Space of modular forms of level 8228, weight 2, and trivial character

• Label: 8228.2.a
• Level: $8228$
• Weight: $2$
• Character orbit: 8228.a
• Rep. character: $\chi_{8228}(1,\cdot)$
• Character field: $\Q$
• Dimension: $146$
• Newform subspaces: $28$
• Sturm bound: $2376$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 8228 = 2^{2} \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8228.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(2376\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8228))\).

	Total	New	Old
Modular forms	1224	146	1078
Cusp forms	1153	146	1007
Eisenstein series	71	0	71

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

\(2\)	\(11\)	\(17\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(40\)
\(-\)	\(+\)	\(-\)	$+$	\(32\)
\(-\)	\(-\)	\(+\)	$+$	\(32\)
\(-\)	\(-\)	\(-\)	$-$	\(42\)
Plus space			\(+\)	\(64\)
Minus space			\(-\)	\(82\)

Trace form

\( 146 q - 2 q^{3} - 4 q^{5} + 6 q^{7} + 142 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8228))\) 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	11	17
8228.2.a.a	$8228$	$2$	8228.a	1.a	$1$	$1$	$1$	$65.701$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}-5q^{13}+q^{17}+\cdots\)
8228.2.a.b	$8228$	$2$	8228.a	1.a	$1$	$1$	$1$	$65.701$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}-2q^{13}-q^{17}+\cdots\)
8228.2.a.c	$8228$	$2$	8228.a	1.a	$1$	$1$	$1$	$65.701$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}+2q^{13}+q^{17}+\cdots\)
8228.2.a.d	$8228$	$2$	8228.a	1.a	$1$	$1$	$1$	$65.701$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+5q^{13}-q^{17}+\cdots\)
8228.2.a.e	$8228$	$2$	8228.a	1.a	$1$	$1$	$1$	$65.701$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}+2q^{7}+6q^{9}+2q^{13}+\cdots\)
8228.2.a.f	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-\beta )q^{5}+2q^{7}-2\beta q^{9}+\cdots\)
8228.2.a.g	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-\beta q^{5}+(-3+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
8228.2.a.h	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}-\beta q^{7}+(-2+\beta )q^{9}+\cdots\)
8228.2.a.i	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}+\beta q^{7}+(-2+\beta )q^{9}+\cdots\)
8228.2.a.j	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{5}+(-1-\beta )q^{7}+\cdots\)
8228.2.a.k	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{5}+(1+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
8228.2.a.l	$8228$	$2$	8228.a	1.a	$1$	$2$	$2$	$65.701$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{5}+(1+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
8228.2.a.m	$8228$	$2$	8228.a	1.a	$1$	$3$	$3$	$65.701$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}-\beta _{1}q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots\)
8228.2.a.n	$8228$	$2$	8228.a	1.a	$1$	$4$	$4$	$65.701$	4.4.2225.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-6\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+\beta _{3})q^{5}+\cdots\)
8228.2.a.o	$8228$	$2$	8228.a	1.a	$1$	$4$	$4$	$65.701$	4.4.2225.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-6\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+\beta _{3})q^{5}+\cdots\)
8228.2.a.p	$8228$	$2$	8228.a	1.a	$1$	$4$	$4$	$65.701$	4.4.19796.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}+(-1-\beta _{2}+\beta _{3})q^{7}+\cdots\)
8228.2.a.q	$8228$	$2$	8228.a	1.a	$1$	$5$	$5$	$65.701$	5.5.503376.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
8228.2.a.r	$8228$	$2$	8228.a	1.a	$1$	$5$	$5$	$65.701$	5.5.503376.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{3})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
8228.2.a.s	$8228$	$2$	8228.a	1.a	$1$	$6$	$6$	$65.701$	6.6.9642625.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{5}q^{5}+(-1+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
8228.2.a.t	$8228$	$2$	8228.a	1.a	$1$	$6$	$6$	$65.701$	6.6.9642625.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{5}q^{5}+(1-\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
8228.2.a.u	$8228$	$2$	8228.a	1.a	$1$	$6$	$6$	$65.701$	6.6.492881873.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{4}q^{5}-\beta _{3}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8228.2.a.v	$8228$	$2$	8228.a	1.a	$1$	$6$	$6$	$65.701$	6.6.492881873.1	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{4}q^{5}+\beta _{3}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8228.2.a.w	$8228$	$2$	8228.a	1.a	$1$	$7$	$7$	$65.701$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{5}q^{7}+(\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
8228.2.a.x	$8228$	$2$	8228.a	1.a	$1$	$7$	$7$	$65.701$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+\beta _{5}q^{7}+(\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
8228.2.a.y	$8228$	$2$	8228.a	1.a	$1$	$12$	$12$	$65.701$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-6\)	\(-4\)		$-$	$+$	$-$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+\beta _{1}q^{5}+(-1-\beta _{1}+\beta _{11})q^{7}+\cdots\)
8228.2.a.z	$8228$	$2$	8228.a	1.a	$1$	$12$	$12$	$65.701$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(-6\)	\(4\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+\beta _{1}q^{5}+(1+\beta _{1}-\beta _{11})q^{7}+\cdots\)
8228.2.a.ba	$8228$	$2$	8228.a	1.a	$1$	$20$	$20$	$65.701$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(7\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{11}q^{5}-\beta _{14}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
8228.2.a.bb	$8228$	$2$	8228.a	1.a	$1$	$20$	$20$	$65.701$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(7\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{11}q^{5}+\beta _{14}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8228))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8228)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(187))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(374))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(748))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2057))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4114))\)\(^{\oplus 2}\)