Space of modular forms of level 62, weight 8, and trivial character

• Label: 62.8.a
• Level: $62$
• Weight: $8$
• Character orbit: 62.a
• Rep. character: $\chi_{62}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $64$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 62 = 2 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	62.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(64\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(62))\).

	Total	New	Old
Modular forms	58	18	40
Cusp forms	54	18	36
Eisenstein series	4	0	4

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

\(2\)	\(31\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(17\)	\(5\)	\(12\)		\(16\)	\(5\)	\(11\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(12\)	\(3\)	\(9\)		\(11\)	\(3\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(15\)	\(4\)	\(11\)		\(14\)	\(4\)	\(10\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(14\)	\(6\)	\(8\)		\(13\)	\(6\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(31\)	\(11\)	\(20\)		\(29\)	\(11\)	\(18\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(27\)	\(7\)	\(20\)		\(25\)	\(7\)	\(18\)		\(2\)	\(0\)	\(2\)

Trace form

18 * q + 16 * q^2 - 26 * q^3 + 1152 * q^4 + 136 * q^5 - 240 * q^6 - 788 * q^7 + 1024 * q^8 + 17266 * q^9 - 3776 * q^10 - 7482 * q^11 - 1664 * q^12 + 5838 * q^13 + 19872 * q^14 - 2356 * q^15 + 73728 * q^16 - 24648 * q^17 - 28144 * q^18 - 17992 * q^19 + 8704 * q^20 + 62420 * q^21 - 70512 * q^22 - 94528 * q^23 - 15360 * q^24 + 552918 * q^25 + 42128 * q^26 + 135136 * q^27 - 50432 * q^28 - 15886 * q^29 + 217248 * q^30 + 65536 * q^32 + 808416 * q^33 + 336064 * q^34 + 862608 * q^35 + 1105024 * q^36 - 207926 * q^37 - 148544 * q^38 - 230956 * q^39 - 241664 * q^40 + 357396 * q^41 - 637792 * q^42 + 2985666 * q^43 - 478848 * q^44 - 770928 * q^45 + 811648 * q^46 - 883628 * q^47 - 106496 * q^48 + 526178 * q^49 + 481520 * q^50 - 1713196 * q^51 + 373632 * q^52 - 1224402 * q^53 - 4467072 * q^54 - 1407220 * q^55 + 1271808 * q^56 + 7125984 * q^57 - 1956496 * q^58 - 2927028 * q^59 - 150784 * q^60 - 10170826 * q^61 + 953312 * q^62 + 105580 * q^63 + 4718592 * q^64 - 13874548 * q^65 - 5766784 * q^66 - 1399088 * q^67 - 1577472 * q^68 - 220944 * q^69 + 3232640 * q^70 + 5188496 * q^71 - 1801216 * q^72 - 3621812 * q^73 + 3571312 * q^74 + 4263258 * q^75 - 1151488 * q^76 - 1213236 * q^77 - 9115168 * q^78 + 4520540 * q^79 + 557056 * q^80 + 8871346 * q^81 + 4507360 * q^82 - 2012110 * q^83 + 3994880 * q^84 - 9320528 * q^85 - 13370320 * q^86 - 29558316 * q^87 - 4512768 * q^88 + 29421684 * q^89 + 5485952 * q^90 - 22581924 * q^91 - 6049792 * q^92 + 1608714 * q^93 + 3172832 * q^94 - 27040880 * q^95 - 983040 * q^96 - 6110564 * q^97 + 15467216 * q^98 + 28181314 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(62))\) 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	31
62.8.a.a	$62$	$8$	62.a	1.a	$1$	$3$	$3$	$19.368$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	62.8.a.a	$1$	$1$	\(-24\)	\(-26\)	\(152\)	\(-1504\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-9-\beta _{1})q^{3}+2^{6}q^{4}+(50+\cdots)q^{5}+\cdots\)
62.8.a.b	$62$	$8$	62.a	1.a	$1$	$4$	$4$	$19.368$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	62.8.a.b	$1$	$1$	\(32\)	\(-68\)	\(-334\)	\(-948\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-17+\beta _{1}+\beta _{3})q^{3}+2^{6}q^{4}+\cdots\)
62.8.a.c	$62$	$8$	62.a	1.a	$1$	$5$	$5$	$19.368$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	62.8.a.c	$1$	$0$	\(-40\)	\(28\)	\(152\)	\(-132\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(6-\beta _{3})q^{3}+2^{6}q^{4}+(33-11\beta _{1}+\cdots)q^{5}+\cdots\)
62.8.a.d	$62$	$8$	62.a	1.a	$1$	$6$	$6$	$19.368$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	62.8.a.d	$1$	$0$	\(48\)	\(40\)	\(166\)	\(1796\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(7-\beta _{1})q^{3}+2^{6}q^{4}+(29-3\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(62))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(62)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 2}\)