LMFDB - Space of modular forms of level 2450, weight 4, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 81 $
Sturm bound:			$1680$
Trace bound:			$11$
Distinguishing $T_p$:			$3$, $11$, $19$, $23$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(2450))$.

	Total	New	Old
Modular forms	1308	195	1113
Cusp forms	1212	195	1017
Eisenstein series	96	0	96

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$	$5$	$7$	Fricke	Dim
$+$	$+$	$+$	$+$	$25$
$+$	$+$	$-$	$-$	$22$
$+$	$-$	$+$	$-$	$24$
$+$	$-$	$-$	$+$	$27$
$-$	$+$	$+$	$-$	$19$
$-$	$+$	$-$	$+$	$26$
$-$	$-$	$+$	$+$	$28$
$-$	$-$	$-$	$-$	$24$
Plus space			$+$	$106$
Minus space			$-$	$89$

Trace form

$ 195 q - 2 q^{2} + 2 q^{3} + 780 q^{4} - 8 q^{8} + 1709 q^{9} + O(q^{10}) $

$ 195 q - 2 q^{2} + 2 q^{3} + 780 q^{4} - 8 q^{8} + 1709 q^{9} + 22 q^{11} + 8 q^{12} - 16 q^{13} + 3120 q^{16} + 38 q^{17} + 22 q^{18} - 28 q^{19} - 56 q^{22} - 64 q^{23} - 264 q^{26} - 124 q^{27} - 102 q^{29} + 8 q^{31} - 32 q^{32} - 840 q^{33} + 64 q^{34} + 6836 q^{36} - 1122 q^{37} + 4 q^{38} - 208 q^{39} + 88 q^{41} - 928 q^{43} + 88 q^{44} + 280 q^{46} + 1216 q^{47} + 32 q^{48} + 554 q^{51} - 64 q^{52} + 1722 q^{53} - 900 q^{54} + 2812 q^{57} - 612 q^{58} + 798 q^{59} + 732 q^{61} - 56 q^{62} + 12480 q^{64} + 116 q^{66} + 196 q^{67} + 152 q^{68} - 2396 q^{69} - 56 q^{71} + 88 q^{72} + 1330 q^{73} + 1132 q^{74} - 112 q^{76} + 704 q^{78} + 2152 q^{79} + 11939 q^{81} + 372 q^{82} + 3702 q^{83} + 3224 q^{86} - 1780 q^{87} - 224 q^{88} + 1800 q^{89} - 256 q^{92} - 2796 q^{93} - 1752 q^{94} - 3050 q^{97} - 1664 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(2450))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	5	7	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
2450.4.a.a	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$-10$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-10q^{3}+4q^{4}+20q^{6}-8q^{8}+\cdots$
2450.4.a.b	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$-8$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-8q^{3}+4q^{4}+2^{4}q^{6}-8q^{8}+\cdots$
2450.4.a.c	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$-7$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-7q^{3}+4q^{4}+14q^{6}-8q^{8}+\cdots$
2450.4.a.d	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$-5$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-5q^{3}+4q^{4}+10q^{6}-8q^{8}+\cdots$
2450.4.a.e	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$-4$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-4q^{3}+4q^{4}+8q^{6}-8q^{8}+\cdots$
2450.4.a.f	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$-4$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-4q^{3}+4q^{4}+8q^{6}-8q^{8}+\cdots$
2450.4.a.g	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$-3$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-8q^{8}+\cdots$
2450.4.a.h	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-8q^{8}+\cdots$
2450.4.a.i	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-8q^{8}+\cdots$
2450.4.a.j	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$-1$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-q^{3}+4q^{4}+2q^{6}-8q^{8}+\cdots$
2450.4.a.k	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$-1$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-q^{3}+4q^{4}+2q^{6}-8q^{8}+\cdots$
2450.4.a.l	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$1$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+q^{3}+4q^{4}-2q^{6}-8q^{8}+\cdots$
2450.4.a.m	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$1$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+q^{3}+4q^{4}-2q^{6}-8q^{8}+\cdots$
2450.4.a.n	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$1$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+q^{3}+4q^{4}-2q^{6}-8q^{8}+\cdots$
2450.4.a.o	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-8q^{8}+\cdots$
2450.4.a.p	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$4$	$0$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{3}+4q^{4}-8q^{6}-8q^{8}+\cdots$
2450.4.a.q	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$5$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+5q^{3}+4q^{4}-10q^{6}-8q^{8}+\cdots$
