LMFDB - Space of modular forms of level 5550, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5550.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 68 $
Sturm bound:			$2280$
Trace bound:			$17$
Distinguishing $T_p$:			$7$, $11$, $13$, $17$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(5550))$.

	Total	New	Old
Modular forms	1164	114	1050
Cusp forms	1117	114	1003
Eisenstein series	47	0	47

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

$2$	$3$	$5$	$37$	Fricke	Total			Cusp			Eisenstein
$2$	$3$	$5$	$37$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$+$	$+$	$60$	$4$	$56$	$58$	$4$	$54$	$2$	$0$	$2$
$+$	$+$	$+$	$-$	$-$	$84$	$10$	$74$	$81$	$10$	$71$	$3$	$0$	$3$
$+$	$+$	$-$	$+$	$-$	$79$	$9$	$70$	$76$	$9$	$67$	$3$	$0$	$3$
$+$	$+$	$-$	$-$	$+$	$68$	$5$	$63$	$65$	$5$	$60$	$3$	$0$	$3$
$+$	$-$	$+$	$+$	$-$	$78$	$8$	$70$	$75$	$8$	$67$	$3$	$0$	$3$
$+$	$-$	$+$	$-$	$+$	$69$	$5$	$64$	$66$	$5$	$61$	$3$	$0$	$3$
$+$	$-$	$-$	$+$	$+$	$69$	$7$	$62$	$66$	$7$	$59$	$3$	$0$	$3$
$+$	$-$	$-$	$-$	$-$	$75$	$9$	$66$	$72$	$9$	$63$	$3$	$0$	$3$
$-$	$+$	$+$	$+$	$-$	$72$	$8$	$64$	$69$	$8$	$61$	$3$	$0$	$3$
$-$	$+$	$+$	$-$	$+$	$72$	$5$	$67$	$69$	$5$	$64$	$3$	$0$	$3$
$-$	$+$	$-$	$+$	$+$	$72$	$7$	$65$	$69$	$7$	$62$	$3$	$0$	$3$
$-$	$+$	$-$	$-$	$-$	$75$	$9$	$66$	$72$	$9$	$63$	$3$	$0$	$3$
$-$	$-$	$+$	$+$	$+$	$81$	$4$	$77$	$78$	$4$	$74$	$3$	$0$	$3$
$-$	$-$	$+$	$-$	$-$	$66$	$10$	$56$	$63$	$10$	$53$	$3$	$0$	$3$
$-$	$-$	$-$	$+$	$-$	$71$	$9$	$62$	$68$	$9$	$59$	$3$	$0$	$3$
$-$	$-$	$-$	$-$	$+$	$73$	$5$	$68$	$70$	$5$	$65$	$3$	$0$	$3$
Plus space				$+$	$564$	$42$	$522$	$541$	$42$	$499$	$23$	$0$	$23$
Minus space				$-$	$600$	$72$	$528$	$576$	$72$	$504$	$24$	$0$	$24$

Trace form

$ 114 q + 114 q^{4} - 2 q^{6} - 12 q^{7} + 114 q^{9} + 4 q^{13} + 8 q^{14} + 114 q^{16} + 8 q^{17} - 12 q^{19} + 8 q^{22} - 8 q^{23} - 2 q^{24} + 8 q^{26} - 12 q^{28} - 8 q^{29} + 12 q^{31} - 12 q^{33} + 114 q^{36}+ \cdots - 4 q^{97}+O(q^{100}) $

114 * q + 114 * q^4 - 2 * q^6 - 12 * q^7 + 114 * q^9 + 4 * q^13 + 8 * q^14 + 114 * q^16 + 8 * q^17 - 12 * q^19 + 8 * q^22 - 8 * q^23 - 2 * q^24 + 8 * q^26 - 12 * q^28 - 8 * q^29 + 12 * q^31 - 12 * q^33 + 114 * q^36 + 2 * q^37 + 20 * q^38 + 20 * q^41 - 12 * q^43 + 12 * q^46 + 32 * q^47 + 126 * q^49 + 4 * q^51 + 4 * q^52 + 28 * q^53 - 2 * q^54 + 8 * q^56 + 4 * q^57 + 20 * q^58 + 8 * q^59 - 12 * q^61 - 12 * q^62 - 12 * q^63 + 114 * q^64 + 8 * q^68 - 8 * q^69 + 8 * q^71 - 12 * q^76 - 8 * q^78 - 36 * q^79 + 114 * q^81 + 24 * q^83 + 4 * q^86 - 28 * q^87 + 8 * q^88 - 48 * q^89 - 32 * q^91 - 8 * q^92 - 4 * q^93 - 16 * q^94 - 2 * q^96 - 4 * q^97

