Space of modular forms of level 74, weight 8, and trivial character

• Label: 74.8.a
• Level: $74$
• Weight: $8$
• Character orbit: 74.a
• Rep. character: $\chi_{74}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $4$
• Sturm bound: $76$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	74.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(76\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(74))\).

	Total	New	Old
Modular forms	69	21	48
Cusp forms	65	21	44
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.

\(2\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(13\)
Minus space		\(-\)	\(8\)

Trace form

21 * q + 8 * q^2 - 26 * q^3 + 1344 * q^4 + 246 * q^5 + 192 * q^6 - 2136 * q^7 + 512 * q^8 + 17995 * q^9 + 416 * q^10 - 1818 * q^11 - 1664 * q^12 + 4798 * q^13 + 25248 * q^14 + 18920 * q^15 + 86016 * q^16 - 21690 * q^17 - 61848 * q^18 - 16704 * q^19 + 15744 * q^20 - 19908 * q^21 + 11648 * q^22 + 87476 * q^23 + 12288 * q^24 + 385581 * q^25 - 125472 * q^26 - 50180 * q^27 - 136704 * q^28 + 285654 * q^29 + 310080 * q^30 + 111556 * q^31 + 32768 * q^32 + 901312 * q^33 + 177456 * q^34 + 513116 * q^35 + 1151680 * q^36 + 50653 * q^37 - 700992 * q^38 - 496796 * q^39 + 26624 * q^40 - 163120 * q^41 + 896128 * q^42 + 3933272 * q^43 - 116352 * q^44 + 2387110 * q^45 - 22704 * q^46 - 301148 * q^47 - 106496 * q^48 + 835101 * q^49 + 2179128 * q^50 - 2368592 * q^51 + 307072 * q^52 + 2381818 * q^53 - 490944 * q^54 - 1834924 * q^55 + 1615872 * q^56 + 835852 * q^57 - 74848 * q^58 - 1989660 * q^59 + 1210880 * q^60 - 5609946 * q^61 + 2264304 * q^62 - 3434540 * q^63 + 5505024 * q^64 + 1519280 * q^65 - 4625088 * q^66 + 2890918 * q^67 - 1388160 * q^68 - 10217032 * q^69 - 11807200 * q^70 + 1851912 * q^71 - 3958272 * q^72 + 8415164 * q^73 + 2026120 * q^74 - 8186600 * q^75 - 1069056 * q^76 - 31574388 * q^77 + 4882128 * q^78 + 1414636 * q^79 + 1007616 * q^80 + 8717021 * q^81 - 8778608 * q^82 - 23694852 * q^83 - 1274112 * q^84 - 8432844 * q^85 - 9178208 * q^86 + 8689068 * q^87 + 745472 * q^88 + 43803746 * q^89 + 6211280 * q^90 - 30395504 * q^91 + 5598464 * q^92 - 8919288 * q^93 + 6820992 * q^94 - 22573244 * q^95 + 786432 * q^96 - 3178498 * q^97 - 7048504 * q^98 + 8248312 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(74))\) 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	37
74.8.a.a	$74$	$8$	74.a	1.a	$1$	$4$	$4$	$23.116$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	74.8.a.a	$1$	$1$	\(-32\)	\(-53\)	\(111\)	\(-1666\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-14+2\beta _{1}+\beta _{3})q^{3}+2^{6}q^{4}+\cdots\)
74.8.a.b	$74$	$8$	74.a	1.a	$1$	$4$	$4$	$23.116$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	74.8.a.b	$1$	$1$	\(32\)	\(-41\)	\(-363\)	\(-774\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-10+\beta _{2})q^{3}+2^{6}q^{4}+(-89+\cdots)q^{5}+\cdots\)
74.8.a.c	$74$	$8$	74.a	1.a	$1$	$6$	$6$	$23.116$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	74.8.a.c	$1$	$0$	\(-48\)	\(28\)	\(-14\)	\(-980\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(5-\beta _{1})q^{3}+2^{6}q^{4}+(-4+\cdots)q^{5}+\cdots\)
74.8.a.d	$74$	$8$	74.a	1.a	$1$	$7$	$7$	$23.116$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	74.8.a.d	$1$	$0$	\(56\)	\(40\)	\(512\)	\(1284\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(6-\beta _{1})q^{3}+2^{6}q^{4}+(74-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(74))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(74)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)