Space of modular forms of level 399 and weight 2

• Label: 399.2
• Level: 399
• Weight: 2
• Dimension: 3951
• Nonzero newspaces: 32
• Newform subspaces: 100
• Sturm bound: 23040
• Trace bound: 12

Defining parameters

Level:	$N$	=	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	=	$2$
Nonzero newspaces:			$32$
Newform subspaces:			$100$
Sturm bound:			$23040$
Trace bound:			$12$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_1(399))$.

	Total	New	Old
Modular forms	6192	4287	1905
Cusp forms	5329	3951	1378
Eisenstein series	863	336	527

Trace form

3951 * q + 9 * q^2 - 29 * q^3 - 55 * q^4 + 6 * q^5 - 39 * q^6 - 79 * q^7 + 9 * q^8 - 41 * q^9 - 66 * q^10 - 59 * q^12 - 106 * q^13 - 39 * q^14 - 108 * q^15 - 183 * q^16 - 6 * q^17 - 45 * q^18 - 131 * q^19 - 90 * q^20 - 59 * q^21 - 228 * q^22 - 12 * q^23 - 99 * q^24 - 123 * q^25 - 30 * q^26 - 65 * q^27 - 195 * q^28 - 54 * q^29 - 156 * q^30 - 124 * q^31 - 123 * q^32 - 144 * q^33 - 210 * q^34 - 66 * q^35 - 209 * q^36 - 146 * q^37 - 189 * q^38 - 166 * q^39 - 294 * q^40 + 6 * q^41 - 114 * q^42 - 272 * q^43 - 96 * q^44 - 84 * q^45 - 180 * q^46 - 36 * q^47 + 13 * q^48 - 123 * q^49 - 9 * q^50 + 66 * q^51 + 38 * q^52 + 66 * q^53 + 81 * q^54 + 21 * q^56 - 5 * q^57 - 42 * q^58 + 12 * q^59 + 120 * q^60 - 130 * q^61 - 156 * q^62 - 50 * q^63 - 355 * q^64 - 144 * q^65 - 288 * q^67 - 138 * q^68 - 102 * q^69 - 252 * q^70 - 48 * q^71 - 153 * q^72 - 262 * q^73 - 150 * q^74 - 161 * q^75 - 299 * q^76 - 162 * q^77 - 348 * q^78 - 396 * q^79 - 306 * q^80 - 89 * q^81 - 666 * q^82 - 168 * q^83 - 248 * q^84 - 528 * q^85 - 252 * q^86 - 282 * q^87 - 564 * q^88 - 174 * q^89 - 132 * q^90 - 256 * q^91 - 300 * q^92 - 160 * q^93 - 348 * q^94 + 12 * q^95 - 159 * q^96 - 94 * q^97 - 147 * q^98 - 96 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_1(399))$

