Space of modular forms of level 13, weight 4, and trivial character

• Label: 13.4.a
• Level: $13$
• Weight: $4$
• Character orbit: 13.a
• Rep. character: $\chi_{13}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $4$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(13))\).

	Total	New	Old
Modular forms	5	3	2
Cusp forms	3	3	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.

\(13\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

3 * q - 4 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 + 12 * q^6 - 22 * q^7 - 48 * q^8 + 57 * q^9 + 42 * q^10 + 54 * q^11 - 162 * q^12 - 13 * q^13 + 154 * q^14 + 16 * q^15 + 50 * q^16 + 96 * q^17 - 220 * q^18 - 210 * q^19 - 100 * q^20 - 212 * q^21 + 272 * q^22 + 100 * q^23 + 588 * q^24 - 313 * q^25 - 78 * q^26 + 370 * q^27 - 96 * q^28 - 126 * q^29 - 202 * q^30 + 110 * q^31 - 208 * q^32 + 76 * q^33 - 520 * q^34 + 198 * q^35 + 124 * q^36 + 78 * q^37 + 316 * q^38 - 156 * q^39 + 226 * q^40 + 106 * q^41 - 258 * q^42 + 86 * q^43 - 620 * q^44 - 334 * q^45 + 476 * q^46 + 330 * q^47 - 338 * q^48 + 209 * q^49 + 236 * q^50 - 58 * q^51 + 312 * q^52 - 550 * q^53 - 84 * q^54 + 164 * q^55 - 430 * q^56 + 1488 * q^57 + 1204 * q^58 - 662 * q^59 + 1008 * q^60 - 1114 * q^61 - 1312 * q^62 - 1846 * q^63 + 482 * q^64 - 52 * q^65 - 1728 * q^66 + 546 * q^67 + 1098 * q^68 + 1468 * q^69 - 580 * q^70 - 122 * q^71 + 360 * q^72 + 554 * q^73 + 802 * q^74 + 16 * q^75 - 2120 * q^76 + 1100 * q^77 + 754 * q^78 + 296 * q^79 - 692 * q^80 - 717 * q^81 - 1608 * q^82 + 1650 * q^83 + 2956 * q^84 - 712 * q^85 + 2364 * q^86 - 1984 * q^87 - 72 * q^88 - 1910 * q^89 + 1020 * q^90 - 52 * q^91 - 2420 * q^92 - 720 * q^93 - 286 * q^94 + 736 * q^95 - 1268 * q^96 - 858 * q^97 + 220 * q^98 - 702 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13
13.4.a.a	$13$	$4$	13.a	1.a	$1$	$1$	$1$	$0.767$	\(\Q\)	None	✓	✓	✓	✓	13.4.a.a	$1$	$1$	\(-5\)	\(-7\)	\(-7\)	\(-13\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-7q^{3}+17q^{4}-7q^{5}+35q^{6}+\cdots\)
13.4.a.b	$13$	$4$	13.a	1.a	$1$	$2$	$2$	$0.767$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	13.4.a.b	$1$	$0$	\(1\)	\(5\)	\(-3\)	\(-9\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)