cdot)\) n/a 720 2
6840.2.fq \(\chi_{6840}(4909, \cdot)\) n/a 2864 2
6840.2.fr \(\chi_{6840}(569, \cdot)\) n/a 720 2
6840.2.ft \(\chi_{6840}(3611, \cdot)\) n/a 1728 2
6840.2.fw \(\chi_{6840}(31, \cdot)\) None 0 2
6840.2.fx \(\chi_{6840}(11, \cdot)\) n/a 1920 2
6840.2.ga \(\chi_{6840}(151, \cdot)\) None 0 2
6840.2.gc \(\chi_{6840}(2729, \cdot)\) n/a 720 2
6840.2.gd \(\chi_{6840}(349, \cdot)\) n/a 2864 2
6840.2.gg \(\chi_{6840}(539, \cdot)\) n/a 960 2
6840.2.gh \(\chi_{6840}(559, \cdot)\) None 0 2
6840.2.gj \(\chi_{6840}(1261, \cdot)\) n/a 800 2
6840.2.gm \(\chi_{6840}(521, \cdot)\) n/a 160 2
6840.2.gp \(\chi_{6840}(2501, \cdot)\) n/a 640 2
6840.2.gr \(\chi_{6840}(2519, \cdot)\) None 0 2
6840.2.gs \(\chi_{6840}(2539, \cdot)\) n/a 1192 2
6840.2.gu \(\chi_{6840}(1081, \cdot)\) n/a 600 6
6840.2.gv \(\chi_{6840}(841, \cdot)\) n/a 1440 6
6840.2.gw \(\chi_{6840}(481, \cdot)\) n/a 1440 6
6840.2.gy \(\chi_{6840}(107, \cdot)\) n/a 1920 4
6840.2.gz \(\chi_{6840}(2177, \cdot)\) n/a 480 4
6840.2.hb \(\chi_{6840}(847, \cdot)\) None 0 4
6840.2.he \(\chi_{6840}(2197, \cdot)\) n/a 2384 4
6840.2.hf \(\chi_{6840}(653, \cdot)\) n/a 5728 4
6840.2.hi \(\chi_{6840}(407, \cdot)\) None 0 4
6840.2.hl \(\chi_{6840}(1483, \cdot)\) n/a 5184 4
6840.2.hm \(\chi_{6840}(2443, \cdot)\) n/a 5728 4
6840.2.hn \(\chi_{6840}(1177, \cdot)\) n/a 1440 4
6840.2.ho \(\chi_{6840}(2497, \cdot)\) n/a 1440 4
6840.2.hr \(\chi_{6840}(1823, \cdot)\) None 0 4
6840.2.hs \(\chi_{6840}(1703, \cdot)\) None 0 4
6840.2.hx \(\chi_{6840}(77, \cdot)\) n/a 5184 4
6840.2.hy \(\chi_{6840}(2477, \cdot)\) n/a 5728 4
6840.2.ia \(\chi_{6840}(673, \cdot)\) n/a 1440 4
6840.2.ib \(\chi_{6840}(1147, \cdot)\) n/a 5728 4
6840.2.id \(\chi_{6840}(373, \cdot)\) n/a 5728 4
6840.2.ig \(\chi_{6840}(463, \cdot)\) None 0 4
6840.2.ij \(\chi_{6840}(227, \cdot)\) n/a 5728 4
6840.2.ik \(\chi_{6840}(2003, \cdot)\) n/a 5728 4
6840.2.il \(\chi_{6840}(1217, \cdot)\) n/a 1296 4
6840.2.im \(\chi_{6840}(1793, \cdot)\) n/a 1440 4
6840.2.ip \(\chi_{6840}(2167, \cdot)\) None 0 4
6840.2.iq \(\chi_{6840}(7, \cdot)\) None 0 4
6840.2.iv \(\chi_{6840}(493, \cdot)\) n/a 5728 4
6840.2.iw \(\chi_{6840}(1813, \cdot)\) n/a 5728 4
6840.2.iy \(\chi_{6840}(353, \cdot)\) n/a 1440 4
6840.2.iz \(\chi_{6840}(563, \cdot)\) n/a 5728 4
6840.2.jc \(\chi_{6840}(163, \cdot)\) n/a 2384 4
6840.2.jd \(\chi_{6840}(217, \cdot)\) n/a 600 4
6840.2.jf \(\chi_{6840}(863, \cdot)\) None 0 4
6840.2.ji \(\chi_{6840}(197, \cdot)\) n/a 1920 4
6840.2.jj \(\chi_{6840}(169, \cdot)\) n/a 2160 6
6840.2.jm \(\chi_{6840}(2131, \cdot)\) n/a 5760 6
6840.2.jn \(\chi_{6840}(509, \cdot)\) n/a 8592 6
6840.2.jq \(\chi_{6840}(671, \cdot)\) None 0 6
6840.2.jr \(\chi_{6840}(751, \cdot)\) None 0 6
6840.2.ju \(\chi_{6840}(1069, \cdot)\) n/a 8592 6
6840.2.jv \(\chi_{6840}(2171, \cdot)\) n/a 5760 6
6840.2.jy \(\chi_{6840}(2009, \cdot)\) n/a 2160 6
6840.2.ka \(\chi_{6840}(41, \cdot)\) n/a 1440 6
6840.2.kb \(\chi_{6840}(1499, \cdot)\) n/a 8592 6
6840.2.ke \(\chi_{6840}(61, \cdot)\) n/a 5760 6
6840.2.kf \(\chi_{6840}(679, \cdot)\) None 0 6
6840.2.ki \(\chi_{6840}(1459, \cdot)\) n/a 3576 6
6840.2.kl \(\chi_{6840}(359, \cdot)\) None 0 6
6840.2.km \(\chi_{6840}(1421, \cdot)\) n/a 1920 6
6840.2.kp \(\chi_{6840}(541, \cdot)\) n/a 2400 6
6840.2.kq \(\chi_{6840}(2359, \cdot)\) None 0 6
6840.2.kt \(\chi_{6840}(2321, \cdot)\) n/a 480 6
6840.2.ku \(\chi_{6840}(899, \cdot)\) n/a 2880 6
6840.2.kx \(\chi_{6840}(479, \cdot)\) None 0 6
6840.2.ky \(\chi_{6840}(3461, \cdot)\) n/a 5760 6
6840.2.lb \(\chi_{6840}(979, \cdot)\) n/a 8592 6
6840.2.le \(\chi_{6840}(131, \cdot)\) n/a 5760 6
6840.2.lf \(\chi_{6840}(1409, \cdot)\) n/a 2160 6
6840.2.li \(\chi_{6840}(1231, \cdot)\) None 0 6