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
399.2.a	$\chi_{399}(1, \cdot)$	399.2.a.a	1	1
		399.2.a.b	1
		399.2.a.c	1
		399.2.a.d	3
		399.2.a.e	3
		399.2.a.f	5
		399.2.a.g	5
399.2.c	$\chi_{399}(265, \cdot)$	399.2.c.a	2	1
		399.2.c.b	2
		399.2.c.c	12
		399.2.c.d	12
399.2.d	$\chi_{399}(20, \cdot)$	399.2.d.a	48	1
399.2.f	$\chi_{399}(113, \cdot)$	399.2.f.a	4	1
		399.2.f.b	16
		399.2.f.c	20
399.2.i	$\chi_{399}(163, \cdot)$	399.2.i.a	2	2
		399.2.i.b	22
		399.2.i.c	28
399.2.j	$\chi_{399}(58, \cdot)$	399.2.j.a	2	2
		399.2.j.b	2
		399.2.j.c	4
		399.2.j.d	8
		399.2.j.e	8
		399.2.j.f	8
		399.2.j.g	16
399.2.k	$\chi_{399}(64, \cdot)$	399.2.k.a	8	2
		399.2.k.b	8
		399.2.k.c	12
		399.2.k.d	12
399.2.l	$\chi_{399}(121, \cdot)$	399.2.l.a	2	2
		399.2.l.b	22
		399.2.l.c	28
399.2.m	$\chi_{399}(145, \cdot)$	399.2.m.a	2	2
		399.2.m.b	4
		399.2.m.c	22
		399.2.m.d	24
399.2.p	$\chi_{399}(26, \cdot)$	399.2.p.a	2	2
		399.2.p.b	2
		399.2.p.c	96
399.2.t	$\chi_{399}(8, \cdot)$	399.2.t.a	40	2
		399.2.t.b	40
399.2.w	$\chi_{399}(170, \cdot)$	399.2.w.a	2	2
		399.2.w.b	2
		399.2.w.c	8
		399.2.w.d	88
399.2.x	$\chi_{399}(65, \cdot)$	399.2.x.a	2	2
		399.2.x.b	2
		399.2.x.c	96
399.2.z	$\chi_{399}(83, \cdot)$	399.2.z.a	96	2
399.2.bc	$\chi_{399}(248, \cdot)$	399.2.bc.a	96	2
399.2.bd	$\chi_{399}(68, \cdot)$	399.2.bd.a	2	2
		399.2.bd.b	2
		399.2.bd.c	96
399.2.bg	$\chi_{399}(31, \cdot)$	399.2.bg.a	2	2
		399.2.bg.b	4
		399.2.bg.c	22
		399.2.bg.d	24
399.2.bh	$\chi_{399}(94, \cdot)$	399.2.bh.a	2	2
		399.2.bh.b	2
		399.2.bh.c	4
		399.2.bh.d	4
		399.2.bh.e	20
		399.2.bh.f	20
399.2.bk	$\chi_{399}(160, \cdot)$	399.2.bk.a	2	2
		399.2.bk.b	2
		399.2.bk.c	2
		399.2.bk.d	2
		399.2.bk.e	24
		399.2.bk.f	24
399.2.bm	$\chi_{399}(107, \cdot)$	399.2.bm.a	2	2
		399.2.bm.b	2
		399.2.bm.c	96
399.2.bo	$\chi_{399}(43, \cdot)$	399.2.bo.a	6	6
		399.2.bo.b	18
		399.2.bo.c	24
		399.2.bo.d	36
		399.2.bo.e	36
399.2.bp	$\chi_{399}(130, \cdot)$	399.2.bp.a	72	6
		399.2.bp.b	90
399.2.bq	$\chi_{399}(4, \cdot)$	399.2.bq.a	72	6
		399.2.bq.b	90
399.2.br	$\chi_{399}(2, \cdot)$	399.2.br.a	6	6
		399.2.br.b	288
399.2.bv	$\chi_{399}(86, \cdot)$	399.2.bv.a	6	6
		399.2.bv.b	288
399.2.bw	$\chi_{399}(29, \cdot)$	399.2.bw.a	120	6
		399.2.bw.b	120
399.2.cb	$\chi_{399}(5, \cdot)$	399.2.cb.a	6	6
		399.2.cb.b	288
399.2.cc	$\chi_{399}(13, \cdot)$	399.2.cc.a	78	6
		399.2.cc.b	78
399.2.cd	$\chi_{399}(52, \cdot)$	399.2.cd.a	78	6
		399.2.cd.b	84
399.2.ci	$\chi_{399}(17, \cdot)$	399.2.ci.a	6	6
		399.2.ci.b	288
399.2.cj	$\chi_{399}(62, \cdot)$	399.2.cj.a	6	6
		399.2.cj.b	6
		399.2.cj.c	288
399.2.ck	$\chi_{399}(10, \cdot)$	399.2.ck.a	78	6
		399.2.ck.b	84

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_1(399))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_1(399)) \cong$ $S_{2}^{\mathrm{new}}(\Gamma_1(1))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_1(7))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_1(133))$$^{\oplus 2}$