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Properties

Label 728.2
Level 728
Weight 2
Dimension 8370
Nonzero newspaces 45
Newform subspaces 94
Sturm bound 64512
Trace bound 11

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Defining parameters

Level: \( N \) = \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 45 \)
Newform subspaces: \( 94 \)
Sturm bound: \(64512\)
Trace bound: \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(728))\).

Total New Old
Modular forms 16992 8810 8182
Cusp forms 15265 8370 6895
Eisenstein series 1727 440 1287

Trace form

\( 8370 q - 36 q^{2} - 36 q^{3} - 36 q^{4} - 36 q^{6} - 48 q^{7} - 96 q^{8} - 60 q^{9} + O(q^{10}) \) \( 8370 q - 36 q^{2} - 36 q^{3} - 36 q^{4} - 36 q^{6} - 48 q^{7} - 96 q^{8} - 60 q^{9} - 36 q^{10} - 24 q^{11} - 36 q^{12} + 6 q^{13} - 108 q^{14} - 72 q^{15} - 36 q^{16} - 54 q^{17} - 72 q^{18} - 12 q^{19} - 72 q^{20} + 24 q^{21} - 156 q^{22} - 48 q^{23} - 108 q^{24} - 78 q^{25} - 72 q^{26} - 72 q^{27} - 144 q^{28} + 30 q^{29} - 132 q^{30} - 48 q^{31} - 96 q^{32} - 36 q^{33} - 108 q^{34} - 36 q^{35} - 228 q^{36} + 18 q^{37} - 72 q^{38} - 72 q^{39} - 240 q^{40} - 42 q^{41} - 48 q^{42} - 192 q^{43} - 108 q^{44} - 78 q^{45} - 216 q^{46} - 144 q^{47} - 108 q^{48} - 120 q^{49} - 228 q^{50} - 288 q^{51} - 204 q^{52} - 84 q^{53} - 108 q^{54} - 240 q^{55} - 72 q^{56} - 264 q^{57} - 168 q^{58} - 180 q^{59} - 144 q^{60} - 90 q^{61} - 48 q^{62} - 144 q^{63} - 156 q^{64} - 30 q^{65} - 132 q^{66} - 48 q^{67} - 36 q^{68} + 48 q^{69} - 156 q^{70} - 72 q^{71} - 72 q^{72} + 60 q^{73} - 204 q^{74} - 108 q^{75} - 180 q^{76} + 12 q^{77} - 192 q^{78} - 72 q^{79} - 264 q^{80} - 24 q^{81} - 180 q^{82} + 12 q^{83} - 180 q^{84} + 78 q^{85} - 156 q^{86} - 168 q^{87} - 60 q^{88} - 84 q^{89} + 120 q^{90} - 114 q^{91} - 228 q^{92} - 84 q^{93} + 84 q^{94} - 216 q^{95} + 348 q^{96} - 240 q^{97} + 12 q^{98} - 336 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(728))\)

