Space of modular forms of level 13, weight 8, and trivial character

• Label: 13.8.a
• Level: $13$
• Weight: $8$
• Character orbit: 13.a
• Rep. character: $\chi_{13}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $3$
• Sturm bound: $9$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	13.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(9\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(13))\).

	Total	New	Old
Modular forms	9	7	2
Cusp forms	7	7	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
\(+\)		\(5\)	\(4\)	\(1\)		\(4\)	\(4\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(4\)	\(3\)	\(1\)		\(3\)	\(3\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

7 * q + 6 * q^2 + 52 * q^3 + 318 * q^4 - 390 * q^5 - 84 * q^6 + 1056 * q^7 - 1320 * q^8 + 4791 * q^9 - 2238 * q^10 - 7620 * q^11 + 10086 * q^12 - 2197 * q^13 + 13014 * q^14 - 21704 * q^15 - 27694 * q^16 - 17694 * q^17 + 23462 * q^18 + 46580 * q^19 - 73740 * q^20 - 70952 * q^21 + 105472 * q^22 + 128712 * q^23 + 53196 * q^24 + 154757 * q^25 - 52728 * q^26 + 33400 * q^27 - 273952 * q^28 + 34218 * q^29 - 150142 * q^30 - 316488 * q^31 - 250704 * q^32 + 169144 * q^33 + 56716 * q^34 + 255168 * q^35 + 100504 * q^36 + 684986 * q^37 - 1099260 * q^38 - 237276 * q^39 + 1315346 * q^40 + 843054 * q^41 - 10158 * q^42 + 5052 * q^43 - 590604 * q^44 - 2624614 * q^45 - 496316 * q^46 - 1610664 * q^47 + 1839262 * q^48 + 678811 * q^49 + 5871426 * q^50 - 3096904 * q^51 - 413036 * q^52 + 2453442 * q^53 + 1043412 * q^54 - 42296 * q^55 + 760050 * q^56 - 2563680 * q^57 - 4502440 * q^58 + 399996 * q^59 - 11818512 * q^60 + 5212914 * q^61 + 1682232 * q^62 + 10585472 * q^63 - 9723598 * q^64 - 1990482 * q^65 + 19895328 * q^66 - 6391324 * q^67 - 2959782 * q^68 + 4028200 * q^69 - 13184060 * q^70 - 4184352 * q^71 - 4021200 * q^72 - 17699298 * q^73 + 18282570 * q^74 + 16262716 * q^75 + 27841480 * q^76 - 14950008 * q^77 - 7122674 * q^78 + 18741312 * q^79 + 4873068 * q^80 - 10908753 * q^81 + 2920092 * q^82 - 13691388 * q^83 - 21686852 * q^84 - 6734372 * q^85 - 31170660 * q^86 + 30644360 * q^87 + 21699000 * q^88 + 33292110 * q^89 - 50053200 * q^90 - 5114616 * q^91 + 35518716 * q^92 - 9696336 * q^93 + 5824414 * q^94 + 1053576 * q^95 - 36116084 * q^96 - 25966114 * q^97 - 12157482 * q^98 - 30302100 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\) 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
13.8.a.a	$13$	$8$	13.a	1.a	$1$	$1$	$1$	$4.061$	\(\Q\)	None	✓	✓			13.8.a.a	$1$	$1$	\(10\)	\(-73\)	\(-295\)	\(1373\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{2}-73q^{3}-28q^{4}-295q^{5}+\cdots\)
13.8.a.b	$13$	$8$	13.a	1.a	$1$	$2$	$2$	$4.061$	\(\Q(\sqrt{337}) \)	None	✓	✓	✓		13.8.a.b	$1$	$1$	\(-19\)	\(45\)	\(-353\)	\(-2009\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-9-\beta )q^{2}+(21+3\beta )q^{3}+(37+19\beta )q^{4}+\cdots\)
13.8.a.c	$13$	$8$	13.a	1.a	$1$	$4$	$4$	$4.061$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	13.8.a.c	$1$	$0$	\(15\)	\(80\)	\(258\)	\(1692\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(21-2\beta _{1}+\beta _{2})q^{3}+\cdots\)