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(5550))$ 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	3	5	37	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
5550.2.a.a	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.a	$1$	$2$	$-1$	$-1$	$0$	$-4$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots$
5550.2.a.b	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.o	$1$	$1$	$-1$	$-1$	$0$	$-4$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots$
5550.2.a.c	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.a	$1$	$1$	$-1$	$-1$	$0$	$-2$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots$
5550.2.a.d	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.c	$1$	$1$	$-1$	$-1$	$0$	$-1$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots$
5550.2.a.e	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.e	$1$	$1$	$-1$	$-1$	$0$	$0$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots$
5550.2.a.f	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.f	$1$	$0$	$-1$	$-1$	$0$	$0$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots$
5550.2.a.g	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.n	$1$	$1$	$-1$	$-1$	$0$	$1$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots$
5550.2.a.h	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				222.2.a.e	$1$	$1$	$-1$	$-1$	$0$	$1$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots$
5550.2.a.i	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.i	$1$	$1$	$-1$	$-1$	$0$	$2$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots$
5550.2.a.j	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.e	$1$	$0$	$-1$	$-1$	$0$	$2$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots$
5550.2.a.k	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.m	$1$	$0$	$-1$	$-1$	$0$	$5$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+5q^{7}-q^{8}+\cdots$
5550.2.a.l	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.b	$1$	$0$	$-1$	$-1$	$0$	$5$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+5q^{7}-q^{8}+\cdots$
5550.2.a.m	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.l	$1$	$0$	$-1$	$1$	$0$	$-3$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots$
5550.2.a.n	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				222.2.a.d	$1$	$1$	$-1$	$1$	$0$	$-3$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots$
5550.2.a.o	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.o	$1$	$1$	$-1$	$1$	$0$	$-2$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-2q^{7}-q^{8}+\cdots$
5550.2.a.p	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.d	$1$	$0$	$-1$	$1$	$0$	$-2$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-2q^{7}-q^{8}+\cdots$
5550.2.a.q	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.i	$1$	$1$	$-1$	$1$	$0$	$0$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots$
5550.2.a.r	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.k	$1$	$0$	$-1$	$1$	$0$	$0$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots$
5550.2.a.s	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.s	$1$	$0$	$-1$	$1$	$0$	$2$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots$
5550.2.a.t	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.j	$1$	$1$	$-1$	$1$	$0$	$3$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+3q^{7}-q^{8}+\cdots$
5550.2.a.u	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.u	$1$	$1$	$-1$	$1$	$0$	$4$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots$
5550.2.a.v	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.u	$1$	$1$	$1$	$-1$	$0$	$-4$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots$
5550.2.a.w	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.g	$1$	$1$	$1$	$-1$	$0$	$-4$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots$
5550.2.a.x	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.s	$1$	$0$	$1$	$-1$	$0$	$-2$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots$
5550.2.a.y	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.f	$1$	$1$	$1$	$-1$	$0$	$-1$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots$
5550.2.a.z	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				222.2.a.c	$1$	$1$	$1$	$-1$	$0$	$1$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots$
5550.2.a.ba	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.e	$1$	$0$	$1$	$-1$	$0$	$1$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots$
5550.2.a.bb	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.o	$1$	$1$	$1$	$-1$	$0$	$2$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+2q^{7}+q^{8}+\cdots$
5550.2.a.bc	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.d	$1$	$1$	$1$	$-1$	$0$	$2$	$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+2q^{7}+q^{8}+\cdots$
5550.2.a.bd	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.h	$1$	$0$	$1$	$-1$	$0$	$4$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots$
5550.2.a.be	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.b	$1$	$1$	$1$	$1$	$0$	$-5$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-5q^{7}+q^{8}+\cdots$
5550.2.a.bf	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.d	$1$	$1$	$1$	$1$	$0$	$-3$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
5550.2.a.bg	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.c	$1$	$0$	$1$	$1$	$0$	$-3$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
5550.2.a.bh	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				222.2.a.a	$1$	$1$	$1$	$1$	$0$	$-3$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
5550.2.a.bi	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.e	$1$	$1$	$1$	$1$	$0$	$-2$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots$
5550.2.a.bj	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.i	$1$	$1$	$1$	$1$	$0$	$-2$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots$
5550.2.a.bk	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.b	$1$	$0$	$1$	$1$	$0$	$-1$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots$