2450.4.a.r	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$5$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+5q^{3}+4q^{4}-10q^{6}-8q^{8}+\cdots$
2450.4.a.s	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$7$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+7q^{3}+4q^{4}-14q^{6}-8q^{8}+\cdots$
2450.4.a.t	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$-2$	$7$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+7q^{3}+4q^{4}-14q^{6}-8q^{8}+\cdots$
2450.4.a.u	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$-2$	$8$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+8q^{3}+4q^{4}-2^{4}q^{6}-8q^{8}+\cdots$
2450.4.a.v	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$-10$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-10q^{3}+4q^{4}-20q^{6}+8q^{8}+\cdots$
2450.4.a.w	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-8$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-8q^{3}+4q^{4}-2^{4}q^{6}+8q^{8}+\cdots$
2450.4.a.x	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-8$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-8q^{3}+4q^{4}-2^{4}q^{6}+8q^{8}+\cdots$
2450.4.a.y	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$-7$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-7q^{3}+4q^{4}-14q^{6}+8q^{8}+\cdots$
2450.4.a.z	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$-4$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-4q^{3}+4q^{4}-8q^{6}+8q^{8}+\cdots$
2450.4.a.ba	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-3$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+8q^{8}+\cdots$
2450.4.a.bb	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$-2$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-2q^{3}+4q^{4}-4q^{6}+8q^{8}+\cdots$
2450.4.a.bc	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots$
2450.4.a.bd	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots$
2450.4.a.be	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots$
2450.4.a.bf	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$-1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots$
2450.4.a.bg	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$1$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+q^{3}+4q^{4}+2q^{6}+8q^{8}+\cdots$
2450.4.a.bh	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$1$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+q^{3}+4q^{4}+2q^{6}+8q^{8}+\cdots$
2450.4.a.bi	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$2$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+8q^{8}+\cdots$
2450.4.a.bj	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$3$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+8q^{8}+\cdots$
2450.4.a.bk	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$4$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{3}+4q^{4}+8q^{6}+8q^{8}+\cdots$
2450.4.a.bl	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$4$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{3}+4q^{4}+8q^{6}+8q^{8}+\cdots$
2450.4.a.bm	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$4$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{3}+4q^{4}+8q^{6}+8q^{8}+\cdots$
2450.4.a.bn	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$1$	$2$	$7$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+7q^{3}+4q^{4}+14q^{6}+8q^{8}+\cdots$
2450.4.a.bo	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$8$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+8q^{3}+4q^{4}+2^{4}q^{6}+8q^{8}+\cdots$
2450.4.a.bp	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$10$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+10q^{3}+4q^{4}+20q^{6}+8q^{8}+\cdots$
2450.4.a.bq	$2450$	$4$	2450.a	1.a	$1$	$1$	$1$	$144.555$	$\Q$	None	✓	$1$	$0$	$2$	$10$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+10q^{3}+4q^{4}+20q^{6}+8q^{8}+\cdots$
2450.4.a.br	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{7}) $	None	✓	$2$	$1$	$-4$	$0$	$0$	$0$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}-8q^{8}+\cdots$
2450.4.a.bs	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{22}) $	None	✓	$2$	$1$	$-4$	$0$	$0$	$0$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}-8q^{8}+\cdots$
2450.4.a.bt	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{177}) $	None	✓	$1$	$0$	$4$	$-5$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-2-\beta )q^{3}+4q^{4}+(-4+\cdots)q^{6}+\cdots$
