LMFDB - Space of modular forms of level 6050, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 92 $
Sturm bound:			$1980$
Trace bound:			$19$
Distinguishing $T_p$:			$3$, $7$, $13$, $17$, $19$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(6050))$.

	Total	New	Old
Modular forms	1062	173	889
Cusp forms	919	173	746
Eisenstein series	143	0	143

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

$2$	$5$	$11$	Fricke	Total			Cusp			Eisenstein
$2$	$5$	$11$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$+$	$126$	$18$	$108$	$109$	$18$	$91$	$17$	$0$	$17$
$+$	$+$	$-$	$-$	$138$	$24$	$114$	$120$	$24$	$96$	$18$	$0$	$18$
$+$	$-$	$+$	$-$	$138$	$23$	$115$	$120$	$23$	$97$	$18$	$0$	$18$
$+$	$-$	$-$	$+$	$129$	$22$	$107$	$111$	$22$	$89$	$18$	$0$	$18$
$-$	$+$	$+$	$-$	$135$	$24$	$111$	$117$	$24$	$93$	$18$	$0$	$18$
$-$	$+$	$-$	$+$	$132$	$16$	$116$	$114$	$16$	$98$	$18$	$0$	$18$
$-$	$-$	$+$	$+$	$129$	$19$	$110$	$111$	$19$	$92$	$18$	$0$	$18$
$-$	$-$	$-$	$-$	$135$	$27$	$108$	$117$	$27$	$90$	$18$	$0$	$18$
Plus space			$+$	$516$	$75$	$441$	$445$	$75$	$370$	$71$	$0$	$71$
Minus space			$-$	$546$	$98$	$448$	$474$	$98$	$376$	$72$	$0$	$72$

Trace form

$ 173 q - q^{2} + 2 q^{3} + 173 q^{4} - 2 q^{6} + 8 q^{7} - q^{8} + 177 q^{9} + 2 q^{12} + 2 q^{13} - 4 q^{14} + 173 q^{16} - 10 q^{17} - 13 q^{18} + 18 q^{19} - 4 q^{21} - 12 q^{23} - 2 q^{24} - 16 q^{27}+ \cdots - 25 q^{98}+O(q^{100}) $