5550.2.a.bl	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				222.2.a.b	$1$	$1$	$1$	$1$	$0$	$0$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots$
5550.2.a.bm	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.e	$1$	$1$	$1$	$1$	$0$	$0$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots$
5550.2.a.bn	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.f	$1$	$0$	$1$	$1$	$0$	$0$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots$
5550.2.a.bo	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.c	$1$	$0$	$1$	$1$	$0$	$1$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots$
5550.2.a.bp	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.d.a	$1$	$0$	$1$	$1$	$0$	$2$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots$
5550.2.a.bq	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓	✓			5550.2.a.a	$1$	$0$	$1$	$1$	$0$	$4$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots$
5550.2.a.br	$5550$	$2$	5550.a	1.a	$1$	$1$	$1$	$44.317$	$\Q$	None	✓				1110.2.a.a	$1$	$0$	$1$	$1$	$0$	$4$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots$
5550.2.a.bs	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{2}) $	None	✓				1110.2.d.g	$1$	$0$	$-2$	$-2$	$0$	$-2$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots$
5550.2.a.bt	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{6}) $	None	✓		✓		1110.2.d.h	$1$	$1$	$-2$	$-2$	$0$	$-2$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots$
5550.2.a.bu	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{5}) $	None	✓				1110.2.d.f	$1$	$1$	$-2$	$2$	$0$	$-4$	$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-2q^{7}-q^{8}+\cdots$
5550.2.a.bv	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{33}) $	None	✓	✓	✓		5550.2.a.bv	$1$	$1$	$-2$	$2$	$0$	$0$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots$
5550.2.a.bw	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{17}) $	None	✓				1110.2.a.r	$1$	$0$	$-2$	$2$	$0$	$2$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots$
5550.2.a.bx	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{113}) $	None	✓				1110.2.a.q	$1$	$0$	$2$	$-2$	$0$	$-6$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-3q^{7}+q^{8}+\cdots$
5550.2.a.by	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{33}) $	None	✓	✓			5550.2.a.bv	$1$	$1$	$2$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots$
5550.2.a.bz	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{5}) $	None	✓				1110.2.d.f	$1$	$0$	$2$	$-2$	$0$	$4$	$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+2q^{7}+q^{8}+\cdots$
5550.2.a.ca	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{33}) $	None	✓				1110.2.a.p	$1$	$0$	$2$	$2$	$0$	$1$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+\beta q^{7}+q^{8}+\cdots$
5550.2.a.cb	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{2}) $	None	✓		✓		1110.2.d.g	$1$	$1$	$2$	$2$	$0$	$2$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots$
5550.2.a.cc	$5550$	$2$	5550.a	1.a	$1$	$2$	$2$	$44.317$	$\Q(\sqrt{6}) $	None	✓				1110.2.d.h	$1$	$0$	$2$	$2$	$0$	$2$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots$
5550.2.a.cd	$5550$	$2$	5550.a	1.a	$1$	$3$	$3$	$44.317$	3.3.993.1	None	✓	✓	✓		5550.2.a.cd	$1$	$0$	$-3$	$-3$	$0$	$0$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+(\beta _{1}-\beta _{2})q^{7}+\cdots$
5550.2.a.ce	$5550$	$2$	5550.a	1.a	$1$	$3$	$3$	$44.317$	3.3.257.1	None	✓	✓			5550.2.a.ce	$1$	$0$	$-3$	$3$	$0$	$-4$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+(-1-\beta _{1}+\cdots)q^{7}+\cdots$
5550.2.a.cf	$5550$	$2$	5550.a	1.a	$1$	$3$	$3$	$44.317$	3.3.568.1	None	✓		✓		1110.2.d.i	$1$	$1$	$-3$	$3$	$0$	$3$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots$
5550.2.a.cg	$5550$	$2$	5550.a	1.a	$1$	$3$	$3$	$44.317$	3.3.568.1	None	✓				1110.2.d.i	$1$	$0$	$3$	$-3$	$0$	$-3$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+(-1-\beta _{1}+\cdots)q^{7}+\cdots$
5550.2.a.ch	$5550$	$2$	5550.a	1.a	$1$	$3$	$3$	$44.317$	3.3.257.1	None	✓	✓	✓		5550.2.a.ce	$1$	$0$	$3$	$-3$	$0$	$4$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta _{1}-\beta _{2})q^{7}+\cdots$
5550.2.a.ci	$5550$	$2$	5550.a	1.a	$1$	$3$	$3$	$44.317$	3.3.993.1	None	✓	✓	✓		5550.2.a.cd	$1$	$0$	$3$	$3$	$0$	$0$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+(-\beta _{1}+\beta _{2})q^{7}+\cdots$
5550.2.a.cj	$5550$	$2$	5550.a	1.a	$1$	$4$	$4$	$44.317$	4.4.54764.1	None	✓				1110.2.a.s	$1$	$0$	$-4$	$-4$	$0$	$-4$	$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+(-1-\beta _{2}+\cdots)q^{7}+\cdots$
5550.2.a.ck	$5550$	$2$	5550.a	1.a	$1$	$4$	$4$	$44.317$	4.4.358377.1	None	✓	✓	✓		5550.2.a.ck	$1$	$0$	$-4$	$4$	$0$	$-4$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+(-1-\beta _{1}+\cdots)q^{7}+\cdots$
5550.2.a.cl	$5550$	$2$	5550.a	1.a	$1$	$4$	$4$	$44.317$	$\Q(\sqrt{22 +2 \sqrt{17}})$	None	✓		✓		1110.2.d.j	$1$	$0$	$-4$	$4$	$0$	$3$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$
5550.2.a.cm	$5550$	$2$	5550.a	1.a	$1$	$4$	$4$	$44.317$	$\Q(\sqrt{22 +2 \sqrt{17}})$	None	✓		✓		1110.2.d.j	$1$	$1$	$4$	$-4$	$0$	$-3$	$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+(-1+\beta _{1}+\cdots)q^{7}+\cdots$
5550.2.a.cn	$5550$	$2$	5550.a	1.a	$1$	$4$	$4$	$44.317$	4.4.358377.1	None	✓	✓	✓		5550.2.a.ck	$1$	$0$	$4$	$-4$	$0$	$4$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots$
5550.2.a.co	$5550$	$2$	5550.a	1.a	$1$	$5$	$5$	$44.317$	5.5.60594425.1	None	✓	✓	✓		5550.2.a.co	$1$	$0$	$-5$	$-5$	$0$	$-2$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-\beta _{4}q^{7}-q^{8}+\cdots$
5550.2.a.cp	$5550$	$2$	5550.a	1.a	$1$	$5$	$5$	$44.317$	5.5.60594425.1	None	✓	✓	✓		5550.2.a.co	$1$	$0$	$5$	$5$	$0$	$2$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}+\beta _{4}q^{7}+q^{8}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(5550))$ 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:	\( N \)	\(=\)	\( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5550.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 68 \)
Sturm bound:			\(2280\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

