Space of modular forms of level 1122, weight 2, and trivial character

• Label: 1122.2.a
• Level: $1122$
• Weight: $2$
• Character orbit: 1122.a
• Rep. character: $\chi_{1122}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $19$
• Sturm bound: $432$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(432\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1122))\).

	Total	New	Old
Modular forms	224	25	199
Cusp forms	209	25	184
Eisenstein series	15	0	15

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

\(2\)	\(3\)	\(11\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(3\)
Plus space				\(+\)	\(9\)
Minus space				\(-\)	\(16\)

Trace form

25 * q + q^2 - 3 * q^3 + 25 * q^4 + 6 * q^5 + q^6 + q^8 + 25 * q^9 + 6 * q^10 + q^11 - 3 * q^12 + 6 * q^13 + 8 * q^14 + 6 * q^15 + 25 * q^16 + q^17 + q^18 - 4 * q^19 + 6 * q^20 - 16 * q^21 - 3 * q^22 - 16 * q^23 + q^24 + 7 * q^25 + 6 * q^26 - 3 * q^27 - 2 * q^29 + 6 * q^30 + q^32 + q^33 - 7 * q^34 + 25 * q^36 + 6 * q^37 - 20 * q^38 - 26 * q^39 + 6 * q^40 + 10 * q^41 + 8 * q^42 - 44 * q^43 + q^44 + 6 * q^45 + 8 * q^46 + 16 * q^47 - 3 * q^48 + 33 * q^49 + 15 * q^50 + q^51 + 6 * q^52 - 18 * q^53 + q^54 + 6 * q^55 + 8 * q^56 - 4 * q^57 + 14 * q^58 - 12 * q^59 + 6 * q^60 - 10 * q^61 + 25 * q^64 + 20 * q^65 + q^66 + 4 * q^67 + q^68 + 8 * q^69 + 32 * q^70 + 16 * q^71 + q^72 + 34 * q^73 + 22 * q^74 + 3 * q^75 - 4 * q^76 + 6 * q^78 - 56 * q^79 + 6 * q^80 + 25 * q^81 + 10 * q^82 + 4 * q^83 - 16 * q^84 + 6 * q^85 - 12 * q^86 - 2 * q^87 - 3 * q^88 + 42 * q^89 + 6 * q^90 + 16 * q^91 - 16 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + q^96 - 14 * q^97 - 23 * q^98 + q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1122))\) 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	3	11	17
1122.2.a.a	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-2\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-q^{8}+\cdots\)
1122.2.a.b	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓	✓	✓	1122.2.a.b	$1$	$1$	\(-1\)	\(-1\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
1122.2.a.c	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.c	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(4\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)
1122.2.a.d	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓	✓	✓	1122.2.a.d	$1$	$1$	\(-1\)	\(1\)	\(-2\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
1122.2.a.e	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓	✓	✓	1122.2.a.e	$1$	$1$	\(1\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}+q^{8}+\cdots\)
1122.2.a.f	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓	✓	✓	1122.2.a.f	$1$	$1$	\(1\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}+q^{8}+\cdots\)
1122.2.a.g	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.g	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(-2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}-2q^{7}+\cdots\)
1122.2.a.h	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.h	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+4q^{7}+\cdots\)
1122.2.a.i	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓	✓	✓	1122.2.a.i	$1$	$1$	\(1\)	\(1\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-2q^{5}+q^{6}-4q^{7}+\cdots\)
1122.2.a.j	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.j	$1$	$0$	\(1\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
1122.2.a.k	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.k	$1$	$0$	\(1\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
1122.2.a.l	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.l	$1$	$0$	\(1\)	\(1\)	\(2\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
1122.2.a.m	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.m	$1$	$0$	\(1\)	\(1\)	\(2\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)
1122.2.a.n	$1122$	$2$	1122.a	1.a	$1$	$1$	$1$	$8.959$	\(\Q\)	None	✓	✓			1122.2.a.n	$1$	$0$	\(1\)	\(1\)	\(4\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}-2q^{7}+\cdots\)
1122.2.a.o	$1122$	$2$	1122.a	1.a	$1$	$2$	$2$	$8.959$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	1122.2.a.o	$1$	$0$	\(-2\)	\(-2\)	\(-2\)	\(6\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
1122.2.a.p	$1122$	$2$	1122.a	1.a	$1$	$2$	$2$	$8.959$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	1122.2.a.p	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2+\beta )q^{7}+\cdots\)
1122.2.a.q	$1122$	$2$	1122.a	1.a	$1$	$2$	$2$	$8.959$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	1122.2.a.q	$1$	$1$	\(-2\)	\(2\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}+(-2+\cdots)q^{7}+\cdots\)
1122.2.a.r	$1122$	$2$	1122.a	1.a	$1$	$2$	$2$	$8.959$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	1122.2.a.r	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(1+\beta )q^{5}-q^{6}+\cdots\)
1122.2.a.s	$1122$	$2$	1122.a	1.a	$1$	$3$	$3$	$8.959$	3.3.148.1	None	✓	✓	✓	✓	1122.2.a.s	$1$	$0$	\(3\)	\(-3\)	\(0\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}-q^{6}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1122))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1122)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(187))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(374))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(561))\)\(^{\oplus 2}\)