173 * q - q^2 + 2 * q^3 + 173 * q^4 - 2 * q^6 + 8 * q^7 - q^8 + 177 * q^9 + 2 * q^12 + 2 * q^13 - 4 * q^14 + 173 * q^16 - 10 * q^17 - 13 * q^18 + 18 * q^19 - 4 * q^21 - 12 * q^23 - 2 * q^24 - 16 * q^27 + 8 * q^28 - 2 * q^29 + 12 * q^31 - q^32 + 177 * q^36 + 16 * q^37 + 2 * q^38 + 24 * q^39 + 4 * q^42 + 12 * q^43 + 12 * q^46 + 2 * q^48 + 189 * q^49 - 18 * q^51 + 2 * q^52 + 12 * q^53 - 2 * q^54 - 4 * q^56 - 4 * q^58 - 38 * q^59 + 26 * q^61 + 8 * q^62 + 8 * q^63 + 173 * q^64 + 50 * q^67 - 10 * q^68 - 52 * q^69 - 36 * q^71 - 13 * q^72 - 2 * q^73 + 18 * q^74 + 18 * q^76 + 28 * q^79 + 173 * q^81 - 6 * q^82 - 12 * q^83 - 4 * q^84 - 18 * q^86 - 32 * q^87 - 52 * q^89 + 120 * q^91 - 12 * q^92 + 52 * q^93 + 16 * q^94 - 2 * q^96 + 50 * q^97 - 25 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(6050))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	5	11	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
6050.2.a.a	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.b.b	$1$	$1$	$-1$	$-3$	$0$	$1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-3q^{3}+q^{4}+3q^{6}+q^{7}-q^{8}+\cdots$
6050.2.a.b	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.b.c	$1$	$1$	$-1$	$-3$	$0$	$1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-3q^{3}+q^{4}+3q^{6}+q^{7}-q^{8}+\cdots$
6050.2.a.c	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.c	$1$	$0$	$-1$	$-2$	$0$	$-3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-3q^{7}-q^{8}+\cdots$
6050.2.a.d	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.g	$1$	$1$	$-1$	$-2$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots$
6050.2.a.e	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.e	$1$	$1$	$-1$	$-2$	$0$	$1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}+q^{7}-q^{8}+\cdots$
6050.2.a.f	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.b.a	$1$	$1$	$-1$	$-1$	$0$	$-3$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-3q^{7}-q^{8}+\cdots$
6050.2.a.g	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.f	$1$	$0$	$-1$	$-1$	$0$	$-1$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots$
6050.2.a.h	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				50.2.a.a	$1$	$0$	$-1$	$-1$	$0$	$2$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots$
6050.2.a.i	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.a.a	$1$	$0$	$-1$	$-1$	$0$	$5$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+5q^{7}-q^{8}+\cdots$
6050.2.a.j	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.j	$1$	$1$	$-1$	$0$	$0$	$-2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{9}+q^{13}+\cdots$
6050.2.a.k	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.j	$1$	$0$	$-1$	$0$	$0$	$-2$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{9}+q^{13}+\cdots$
6050.2.a.l	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.c	$1$	$0$	$-1$	$1$	$0$	$3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+3q^{7}-q^{8}+\cdots$
6050.2.a.m	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.d	$1$	$1$	$-1$	$1$	$0$	$3$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+3q^{7}-q^{8}+\cdots$
6050.2.a.n	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				550.2.a.a	$1$	$0$	$-1$	$2$	$0$	$-4$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-4q^{7}-q^{8}+\cdots$
6050.2.a.o	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.c	$1$	$1$	$-1$	$2$	$0$	$-3$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-3q^{7}-q^{8}+\cdots$
6050.2.a.p	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				550.2.a.b	$1$	$1$	$-1$	$2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots$
6050.2.a.q	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.b.b	$1$	$1$	$-1$	$2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots$
6050.2.a.r	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.e	$1$	$0$	$-1$	$2$	$0$	$1$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}+q^{7}-q^{8}+\cdots$
6050.2.a.s	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				242.2.a.a	$1$	$0$	$-1$	$2$	$0$	$2$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}+2q^{7}-q^{8}+\cdots$
6050.2.a.t	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.a	$1$	$1$	$-1$	$3$	$0$	$-5$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+3q^{3}+q^{4}-3q^{6}-5q^{7}-q^{8}+\cdots$
6050.2.a.u	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.b.b	$1$	$1$	$-1$	$3$	$0$	$1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+3q^{3}+q^{4}-3q^{6}+q^{7}-q^{8}+\cdots$
6050.2.a.v	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.b.b	$1$	$0$	$1$	$-3$	$0$	$-1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-3q^{3}+q^{4}-3q^{6}-q^{7}+q^{8}+\cdots$
6050.2.a.w	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.e	$1$	$0$	$1$	$-2$	$0$	$-1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots$
6050.2.a.x	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.b.b	$1$	$0$	$1$	$-2$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots$
6050.2.a.y	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				550.2.a.b	$1$	$1$	$1$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots$
6050.2.a.z	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.g	$1$	$0$	$1$	$-2$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots$
6050.2.a.ba	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.c	$1$	$1$	$1$	$-2$	$0$	$3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+3q^{7}+q^{8}+\cdots$
6050.2.a.bb	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				550.2.a.a	$1$	$0$	$1$	$-2$	$0$	$4$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+4q^{7}+q^{8}+\cdots$