\(2\)	\(3\)	\(5\)	\(37\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(3\)	\(5\)	\(37\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(60\)	\(4\)	\(56\)	\(58\)	\(4\)	\(54\)	\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(84\)	\(10\)	\(74\)	\(81\)	\(10\)	\(71\)	\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(79\)	\(9\)	\(70\)	\(76\)	\(9\)	\(67\)	\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(68\)	\(5\)	\(63\)	\(65\)	\(5\)	\(60\)	\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(78\)	\(8\)	\(70\)	\(75\)	\(8\)	\(67\)	\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(69\)	\(5\)	\(64\)	\(66\)	\(5\)	\(61\)	\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(69\)	\(7\)	\(62\)	\(66\)	\(7\)	\(59\)	\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(75\)	\(9\)	\(66\)	\(72\)	\(9\)	\(63\)	\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(72\)	\(8\)	\(64\)	\(69\)	\(8\)	\(61\)	\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(72\)	\(5\)	\(67\)	\(69\)	\(5\)	\(64\)	\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(72\)	\(7\)	\(65\)	\(69\)	\(7\)	\(62\)	\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(75\)	\(9\)	\(66\)	\(72\)	\(9\)	\(63\)	\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(81\)	\(4\)	\(77\)	\(78\)	\(4\)	\(74\)	\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(66\)	\(10\)	\(56\)	\(63\)	\(10\)	\(53\)	\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(71\)	\(9\)	\(62\)	\(68\)	\(9\)	\(59\)	\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(73\)	\(5\)	\(68\)	\(70\)	\(5\)	\(65\)	\(3\)	\(0\)	\(3\)
Plus space				\(+\)	\(564\)	\(42\)	\(522\)	\(541\)	\(42\)	\(499\)	\(23\)	\(0\)	\(23\)
Minus space				\(-\)	\(600\)	\(72\)	\(528\)	\(576\)	\(72\)	\(504\)	\(24\)	\(0\)	\(24\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5550))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5550))\) 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	5550.2.a

Level	$5550$
Weight	$2$
Character orbit	5550.a
Rep. character	$\chi_{5550}(1,\cdot)$
Character field	$\Q$
Dimension	$114$
Newform subspaces	$68$
Sturm bound	$2280$
Trace bound	$17$