Space of modular forms of level 127, weight 4, and trivial character

• Label: 127.4.a
• Level: $127$
• Weight: $4$
• Character orbit: 127.a
• Rep. character: $\chi_{127}(1,\cdot)$
• Character field: $\Q$
• Dimension: $31$
• Newform subspaces: $3$
• Sturm bound: $42$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 127 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	127.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(42\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(127))\).

	Total	New	Old
Modular forms	33	31	2
Cusp forms	31	31	0
Eisenstein series	2	0	2

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

\(127\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(19\)	\(18\)	\(1\)		\(18\)	\(18\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(14\)	\(13\)	\(1\)		\(13\)	\(13\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

31 * q - 2 * q^3 + 108 * q^4 + 8 * q^5 + 32 * q^6 + 4 * q^7 - 42 * q^8 + 231 * q^9 + 50 * q^10 + 26 * q^11 - 150 * q^12 + 58 * q^13 - 24 * q^14 + 56 * q^15 + 484 * q^16 + 62 * q^17 - 152 * q^18 - 190 * q^19 - 66 * q^20 - 12 * q^21 - 550 * q^22 - 16 * q^23 + 648 * q^24 + 893 * q^25 - 412 * q^26 - 44 * q^27 + 156 * q^28 + 264 * q^29 + 26 * q^30 - 242 * q^31 + 99 * q^32 + 52 * q^33 + 115 * q^34 - 640 * q^35 - 548 * q^36 - 174 * q^37 + 589 * q^38 - 564 * q^39 - 226 * q^40 - 18 * q^41 - 102 * q^42 - 514 * q^43 + 505 * q^44 - 1220 * q^45 + 766 * q^46 + 430 * q^47 - 1476 * q^48 + 1047 * q^49 - 1004 * q^50 - 172 * q^51 + 651 * q^52 - 564 * q^53 - 528 * q^54 + 892 * q^55 - 1076 * q^56 + 1732 * q^57 - 484 * q^58 - 158 * q^59 + 1444 * q^60 - 306 * q^61 + 879 * q^62 - 1080 * q^63 + 1854 * q^64 - 484 * q^65 - 674 * q^66 + 2366 * q^67 + 402 * q^68 - 3060 * q^69 - 3922 * q^70 - 470 * q^71 - 4264 * q^72 - 2042 * q^73 + 159 * q^74 + 614 * q^75 - 2952 * q^76 - 384 * q^77 + 1032 * q^78 + 66 * q^79 + 62 * q^80 + 4255 * q^81 - 1652 * q^82 + 1710 * q^83 - 1470 * q^84 + 480 * q^85 - 2130 * q^86 + 468 * q^87 - 6041 * q^88 + 2790 * q^89 + 7810 * q^90 - 1280 * q^91 + 5110 * q^92 + 476 * q^93 + 566 * q^94 - 504 * q^95 + 3468 * q^96 + 2418 * q^97 + 1894 * q^98 + 5310 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(127))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		127
127.4.a.a	$127$	$4$	127.a	1.a	$1$	$1$	$1$	$7.493$	\(\Q\)	None	✓	✓			127.4.a.a	$1$	$2$	\(-1\)	\(-8\)	\(-15\)	\(-25\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-8q^{3}-7q^{4}-15q^{5}+8q^{6}+\cdots\)
127.4.a.b	$127$	$4$	127.a	1.a	$1$	$13$	$13$	$7.493$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	127.4.a.b	$1$	$1$	\(-8\)	\(-16\)	\(-46\)	\(-26\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{8})q^{3}+(3+\cdots)q^{4}+\cdots\)
127.4.a.c	$127$	$4$	127.a	1.a	$1$	$17$	$17$	$7.493$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	✓	✓		127.4.a.c	$1$	$0$	\(9\)	\(22\)	\(69\)	\(55\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{7})q^{3}+(5+\beta _{2}+\cdots)q^{4}+\cdots\)