6050.2.a.bc	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.a.c	$1$	$1$	$1$	$-1$	$0$	$-1$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots$
6050.2.a.bd	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.f	$1$	$1$	$1$	$-1$	$0$	$1$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots$
6050.2.a.be	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.j	$1$	$1$	$1$	$0$	$0$	$2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{9}-q^{13}+\cdots$
6050.2.a.bf	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.j	$1$	$0$	$1$	$0$	$0$	$2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{9}-q^{13}+\cdots$
6050.2.a.bg	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.d	$1$	$0$	$1$	$1$	$0$	$-3$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
6050.2.a.bh	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.c	$1$	$1$	$1$	$1$	$0$	$-3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
6050.2.a.bi	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				50.2.a.a	$1$	$0$	$1$	$1$	$0$	$-2$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots$
6050.2.a.bj	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.a.b	$1$	$1$	$1$	$1$	$0$	$3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots$
6050.2.a.bk	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.b.a	$1$	$0$	$1$	$1$	$0$	$3$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots$
6050.2.a.bl	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				242.2.a.a	$1$	$1$	$1$	$2$	$0$	$-2$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}-2q^{7}+q^{8}+\cdots$
6050.2.a.bm	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.e	$1$	$1$	$1$	$2$	$0$	$-1$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots$
6050.2.a.bn	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓	✓			6050.2.a.c	$1$	$0$	$1$	$2$	$0$	$3$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+3q^{7}+q^{8}+\cdots$
6050.2.a.bo	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				110.2.b.c	$1$	$0$	$1$	$3$	$0$	$-1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+3q^{3}+q^{4}+3q^{6}-q^{7}+q^{8}+\cdots$
6050.2.a.bp	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.b.b	$1$	$0$	$1$	$3$	$0$	$-1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+3q^{3}+q^{4}+3q^{6}-q^{7}+q^{8}+\cdots$
6050.2.a.bq	$6050$	$2$	6050.a	1.a	$1$	$1$	$1$	$48.309$	$\Q$	None	✓				1210.2.a.a	$1$	$0$	$1$	$3$	$0$	$5$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+3q^{3}+q^{4}+3q^{6}+5q^{7}+q^{8}+\cdots$
6050.2.a.br	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.a	$1$	$2$	$-2$	$-3$	$0$	$-6$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.bs	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				22.2.c.a	$1$	$1$	$-2$	$-3$	$0$	$4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.bt	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				1210.2.b.i	$1$	$1$	$-2$	$-2$	$0$	$-6$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots$
6050.2.a.bu	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.g.a	$1$	$1$	$-2$	$-2$	$0$	$-3$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2\beta q^{3}+q^{4}+2\beta q^{6}+(-1+\cdots)q^{7}+\cdots$
6050.2.a.bv	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.j.a	$1$	$1$	$-2$	$-2$	$0$	$-1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2\beta q^{3}+q^{4}+2\beta q^{6}-\beta q^{7}+\cdots$
6050.2.a.bw	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓	✓			6050.2.a.bw	$1$	$0$	$-2$	$-2$	$0$	$4$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots$
6050.2.a.bx	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.c	$1$	$1$	$-2$	$-1$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(-1+2\beta )q^{7}+\cdots$
6050.2.a.by	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				1210.2.a.o	$1$	$1$	$-2$	$0$	$0$	$2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(1+2\beta )q^{7}+\cdots$
6050.2.a.bz	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				1210.2.b.h	$2$	$0$	$-2$	$0$	$0$	$6$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+3q^{7}-q^{8}+\cdots$
6050.2.a.ca	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.c	$1$	$1$	$-2$	$1$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-1+2\beta )q^{7}+\cdots$
6050.2.a.cb	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{33}) $	None	✓				110.2.a.d	$1$	$0$	$-2$	$1$	$0$	$1$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+\beta q^{7}-q^{8}+\cdots$
6050.2.a.cc	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				242.2.a.c	$1$	$1$	$-2$	$2$	$0$	$-6$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.cd	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				1210.2.b.i	$1$	$1$	$-2$	$2$	$0$	$-6$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.ce	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.j.a	$1$	$0$	$-2$	$2$	$0$	$-1$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2\beta q^{3}+q^{4}-2\beta q^{6}-\beta q^{7}+\cdots$
6050.2.a.cf	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓	✓			6050.2.a.bw	$1$	$1$	$-2$	$2$	$0$	$4$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.cg	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.a	$1$	$0$	$-2$	$3$	$0$	$-6$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.ch	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.g.b	$1$	$0$	$-2$	$3$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.ci	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				22.2.c.a	$1$	$1$	$2$	$-3$	$0$	$-4$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.cj	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.a	$1$	$0$	$2$	$-3$	$0$	$6$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
