Space of modular forms of level 13, weight 10, and trivial character

• Label: 13.10.a
• Level: $13$
• Weight: $10$
• Character orbit: 13.a
• Rep. character: $\chi_{13}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $2$
• Sturm bound: $11$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	13.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(11\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(13))\).

	Total	New	Old
Modular forms	11	9	2
Cusp forms	9	9	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
\(+\)		\(5\)	\(4\)	\(1\)		\(4\)	\(4\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(6\)	\(5\)	\(1\)		\(5\)	\(5\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

9 * q - 18 * q^2 - 2 * q^3 + 1790 * q^4 + 2274 * q^5 + 1164 * q^6 - 1142 * q^7 - 22392 * q^8 + 31107 * q^9 + 16674 * q^10 + 81606 * q^11 - 42090 * q^12 + 28561 * q^13 - 269178 * q^14 + 189280 * q^15 + 404146 * q^16 - 416952 * q^17 + 1034798 * q^18 - 631074 * q^19 - 724764 * q^20 - 461036 * q^21 - 659504 * q^22 - 4849596 * q^23 + 1265916 * q^24 - 1229795 * q^25 + 1370928 * q^26 + 4770082 * q^27 + 6623136 * q^28 + 9030246 * q^29 + 4747058 * q^30 + 1519294 * q^31 - 26176032 * q^32 + 2865076 * q^33 + 16323580 * q^34 - 5409066 * q^35 - 32071016 * q^36 + 11808714 * q^37 - 396348 * q^38 + 9253764 * q^39 - 31822654 * q^40 + 23153550 * q^41 + 6044466 * q^42 - 20536106 * q^43 + 25010148 * q^44 + 8711270 * q^45 + 12049636 * q^46 + 39411210 * q^47 - 87495458 * q^48 + 48544411 * q^49 - 20488518 * q^50 - 2758570 * q^51 - 30503148 * q^52 - 77787546 * q^53 - 5963196 * q^54 - 21711692 * q^55 - 81953742 * q^56 + 42344352 * q^57 + 230985704 * q^58 + 183106794 * q^59 + 127806048 * q^60 - 55182422 * q^61 + 62256984 * q^62 - 572650678 * q^63 + 859981570 * q^64 + 38043252 * q^65 - 366129504 * q^66 - 410562606 * q^67 - 479443350 * q^68 - 33344204 * q^69 + 264198964 * q^70 + 464611446 * q^71 + 146804976 * q^72 + 341685262 * q^73 - 764952582 * q^74 - 219139040 * q^75 - 1280673160 * q^76 + 1011616572 * q^77 + 291950542 * q^78 - 1131272744 * q^79 + 391542492 * q^80 + 120090585 * q^81 - 705588852 * q^82 + 1617316434 * q^83 + 1356883180 * q^84 - 225900296 * q^85 + 2701696572 * q^86 - 1702056976 * q^87 - 2547791592 * q^88 - 1906094658 * q^89 + 3173815920 * q^90 + 609491740 * q^91 - 4199605332 * q^92 + 725481888 * q^93 + 2131980718 * q^94 + 168927792 * q^95 + 4102066876 * q^96 + 1052922690 * q^97 + 1991147070 * q^98 + 5148380898 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\) 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
13.10.a.a	$13$	$10$	13.a	1.a	$1$	$4$	$4$	$6.695$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	13.10.a.a	$1$	$1$	\(-33\)	\(-163\)	\(471\)	\(-11241\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-8-\beta _{1})q^{2}+(-41+2\beta _{1}-\beta _{3})q^{3}+\cdots\)
13.10.a.b	$13$	$10$	13.a	1.a	$1$	$5$	$5$	$6.695$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	13.10.a.b	$1$	$0$	\(15\)	\(161\)	\(1803\)	\(10099\)		$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(2^{5}+2\beta _{1}-\beta _{2})q^{3}+\cdots\)