6840.2.lj \(\chi_{6840}(709, \cdot)\) n/a 8592 6
6840.2.lm \(\chi_{6840}(289, \cdot)\) n/a 900 6
6840.2.ln \(\chi_{6840}(91, \cdot)\) n/a 2400 6
6840.2.lq \(\chi_{6840}(269, \cdot)\) n/a 2880 6
6840.2.lr \(\chi_{6840}(1871, \cdot)\) None 0 6
6840.2.lu \(\chi_{6840}(991, \cdot)\) None 0 6
6840.2.lv \(\chi_{6840}(1909, \cdot)\) n/a 3576 6
6840.2.ly \(\chi_{6840}(251, \cdot)\) n/a 1920 6
6840.2.lz \(\chi_{6840}(89, \cdot)\) n/a 720 6
6840.2.mc \(\chi_{6840}(29, \cdot)\) n/a 8592 6
6840.2.md \(\chi_{6840}(1031, \cdot)\) None 0 6
6840.2.mg \(\chi_{6840}(2209, \cdot)\) n/a 2160 6
6840.2.mh \(\chi_{6840}(211, \cdot)\) n/a 5760 6
6840.2.mj \(\chi_{6840}(3499, \cdot)\) n/a 8592 6
6840.2.mm \(\chi_{6840}(119, \cdot)\) None 0 6
6840.2.mp \(\chi_{6840}(941, \cdot)\) n/a 5760 6
6840.2.mq \(\chi_{6840}(2221, \cdot)\) n/a 5760 6
6840.2.mt \(\chi_{6840}(79, \cdot)\) None 0 6
6840.2.mu \(\chi_{6840}(641, \cdot)\) n/a 1440 6
6840.2.mx \(\chi_{6840}(3539, \cdot)\) n/a 8592 6
6840.2.mz \(\chi_{6840}(367, \cdot)\) None 0 12
6840.2.na \(\chi_{6840}(637, \cdot)\) n/a 17184 12
6840.2.nd \(\chi_{6840}(137, \cdot)\) n/a 4320 12
6840.2.ne \(\chi_{6840}(1067, \cdot)\) n/a 17184 12
6840.2.ng \(\chi_{6840}(167, \cdot)\) None 0 12
6840.2.nj \(\chi_{6840}(1013, \cdot)\) n/a 17184 12
6840.2.nl \(\chi_{6840}(143, \cdot)\) None 0 12
6840.2.nm \(\chi_{6840}(557, \cdot)\) n/a 5760 12
6840.2.no \(\chi_{6840}(97, \cdot)\) n/a 4320 12
6840.2.nr \(\chi_{6840}(43, \cdot)\) n/a 17184 12
6840.2.nt \(\chi_{6840}(433, \cdot)\) n/a 1800 12
6840.2.nu \(\chi_{6840}(883, \cdot)\) n/a 7152 12
6840.2.nx \(\chi_{6840}(617, \cdot)\) n/a 4320 12
6840.2.ny \(\chi_{6840}(203, \cdot)\) n/a 17184 12
6840.2.oa \(\chi_{6840}(17, \cdot)\) n/a 1440 12
6840.2.od \(\chi_{6840}(827, \cdot)\) n/a 5760 12
6840.2.of \(\chi_{6840}(727, \cdot)\) None 0 12
6840.2.og \(\chi_{6840}(13, \cdot)\) n/a 17184 12
6840.2.oi \(\chi_{6840}(1423, \cdot)\) None 0 12
6840.2.ol \(\chi_{6840}(1117, \cdot)\) n/a 7152 12
6840.2.om \(\chi_{6840}(193, \cdot)\) n/a 4320 12
6840.2.op \(\chi_{6840}(283, \cdot)\) n/a 17184 12
6840.2.oq \(\chi_{6840}(383, \cdot)\) None 0 12
6840.2.ot \(\chi_{6840}(1157, \cdot)\) n/a 17184 12

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(6840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(6840)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 48}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 36}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 32}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(38))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(57))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(76))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(114))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(152))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(171))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(190))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(228))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(285))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(342))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(380))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(456))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(570))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(684))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(760))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(855))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1368))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1710))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3420))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6840))\)\(^{\oplus 1}\)