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
728.2.a \(\chi_{728}(1, \cdot)\) 728.2.a.a 1 1
728.2.a.b 1
728.2.a.c 1
728.2.a.d 1
728.2.a.e 2
728.2.a.f 2
728.2.a.g 2
728.2.a.h 4
728.2.a.i 4
728.2.b \(\chi_{728}(363, \cdot)\) 728.2.b.a 12 1
728.2.b.b 96
728.2.c \(\chi_{728}(365, \cdot)\) 728.2.c.a 34 1
728.2.c.b 38
728.2.h \(\chi_{728}(27, \cdot)\) 728.2.h.a 48 1
728.2.h.b 48
728.2.i \(\chi_{728}(701, \cdot)\) 728.2.i.a 84 1
728.2.j \(\chi_{728}(391, \cdot)\) None 0 1
728.2.k \(\chi_{728}(337, \cdot)\) 728.2.k.a 8 1
728.2.k.b 12
728.2.p \(\chi_{728}(727, \cdot)\) None 0 1
728.2.q \(\chi_{728}(289, \cdot)\) 728.2.q.a 2 2
728.2.q.b 6
728.2.q.c 8
728.2.q.d 18
728.2.q.e 22
728.2.r \(\chi_{728}(417, \cdot)\) 728.2.r.a 2 2
728.2.r.b 2
728.2.r.c 2
728.2.r.d 4
728.2.r.e 8
728.2.r.f 14
728.2.r.g 16
728.2.s \(\chi_{728}(113, \cdot)\) 728.2.s.a 2 2
728.2.s.b 4
728.2.s.c 4
728.2.s.d 4
728.2.s.e 8
728.2.s.f 8
728.2.s.g 14
728.2.t \(\chi_{728}(9, \cdot)\) 728.2.t.a 2 2
728.2.t.b 6
728.2.t.c 8
728.2.t.d 18
728.2.t.e 22
728.2.v \(\chi_{728}(239, \cdot)\) None 0 2
728.2.w \(\chi_{728}(265, \cdot)\) 728.2.w.a 56 2
728.2.z \(\chi_{728}(99, \cdot)\) 728.2.z.a 168 2
728.2.ba \(\chi_{728}(125, \cdot)\) 728.2.ba.a 4 2
728.2.ba.b 4
728.2.ba.c 8
728.2.ba.d 8
728.2.ba.e 192
728.2.be \(\chi_{728}(485, \cdot)\) 728.2.be.a 216 2
728.2.bf \(\chi_{728}(3, \cdot)\) 728.2.bf.a 4 2
728.2.bf.b 212
728.2.bg \(\chi_{728}(373, \cdot)\) 728.2.bg.a 8 2
728.2.bg.b 12
728.2.bg.c 196
728.2.bh \(\chi_{728}(283, \cdot)\) 728.2.bh.a 216 2
728.2.bm \(\chi_{728}(225, \cdot)\) 728.2.bm.a 4 2
728.2.bm.b 12
728.2.bm.c 24
728.2.bn \(\chi_{728}(55, \cdot)\) None 0 2
728.2.bo \(\chi_{728}(199, \cdot)\) None 0 2
728.2.bp \(\chi_{728}(103, \cdot)\) None 0 2
728.2.by \(\chi_{728}(25, \cdot)\) 728.2.by.a 56 2
728.2.bz \(\chi_{728}(495, \cdot)\) None 0 2
728.2.ca \(\chi_{728}(87, \cdot)\) None 0 2
728.2.cb \(\chi_{728}(569, \cdot)\) 728.2.cb.a 56 2
728.2.cc \(\chi_{728}(335, \cdot)\) None 0 2
728.2.ch \(\chi_{728}(29, \cdot)\) 728.2.ch.a 84 2
728.2.ch.b 84
728.2.ci \(\chi_{728}(251, \cdot)\) 728.2.ci.a 216 2
728.2.cj \(\chi_{728}(389, \cdot)\) 728.2.cj.a 216 2
728.2.ck \(\chi_{728}(131, \cdot)\) 728.2.ck.a 96 2
728.2.ck.b 96
728.2.cl \(\chi_{728}(451, \cdot)\) 728.2.cl.a 4 2
728.2.cl.b 212
728.2.cm \(\chi_{728}(205, \cdot)\) 728.2.cm.a 216 2
728.2.cv \(\chi_{728}(75, \cdot)\) 728.2.cv.a 216 2
728.2.cw \(\chi_{728}(165, \cdot)\) 728.2.cw.a 8 2
728.2.cw.b 12
728.2.cw.c 196
728.2.cx \(\chi_{728}(53, \cdot)\) 728.2.cx.a 192 2
728.2.cy \(\chi_{728}(467, \cdot)\) 728.2.cy.a 24 2
728.2.cy.b 192
728.2.cz \(\chi_{728}(309, \cdot)\) 728.2.cz.a 168 2
728.2.da \(\chi_{728}(139, \cdot)\) 728.2.da.a 216 2
728.2.df \(\chi_{728}(647, \cdot)\) None 0 2
728.2.dg \(\chi_{728}(121, \cdot)\) 728.2.dg.a 56 2
728.2.dh \(\chi_{728}(367, \cdot)\) None 0 2
728.2.dl \(\chi_{728}(33, \cdot)\) 728.2.dl.a 112 4
728.2.dm \(\chi_{728}(375, \cdot)\) None 0 4
728.2.dp \(\chi_{728}(123, \cdot)\) 728.2.dp.a 432 4
728.2.ds \(\chi_{728}(293, \cdot)\) 728.2.ds.a 16 4
728.2.ds.b 16
728.2.ds.c 400
728.2.dt \(\chi_{728}(5, \cdot)\) 728.2.dt.a 432 4
728.2.du \(\chi_{728}(267, \cdot)\) 728.2.du.a 336 4
728.2.dv \(\chi_{728}(291, \cdot)\) 728.2.dv.a 432 4
728.2.dy \(\chi_{728}(45, \cdot)\) 728.2.dy.a 432 4
728.2.eb \(\chi_{728}(319, \cdot)\) None 0 4
728.2.ee \(\chi_{728}(41, \cdot)\) 728.2.ee.a 112 4
728.2.ef \(\chi_{728}(73, \cdot)\) 728.2.ef.a 112 4
728.2.eg \(\chi_{728}(15, \cdot)\) None 0 4
728.2.eh \(\chi_{728}(135, \cdot)\) None 0 4
728.2.ek \(\chi_{728}(89, \cdot)\) 728.2.ek.a 112 4
728.2.en \(\chi_{728}(397, \cdot)\) 728.2.en.a 432 4
728.2.eo \(\chi_{728}(11, \cdot)\) 728.2.eo.a 432 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(728))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(728)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(91))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(182))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(364))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(728))\)\(^{\oplus 1}\)