Space of modular forms of level 605, weight 2, and trivial character

• Label: 605.2.a
• Level: $605$
• Weight: $2$
• Character orbit: 605.a
• Rep. character: $\chi_{605}(1,\cdot)$
• Character field: $\Q$
• Dimension: $37$
• Newform subspaces: $13$
• Sturm bound: $132$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(132\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(605))\).

	Total	New	Old
Modular forms	78	37	41
Cusp forms	55	37	18
Eisenstein series	23	0	23

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

\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(15\)	\(6\)	\(9\)		\(10\)	\(6\)	\(4\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(24\)	\(12\)	\(12\)		\(18\)	\(12\)	\(6\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(21\)	\(12\)	\(9\)		\(15\)	\(12\)	\(3\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(18\)	\(7\)	\(11\)		\(12\)	\(7\)	\(5\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(33\)	\(13\)	\(20\)		\(22\)	\(13\)	\(9\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(45\)	\(24\)	\(21\)		\(33\)	\(24\)	\(9\)		\(12\)	\(0\)	\(12\)

Trace form

37 * q - 3 * q^2 + 39 * q^4 + q^5 + 8 * q^6 + 4 * q^7 - 3 * q^8 + 33 * q^9 + q^10 + 16 * q^12 + 6 * q^13 + 31 * q^16 - 14 * q^17 - 7 * q^18 + 4 * q^19 - q^20 - 8 * q^23 + 8 * q^24 + 37 * q^25 - 10 * q^26 + 4 * q^28 - 10 * q^29 - 8 * q^30 + 4 * q^31 + q^32 - 34 * q^34 - 4 * q^35 + 7 * q^36 - 2 * q^37 - 20 * q^38 + 16 * q^39 + 9 * q^40 - 14 * q^41 - 44 * q^42 + 8 * q^43 + 13 * q^45 + 4 * q^46 + 4 * q^47 - 20 * q^48 + 41 * q^49 - 3 * q^50 + 16 * q^51 - 6 * q^52 - 14 * q^53 + 16 * q^54 - 20 * q^56 - 6 * q^58 - 8 * q^59 - 16 * q^60 + 6 * q^61 + 8 * q^62 + 20 * q^63 + 39 * q^64 - 10 * q^65 + 4 * q^67 - 18 * q^68 - 24 * q^69 - 8 * q^70 - 20 * q^71 - 39 * q^72 - 6 * q^73 + 22 * q^74 - 4 * q^76 - 52 * q^78 - 16 * q^79 - q^80 + 13 * q^81 - 22 * q^82 + 16 * q^83 - 32 * q^84 + 2 * q^85 - 16 * q^86 - 32 * q^87 - 22 * q^89 + 13 * q^90 + 48 * q^91 - 12 * q^92 + 56 * q^93 + 4 * q^94 + 4 * q^95 + 8 * q^96 + 46 * q^97 + 13 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
605.2.a.a	$605$	$2$	605.a	1.a	$1$	$1$	$1$	$4.831$	\(\Q\)	None	✓	✓			605.2.a.a	$1$	$1$	\(-1\)	\(-3\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-q^{4}+q^{5}+3q^{6}-3q^{7}+\cdots\)
605.2.a.b	$605$	$2$	605.a	1.a	$1$	$1$	$1$	$4.831$	\(\Q\)	None	✓				55.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}+3q^{8}-3q^{9}-q^{10}+\cdots\)
605.2.a.c	$605$	$2$	605.a	1.a	$1$	$1$	$1$	$4.831$	\(\Q\)	None	✓	✓			605.2.a.a	$1$	$1$	\(1\)	\(-3\)	\(1\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-q^{4}+q^{5}-3q^{6}+3q^{7}+\cdots\)
605.2.a.d	$605$	$2$	605.a	1.a	$1$	$2$	$2$	$4.831$	\(\Q(\sqrt{2}) \)	None	✓				55.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
605.2.a.e	$605$	$2$	605.a	1.a	$1$	$2$	$2$	$4.831$	\(\Q(\sqrt{3}) \)	None	✓	✓			605.2.a.e	$2$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+q^{4}-q^{5}-\beta q^{6}-\beta q^{7}+\cdots\)
605.2.a.f	$605$	$2$	605.a	1.a	$1$	$2$	$2$	$4.831$	\(\Q(\sqrt{3}) \)	None	✓	✓			605.2.a.f	$2$	$0$	\(0\)	\(4\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+2q^{3}+q^{4}+q^{5}+2\beta q^{6}+\cdots\)
605.2.a.g	$605$	$2$	605.a	1.a	$1$	$3$	$3$	$4.831$	3.3.404.1	None	✓	✓			605.2.a.g	$1$	$0$	\(-1\)	\(1\)	\(-3\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+\cdots\)
605.2.a.h	$605$	$2$	605.a	1.a	$1$	$3$	$3$	$4.831$	3.3.404.1	None	✓	✓			605.2.a.g	$1$	$0$	\(1\)	\(1\)	\(-3\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+\cdots\)
605.2.a.i	$605$	$2$	605.a	1.a	$1$	$4$	$4$	$4.831$	4.4.2525.1	None	✓		✓		55.2.g.a	$1$	$1$	\(-3\)	\(-2\)	\(4\)	\(-11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
605.2.a.j	$605$	$2$	605.a	1.a	$1$	$4$	$4$	$4.831$	4.4.725.1	None	✓		✓		55.2.g.b	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
605.2.a.k	$605$	$2$	605.a	1.a	$1$	$4$	$4$	$4.831$	4.4.725.1	None	✓		✓		55.2.g.b	$1$	$0$	\(1\)	\(0\)	\(-4\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
605.2.a.l	$605$	$2$	605.a	1.a	$1$	$4$	$4$	$4.831$	4.4.2525.1	None	✓				55.2.g.a	$1$	$0$	\(3\)	\(-2\)	\(4\)	\(11\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2}-\beta _{3})q^{2}+(-1+\beta _{1})q^{3}+\cdots\)
605.2.a.m	$605$	$2$	605.a	1.a	$1$	$6$	$6$	$4.831$	6.6.27433728.1	None	✓	✓	✓		605.2.a.m	$2$	$0$	\(0\)	\(6\)	\(6\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(605))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(605)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)