2450.4.a.bu	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{2}) $	None	✓	$1$	$1$	$4$	$-2$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1+3\beta )q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots$
2450.4.a.bv	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{46}) $	None	✓	$1$	$0$	$4$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1+\beta )q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots$
2450.4.a.bw	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{7}) $	None	✓	$2$	$1$	$4$	$0$	$0$	$0$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta q^{3}+4q^{4}+2\beta q^{6}+8q^{8}+\cdots$
2450.4.a.bx	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{2}) $	None	✓	$2$	$1$	$4$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+5\beta q^{3}+4q^{4}+10\beta q^{6}+8q^{8}+\cdots$
2450.4.a.by	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{2}) $	None	✓	$1$	$1$	$4$	$2$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(1+3\beta )q^{3}+4q^{4}+(2+6\beta )q^{6}+\cdots$
2450.4.a.bz	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{46}) $	None	✓	$1$	$1$	$4$	$2$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(1+\beta )q^{3}+4q^{4}+(2+2\beta )q^{6}+\cdots$
2450.4.a.ca	$2450$	$4$	2450.a	1.a	$1$	$2$	$2$	$144.555$	$\Q(\sqrt{177}) $	None	✓	$1$	$0$	$4$	$5$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(3-\beta )q^{3}+4q^{4}+(6-2\beta )q^{6}+\cdots$
2450.4.a.cb	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.115880.1	None	✓	$1$	$0$	$-6$	$-4$	$0$	$0$	$+$	$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cc	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.238585.1	None	✓	$1$	$1$	$-6$	$-3$	$0$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(2-2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cd	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.238585.1	None	✓	$1$	$0$	$-6$	$3$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(-2+2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.ce	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.115880.1	None	✓	$1$	$1$	$-6$	$4$	$0$	$0$	$+$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	$q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-2-2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cf	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.51960.1	None	✓	$1$	$0$	$-6$	$7$	$0$	$0$	$+$	$-$	$-$	$+$	$5$	$\mathrm{SU}(2)$	$q-2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+(-4-2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cg	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.51960.1	None	✓	$1$	$1$	$6$	$-7$	$0$	$0$	$-$	$-$	$-$	$-$	$5$	$\mathrm{SU}(2)$	$q+2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}+(-4+\cdots)q^{6}+\cdots$
2450.4.a.ch	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.238585.1	None	✓	$1$	$0$	$6$	$-3$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots$
2450.4.a.ci	$2450$	$4$	2450.a	1.a	$1$	$3$	$3$	$144.555$	3.3.238585.1	None	✓	$1$	$1$	$6$	$3$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(2-2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cj	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\Q(\sqrt{2}, \sqrt{113})$	None	✓	$1$	$0$	$-8$	$-10$	$0$	$0$	$+$	$+$	$+$	$+$	$7$	$\mathrm{SU}(2)$	$q-2q^{2}+(-3-\beta _{1})q^{3}+4q^{4}+(6+2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.ck	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$1$	$-8$	$-1$	$0$	$0$	$+$	$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}-\beta _{2}q^{3}+4q^{4}+2\beta _{2}q^{6}-8q^{8}+\cdots$
2450.4.a.cl	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\Q(\sqrt{2}, \sqrt{29})$	None	✓	$2$	$1$	$-8$	$0$	$0$	$0$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+(2\beta _{1}+\beta _{2})q^{3}+4q^{4}+(-4\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cm	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	4.4.10197128.1	None	✓	$2$	$0$	$-8$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}-8q^{8}+\cdots$
2450.4.a.cn	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	4.4.1555279308.1	None	✓	$2$	$0$	$-8$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}-8q^{8}+\cdots$
