Space of modular forms of level 1100, weight 4, and trivial character

• Label: 1100.4.a
• Level: $1100$
• Weight: $4$
• Character orbit: 1100.a
• Rep. character: $\chi_{1100}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $15$
• Sturm bound: $720$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(720\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1100))\).

	Total	New	Old
Modular forms	558	47	511
Cusp forms	522	47	475
Eisenstein series	36	0	36

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
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(75\)	\(0\)	\(75\)		\(69\)	\(0\)	\(69\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(66\)	\(0\)	\(66\)		\(60\)	\(0\)	\(60\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(69\)	\(0\)	\(69\)		\(63\)	\(0\)	\(63\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(72\)	\(0\)	\(72\)		\(66\)	\(0\)	\(66\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(69\)	\(11\)	\(58\)		\(66\)	\(11\)	\(55\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(69\)	\(12\)	\(57\)		\(66\)	\(12\)	\(54\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(69\)	\(13\)	\(56\)		\(66\)	\(13\)	\(53\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(69\)	\(11\)	\(58\)		\(66\)	\(11\)	\(55\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(285\)	\(25\)	\(260\)		\(267\)	\(25\)	\(242\)		\(18\)	\(0\)	\(18\)
Minus space			\(-\)		\(273\)	\(22\)	\(251\)		\(255\)	\(22\)	\(233\)		\(18\)	\(0\)	\(18\)

Trace form

47 * q + 8 * q^3 - 16 * q^7 + 417 * q^9 - 11 * q^11 + 106 * q^13 + 50 * q^17 - 156 * q^19 - 4 * q^21 + 176 * q^23 - 64 * q^27 + 462 * q^29 + 392 * q^31 - 22 * q^33 + 124 * q^37 + 1296 * q^39 + 370 * q^41 - 76 * q^43 + 612 * q^47 + 2447 * q^49 + 584 * q^51 - 638 * q^53 - 528 * q^57 + 36 * q^59 - 530 * q^61 + 344 * q^63 - 1416 * q^67 - 534 * q^69 - 2544 * q^71 + 270 * q^73 + 396 * q^77 + 2048 * q^79 + 2343 * q^81 + 1644 * q^83 - 4176 * q^87 - 2864 * q^89 + 2440 * q^91 - 1094 * q^93 + 1592 * q^97 + 33 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1100))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	11
1100.4.a.a	$1100$	$4$	1100.a	1.a	$1$	$1$	$1$	$64.902$	\(\Q\)	None	✓				220.4.a.c	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-24\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-24q^{7}+37q^{9}-11q^{11}+\cdots\)
1100.4.a.b	$1100$	$4$	1100.a	1.a	$1$	$1$	$1$	$64.902$	\(\Q\)	None	✓				220.4.a.b	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(19\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+19q^{7}-2q^{9}-11q^{11}+62q^{13}+\cdots\)
1100.4.a.c	$1100$	$4$	1100.a	1.a	$1$	$1$	$1$	$64.902$	\(\Q\)	None	✓				220.4.a.a	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-11q^{7}-2q^{9}-11q^{11}+22q^{13}+\cdots\)
1100.4.a.d	$1100$	$4$	1100.a	1.a	$1$	$1$	$1$	$64.902$	\(\Q\)	None	✓				44.4.a.a	$1$	$1$	\(0\)	\(5\)	\(0\)	\(26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+26q^{7}-2q^{9}-11q^{11}-52q^{13}+\cdots\)
1100.4.a.e	$1100$	$4$	1100.a	1.a	$1$	$2$	$2$	$64.902$	\(\Q(\sqrt{97}) \)	None	✓				44.4.a.b	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(-10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(-8+6\beta )q^{7}+(13+\cdots)q^{9}+\cdots\)
1100.4.a.f	$1100$	$4$	1100.a	1.a	$1$	$2$	$2$	$64.902$	\(\Q(\sqrt{19}) \)	None	✓				220.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{7}-8q^{9}-11q^{11}+2^{4}\beta q^{13}+\cdots\)
1100.4.a.g	$1100$	$4$	1100.a	1.a	$1$	$2$	$2$	$64.902$	\(\Q(\sqrt{6}) \)	None	✓				220.4.a.e	$1$	$1$	\(0\)	\(8\)	\(0\)	\(-36\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}+(-18-2\beta )q^{7}+(13+8\beta )q^{9}+\cdots\)
1100.4.a.h	$1100$	$4$	1100.a	1.a	$1$	$2$	$2$	$64.902$	\(\Q(\sqrt{97}) \)	None	✓				220.4.a.d	$1$	$0$	\(0\)	\(9\)	\(0\)	\(15\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{3}+(7+\beta )q^{7}+(22-9\beta )q^{9}+\cdots\)
1100.4.a.i	$1100$	$4$	1100.a	1.a	$1$	$3$	$3$	$64.902$	3.3.9192.1	None	✓				220.4.a.f	$1$	$0$	\(0\)	\(3\)	\(0\)	\(5\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(2+\beta _{1}-2\beta _{2})q^{7}+(9+\cdots)q^{9}+\cdots\)
1100.4.a.j	$1100$	$4$	1100.a	1.a	$1$	$5$	$5$	$64.902$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			1100.4.a.j	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-50\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-10-\beta _{1}-\beta _{4})q^{7}+\cdots\)
1100.4.a.k	$1100$	$4$	1100.a	1.a	$1$	$5$	$5$	$64.902$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		1100.4.a.k	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+\beta _{2}q^{7}+(11+2\beta _{1}+\cdots)q^{9}+\cdots\)
1100.4.a.l	$1100$	$4$	1100.a	1.a	$1$	$5$	$5$	$64.902$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			1100.4.a.k	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}-\beta _{2}q^{7}+(11+2\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
1100.4.a.m	$1100$	$4$	1100.a	1.a	$1$	$5$	$5$	$64.902$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		1100.4.a.j	$1$	$0$	\(0\)	\(6\)	\(0\)	\(50\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(10+\beta _{1}+\beta _{4})q^{7}+(11+\cdots)q^{9}+\cdots\)
1100.4.a.n	$1100$	$4$	1100.a	1.a	$1$	$6$	$6$	$64.902$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		220.4.b.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2\beta _{1}-\beta _{2}+\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots\)
1100.4.a.o	$1100$	$4$	1100.a	1.a	$1$	$6$	$6$	$64.902$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		220.4.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{5}q^{7}+(8-\beta _{2}-\beta _{4})q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1100))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1100)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 2}\)