6050.2.a.ck	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓	✓			6050.2.a.bw	$1$	$1$	$2$	$-2$	$0$	$-4$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots$
6050.2.a.cl	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.j.a	$1$	$1$	$2$	$-2$	$0$	$1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2\beta q^{3}+q^{4}-2\beta q^{6}+\beta q^{7}+\cdots$
6050.2.a.cm	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.g.a	$1$	$1$	$2$	$-2$	$0$	$3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2\beta q^{3}+q^{4}-2\beta q^{6}+(1+\beta )q^{7}+\cdots$
6050.2.a.cn	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				1210.2.b.i	$1$	$0$	$2$	$-2$	$0$	$6$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots$
6050.2.a.co	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.c	$1$	$1$	$2$	$-1$	$0$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(1-2\beta )q^{7}+\cdots$
6050.2.a.cp	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				1210.2.b.h	$2$	$1$	$2$	$0$	$0$	$-6$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}-3q^{7}+q^{8}+\cdots$
6050.2.a.cq	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				1210.2.a.o	$1$	$0$	$2$	$0$	$0$	$-2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(-1-2\beta )q^{7}+\cdots$
6050.2.a.cr	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.c	$1$	$1$	$2$	$1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(1-2\beta )q^{7}+\cdots$
6050.2.a.cs	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓	✓			6050.2.a.bw	$1$	$0$	$2$	$2$	$0$	$-4$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.ct	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.j.a	$1$	$0$	$2$	$2$	$0$	$1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2\beta q^{3}+q^{4}+2\beta q^{6}+\beta q^{7}+\cdots$
6050.2.a.cu	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				1210.2.b.i	$1$	$0$	$2$	$2$	$0$	$6$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.cv	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{3}) $	None	✓				242.2.a.c	$1$	$0$	$2$	$2$	$0$	$6$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.cw	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				110.2.g.b	$1$	$0$	$2$	$3$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.cx	$6050$	$2$	6050.a	1.a	$1$	$2$	$2$	$48.309$	$\Q(\sqrt{5}) $	None	✓				550.2.h.a	$1$	$0$	$2$	$3$	$0$	$6$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
6050.2.a.cy	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{62 +10 \sqrt{5}})$	None	✓				110.2.g.c	$1$	$0$	$-4$	$-3$	$0$	$-3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots$
6050.2.a.cz	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{11 +2 \sqrt{5}})$	None	✓				550.2.h.k	$1$	$0$	$-4$	$-3$	$0$	$6$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{3}+q^{4}+\cdots$
6050.2.a.da	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{34 +2 \sqrt{5}})$	None	✓		✓		110.2.j.b	$1$	$1$	$-4$	$-2$	$0$	$2$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(1+\beta _{2}+\cdots)q^{7}+\cdots$
6050.2.a.db	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{16 +2 \sqrt{3}})$	None	✓	✓			6050.2.a.db	$1$	$0$	$-4$	$-2$	$0$	$2$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots$
6050.2.a.dc	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓				1210.2.b.m	$2$	$0$	$-4$	$0$	$0$	$6$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(2+\beta _{3})q^{7}+\cdots$
6050.2.a.dd	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{34 +2 \sqrt{5}})$	None	✓				110.2.j.b	$1$	$0$	$-4$	$2$	$0$	$2$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(1+\beta _{2}+\cdots)q^{7}+\cdots$
6050.2.a.de	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{16 +2 \sqrt{3}})$	None	✓	✓	✓		6050.2.a.db	$1$	$1$	$-4$	$2$	$0$	$2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(\beta _{1}-\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{6}+\cdots$
6050.2.a.df	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{11 +2 \sqrt{5}})$	None	✓				550.2.h.k	$1$	$0$	$-4$	$3$	$0$	$6$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$
6050.2.a.dg	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{11 +2 \sqrt{5}})$	None	✓				550.2.h.k	$1$	$0$	$4$	$-3$	$0$	$-6$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{3}+q^{4}+\cdots$
6050.2.a.dh	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{62 +10 \sqrt{5}})$	None	✓				110.2.g.c	$1$	$0$	$4$	$-3$	$0$	$3$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots$
6050.2.a.di	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{34 +2 \sqrt{5}})$	None	✓				110.2.j.b	$1$	$1$	$4$	$-2$	$0$	$-2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(-1+\cdots)q^{7}+\cdots$
6050.2.a.dj	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{16 +2 \sqrt{3}})$	None	✓	✓			6050.2.a.db	$1$	$1$	$4$	$-2$	$0$	$-2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$
6050.2.a.dk	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓				1210.2.b.m	$2$	$1$	$4$	$0$	$0$	$-6$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots$
6050.2.a.dl	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{34 +2 \sqrt{5}})$	None	✓				110.2.j.b	$1$	$0$	$4$	$2$	$0$	$-2$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(-1+\cdots)q^{7}+\cdots$
6050.2.a.dm	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{16 +2 \sqrt{3}})$	None	✓	✓			6050.2.a.db	$1$	$0$	$4$	$2$	$0$	$-2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(\beta _{1}-\beta _{2})q^{3}+q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots$
6050.2.a.dn	$6050$	$2$	6050.a	1.a	$1$	$4$	$4$	$48.309$	$\Q(\sqrt{11 +2 \sqrt{5}})$	None	✓				550.2.h.k	$1$	$0$	$4$	$3$	$0$	$-6$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(6050))$ into lower level spaces

