Space of modular forms of level 116, weight 4, and trivial character

• Label: 116.4.a
• Level: $116$
• Weight: $4$
• Character orbit: 116.a
• Rep. character: $\chi_{116}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $3$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$116 = 2^{2} \cdot 29$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	116.a (trivial)
Character field:			$\Q$
Newform subspaces:			$3$
Sturm bound:			$60$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(116))$.

	Total	New	Old
Modular forms	48	7	41
Cusp forms	42	7	35
Eisenstein series	6	0	6

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

$2$	$29$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$14$	$0$	$14$		$12$	$0$	$12$		$2$	$0$	$2$
$+$	$-$	$-$		$11$	$0$	$11$		$9$	$0$	$9$		$2$	$0$	$2$
$-$	$+$	$-$		$10$	$2$	$8$		$9$	$2$	$7$		$1$	$0$	$1$
$-$	$-$	$+$		$13$	$5$	$8$		$12$	$5$	$7$		$1$	$0$	$1$
Plus space		$+$		$27$	$5$	$22$		$24$	$5$	$19$		$3$	$0$	$3$
Minus space		$-$		$21$	$2$	$19$		$18$	$2$	$16$		$3$	$0$	$3$

Trace form

7 * q - 12 * q^7 + 105 * q^9 + 8 * q^11 + 60 * q^13 + 152 * q^15 + 110 * q^17 - 4 * q^19 + 88 * q^21 + 92 * q^23 + 215 * q^25 + 36 * q^27 + 87 * q^29 + 288 * q^31 - 366 * q^33 - 476 * q^35 - 98 * q^37 - 1016 * q^39 + 410 * q^41 - 856 * q^43 + 498 * q^45 + 280 * q^47 + 487 * q^49 - 772 * q^51 - 688 * q^53 - 384 * q^55 - 624 * q^57 - 64 * q^59 + 234 * q^61 - 1696 * q^63 - 1346 * q^65 + 396 * q^67 + 164 * q^69 - 632 * q^71 + 1762 * q^73 - 348 * q^75 + 552 * q^77 + 1576 * q^79 + 1503 * q^81 - 1264 * q^83 + 504 * q^85 + 1738 * q^89 + 1244 * q^91 + 2134 * q^93 - 1296 * q^95 - 422 * q^97 - 132 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(116))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	29
116.4.a.a	$116$	$4$	116.a	1.a	$1$	$2$	$2$	$6.844$	$\Q(\sqrt{13})$	None	✓	✓	✓	✓	116.4.a.a	$1$	$1$	$0$	$0$	$-10$	$-20$		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta q^{3}+(-5+2\beta )q^{5}+(-10+4\beta )q^{7}+\cdots$
116.4.a.b	$116$	$4$	116.a	1.a	$1$	$2$	$2$	$6.844$	$\Q(\sqrt{22})$	None	✓	✓			116.4.a.b	$1$	$0$	$0$	$10$	$30$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(5+\beta )q^{3}+15q^{5}-2\beta q^{7}+(20+10\beta )q^{9}+\cdots$
116.4.a.c	$116$	$4$	116.a	1.a	$1$	$3$	$3$	$6.844$	3.3.148344.1	None	✓	✓	✓		116.4.a.c	$1$	$0$	$0$	$-10$	$-20$	$8$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-3-\beta _{1})q^{3}+(-7+\beta _{2})q^{5}+(4+\cdots)q^{7}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(116))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(116)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(29))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(58))$$^{\oplus 2}$