2450.4.a.co	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$-8$	$1$	$0$	$0$	$+$	$+$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta _{2}q^{3}+4q^{4}-2\beta _{2}q^{6}-8q^{8}+\cdots$
2450.4.a.cp	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\Q(\sqrt{2}, \sqrt{113})$	None	✓	$1$	$0$	$-8$	$10$	$0$	$0$	$+$	$+$	$+$	$+$	$7$	$\mathrm{SU}(2)$	$q-2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(-4+2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cq	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$8$	$-1$	$0$	$0$	$-$	$-$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}-\beta _{2}q^{3}+4q^{4}-2\beta _{2}q^{6}+8q^{8}+\cdots$
2450.4.a.cr	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\Q(\sqrt{2}, \sqrt{29})$	None	✓	$2$	$1$	$8$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(2\beta _{1}+\beta _{2})q^{3}+4q^{4}+(4\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cs	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	4.4.10197128.1	None	✓	$2$	$1$	$8$	$0$	$0$	$0$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+8q^{8}+\cdots$
2450.4.a.ct	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	4.4.1555279308.1	None	✓	$2$	$0$	$8$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+8q^{8}+\cdots$
2450.4.a.cu	$2450$	$4$	2450.a	1.a	$1$	$4$	$4$	$144.555$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$1$	$8$	$1$	$0$	$0$	$-$	$-$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta _{2}q^{3}+4q^{4}+2\beta _{2}q^{6}+8q^{8}+\cdots$
2450.4.a.cv	$2450$	$4$	2450.a	1.a	$1$	$6$	$6$	$144.555$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$1$	$-12$	$-7$	$0$	$0$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cw	$2450$	$4$	2450.a	1.a	$1$	$6$	$6$	$144.555$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$0$	$-12$	$7$	$0$	$0$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-2-2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cx	$2450$	$4$	2450.a	1.a	$1$	$6$	$6$	$144.555$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$1$	$12$	$-7$	$0$	$0$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots$
2450.4.a.cy	$2450$	$4$	2450.a	1.a	$1$	$6$	$6$	$144.555$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$0$	$12$	$7$	$0$	$0$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots$
2450.4.a.cz	$2450$	$4$	2450.a	1.a	$1$	$8$	$8$	$144.555$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$0$	$-16$	$0$	$0$	$0$	$+$	$+$	$+$	$+$	$2\cdot 7^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}-\beta _{3}q^{3}+4q^{4}+2\beta _{3}q^{6}-8q^{8}+\cdots$
2450.4.a.da	$2450$	$4$	2450.a	1.a	$1$	$8$	$8$	$144.555$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$0$	$16$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$2\cdot 7^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}-\beta _{3}q^{3}+4q^{4}-2\beta _{3}q^{6}+8q^{8}+\cdots$
2450.4.a.db	$2450$	$4$	2450.a	1.a	$1$	$10$	$10$	$144.555$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓	$2$	$1$	$-20$	$0$	$0$	$0$	$+$	$-$	$+$	$-$	$2^{6}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}-8q^{8}+\cdots$
2450.4.a.dc	$2450$	$4$	2450.a	1.a	$1$	$10$	$10$	$144.555$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓	$2$	$0$	$20$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$2^{6}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+8q^{8}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(2450))$ into lower level spaces

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 81 \)
Sturm bound:			\(1680\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\), \(19\), \(23\)

\(2\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(25\)
\(+\)	\(+\)	\(-\)	$-$	\(22\)
\(+\)	\(-\)	\(+\)	$-$	\(24\)
\(+\)	\(-\)	\(-\)	$+$	\(27\)
\(-\)	\(+\)	\(+\)	$-$	\(19\)
\(-\)	\(+\)	\(-\)	$+$	\(26\)
\(-\)	\(-\)	\(+\)	$+$	\(28\)
\(-\)	\(-\)	\(-\)	$-$	\(24\)
Plus space			\(+\)	\(106\)
Minus space			\(-\)	\(89\)

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\) into newform subspaces

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2450))\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2450.4.a

Level	$2450$
Weight	$4$
Character orbit	2450.a
Rep. character	$\chi_{2450}(1,\cdot)$
Character field	$\Q$
Dimension	$195$
Newform subspaces	$81$
Sturm bound	$1680$
Trace bound	$11$