Space of modular forms of level 3, weight 10, and trivial character

• Label: 3.10.a
• Level: $3$
• Weight: $10$
• Character orbit: 3.a
• Rep. character: $\chi_{3}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $3$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	3.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(3\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(3))\).

	Total	New	Old
Modular forms	4	2	2
Cusp forms	2	2	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.

\(3\)	Dim
\(+\)	\(1\)
\(-\)	\(1\)

Trace form

2 * q - 18 * q^2 + 596 * q^4 - 2844 * q^5 + 4374 * q^6 + 4648 * q^7 - 22392 * q^8 + 13122 * q^9 + 19764 * q^10 + 22608 * q^11 - 78732 * q^12 - 120308 * q^13 + 325584 * q^14 - 17496 * q^15 - 179440 * q^16 - 171180 * q^17 - 118098 * q^18 + 737104 * q^19 - 742536 * q^20 + 1102248 * q^21 + 327240 * q^22 - 2732976 * q^23 - 227448 * q^24 + 161246 * q^25 + 6016644 * q^26 - 5228384 * q^28 - 1950012 * q^29 - 6062364 * q^30 + 6701512 * q^31 + 10875168 * q^32 + 1592136 * q^33 - 8921988 * q^34 - 8079120 * q^35 + 3910356 * q^36 + 1874668 * q^37 - 18758664 * q^38 + 14801616 * q^39 + 32144688 * q^40 - 11576988 * q^41 + 244944 * q^42 - 27506192 * q^43 - 2815632 * q^44 - 18659484 * q^45 + 3158352 * q^46 + 51281136 * q^47 - 6613488 * q^48 + 22683570 * q^49 + 15134994 * q^50 - 31387824 * q^51 - 124661480 * q^52 + 94588884 * q^53 + 28697814 * q^54 - 34271424 * q^55 - 71144640 * q^56 - 36374184 * q^57 + 323595972 * q^58 - 106386624 * q^59 + 106743096 * q^60 - 147126020 * q^61 - 261185184 * q^62 + 30495528 * q^63 - 78156736 * q^64 + 151342488 * q^65 + 35114472 * q^66 + 374059840 * q^67 + 137315304 * q^68 - 64315296 * q^69 - 463307040 * q^70 - 80513136 * q^71 - 146913912 * q^72 - 270650156 * q^73 + 437254164 * q^74 + 49758624 * q^75 + 437902096 * q^76 + 186280416 * q^77 - 396328140 * q^78 - 209879960 * q^79 + 263981664 * q^80 + 86093442 * q^81 + 26522316 * q^82 - 1020687984 * q^83 + 145496736 * q^84 + 285268392 * q^85 - 259740792 * q^86 + 918137592 * q^87 - 280716192 * q^88 + 90873684 * q^89 + 129671604 * q^90 + 963739952 * q^91 - 428535072 * q^92 - 602614728 * q^93 - 605638944 * q^94 - 999662976 * q^95 - 216460512 * q^96 + 339473380 * q^97 + 1503597438 * q^98 + 148331088 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
3.10.a.a	$3$	$10$	3.a	1.a	$1$	$1$	$1$	$1.545$	\(\Q\)	None	✓	✓	✓	✓	3.10.a.a	$1$	$1$	\(-36\)	\(-81\)	\(-1314\)	\(-4480\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6^{2}q^{2}-3^{4}q^{3}+28^{2}q^{4}-1314q^{5}+\cdots\)
3.10.a.b	$3$	$10$	3.a	1.a	$1$	$1$	$1$	$1.545$	\(\Q\)	None	✓	✓	✓	✓	3.10.a.b	$1$	$0$	\(18\)	\(81\)	\(-1530\)	\(9128\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+18q^{2}+3^{4}q^{3}-188q^{4}-1530q^{5}+\cdots\)