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 92 \)
Sturm bound:			\(1980\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\), \(17\), \(19\)

\(2\)	\(5\)	\(11\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(5\)	\(11\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(126\)	\(18\)	\(108\)	\(109\)	\(18\)	\(91\)	\(17\)	\(0\)	\(17\)
\(+\)	\(+\)	\(-\)	\(-\)	\(138\)	\(24\)	\(114\)	\(120\)	\(24\)	\(96\)	\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(+\)	\(-\)	\(138\)	\(23\)	\(115\)	\(120\)	\(23\)	\(97\)	\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)	\(+\)	\(129\)	\(22\)	\(107\)	\(111\)	\(22\)	\(89\)	\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(+\)	\(-\)	\(135\)	\(24\)	\(111\)	\(117\)	\(24\)	\(93\)	\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)	\(132\)	\(16\)	\(116\)	\(114\)	\(16\)	\(98\)	\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(+\)	\(+\)	\(129\)	\(19\)	\(110\)	\(111\)	\(19\)	\(92\)	\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(-\)	\(-\)	\(135\)	\(27\)	\(108\)	\(117\)	\(27\)	\(90\)	\(18\)	\(0\)	\(18\)
Plus space			\(+\)	\(516\)	\(75\)	\(441\)	\(445\)	\(75\)	\(370\)	\(71\)	\(0\)	\(71\)
Minus space			\(-\)	\(546\)	\(98\)	\(448\)	\(474\)	\(98\)	\(376\)	\(72\)	\(0\)	\(72\)

Introduction

L-functions

Modular forms

Varieties

Fields

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Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6050))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6050))\) 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	6050.2.a

Level	$6050$
Weight	$2$
Character orbit	6050.a
Rep. character	$\chi_{6050}(1,\cdot)$
Character field	$\Q$
Dimension	$173$
Newform subspaces	$92$
Sturm bound	$1980$
Trace bound	$19$