Space of modular forms of level 198, weight 10, and trivial character

• Label: 198.10.a
• Level: $198$
• Weight: $10$
• Character orbit: 198.a
• Rep. character: $\chi_{198}(1,\cdot)$
• Character field: $\Q$
• Dimension: $37$
• Newform subspaces: $19$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(198))\).

	Total	New	Old
Modular forms	332	37	295
Cusp forms	316	37	279
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	$-$	\(6\)
\(+\)	\(-\)	\(-\)	$+$	\(6\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(-\)	$-$	\(6\)
Plus space			\(+\)	\(17\)
Minus space			\(-\)	\(20\)

Trace form

\( 37 q - 16 q^{2} + 9472 q^{4} + 4400 q^{5} + 2964 q^{7} - 4096 q^{8} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(198))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	11
198.10.a.a	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(-2349\)	\(-8806\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}-2349q^{5}-8806q^{7}+\cdots\)
198.10.a.b	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(595\)	\(11354\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+595q^{5}+11354q^{7}+\cdots\)
198.10.a.c	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(636\)	\(-7714\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+636q^{5}-7714q^{7}+\cdots\)
198.10.a.d	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$0$	\(-16\)	\(0\)	\(1039\)	\(-3482\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+1039q^{5}-3482q^{7}+\cdots\)
198.10.a.e	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(1054\)	\(-3640\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+1054q^{5}-3640q^{7}+\cdots\)
198.10.a.f	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(0\)	\(-636\)	\(-7714\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}-636q^{5}-7714q^{7}+\cdots\)
198.10.a.g	$198$	$10$	198.a	1.a	$1$	$1$	$1$	$101.977$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(0\)	\(350\)	\(1022\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+350q^{5}+1022q^{7}+\cdots\)
198.10.a.h	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{77119}) \)	None	✓	$1$	$1$	\(-32\)	\(0\)	\(-76\)	\(-1504\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-38+\beta )q^{5}+(-752+\cdots)q^{7}+\cdots\)
198.10.a.i	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{7561}) \)	None	✓	$1$	$0$	\(-32\)	\(0\)	\(1054\)	\(1638\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(527-17\beta )q^{5}+\cdots\)
198.10.a.j	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{77119}) \)	None	✓	$1$	$1$	\(32\)	\(0\)	\(76\)	\(-1504\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(38+\beta )q^{5}+(-752+\cdots)q^{7}+\cdots\)
198.10.a.k	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{119497}) \)	None	✓	$1$	$0$	\(32\)	\(0\)	\(350\)	\(-3024\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(175-5\beta )q^{5}+(-1512+\cdots)q^{7}+\cdots\)
198.10.a.l	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{88705}) \)	None	✓	$1$	$0$	\(32\)	\(0\)	\(350\)	\(-2646\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(175-\beta )q^{5}+(-1323+\cdots)q^{7}+\cdots\)
198.10.a.m	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{80521}) \)	None	✓	$1$	$1$	\(32\)	\(0\)	\(350\)	\(5446\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(175-\beta )q^{5}+(2723+\cdots)q^{7}+\cdots\)
198.10.a.n	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{889}) \)	None	✓	$1$	$1$	\(32\)	\(0\)	\(521\)	\(-7490\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(309-97\beta )q^{5}+\cdots\)
198.10.a.o	$198$	$10$	198.a	1.a	$1$	$2$	$2$	$101.977$	\(\Q(\sqrt{463}) \)	None	✓	$1$	$0$	\(32\)	\(0\)	\(1478\)	\(8196\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(739+10\beta )q^{5}+\cdots\)
198.10.a.p	$198$	$10$	198.a	1.a	$1$	$3$	$3$	$101.977$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-48\)	\(0\)	\(-196\)	\(-2004\)		$+$	$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-65+\beta _{2})q^{5}+\cdots\)
198.10.a.q	$198$	$10$	198.a	1.a	$1$	$3$	$3$	$101.977$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(0\)	\(-196\)	\(4804\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-65-\beta _{1})q^{5}+\cdots\)
198.10.a.r	$198$	$10$	198.a	1.a	$1$	$4$	$4$	$101.977$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-64\)	\(0\)	\(560\)	\(10016\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(140+\beta _{1})q^{5}+(2504+\cdots)q^{7}+\cdots\)
198.10.a.s	$198$	$10$	198.a	1.a	$1$	$4$	$4$	$101.977$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(64\)	\(0\)	\(-560\)	\(10016\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(-140-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(198))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(198)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)