Space of modular forms of level 2116, weight 4, and trivial character

• Label: 2116.4.a
• Level: $2116$
• Weight: $4$
• Character orbit: 2116.a
• Rep. character: $\chi_{2116}(1,\cdot)$
• Character field: $\Q$
• Dimension: $126$
• Newform subspaces: $10$
• Sturm bound: $1104$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$2116 = 2^{2} \cdot 23^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2116.a (trivial)
Character field:			$\Q$
Newform subspaces:			$10$
Sturm bound:			$1104$
Trace bound:			$5$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(2116))$.

	Total	New	Old
Modular forms	864	126	738
Cusp forms	792	126	666
Eisenstein series	72	0	72

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

$2$	$23$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$222$	$0$	$222$		$198$	$0$	$198$		$24$	$0$	$24$
$+$	$-$	$-$		$216$	$0$	$216$		$192$	$0$	$192$		$24$	$0$	$24$
$-$	$+$	$-$		$210$	$60$	$150$		$198$	$60$	$138$		$12$	$0$	$12$
$-$	$-$	$+$		$216$	$66$	$150$		$204$	$66$	$138$		$12$	$0$	$12$
Plus space		$+$		$438$	$66$	$372$		$402$	$66$	$336$		$36$	$0$	$36$
Minus space		$-$		$426$	$60$	$366$		$390$	$60$	$330$		$36$	$0$	$36$

Trace form

126 * q - 4 * q^3 + 10 * q^5 + 4 * q^7 + 1138 * q^9 + 58 * q^11 - 56 * q^13 + 128 * q^15 + 116 * q^17 - 26 * q^19 - 100 * q^21 + 3222 * q^25 - 424 * q^27 - 236 * q^29 + 172 * q^31 + 300 * q^33 + 60 * q^35 + 186 * q^37 - 288 * q^39 + 12 * q^41 + 498 * q^43 + 158 * q^45 + 596 * q^47 + 6006 * q^49 - 8 * q^51 - 234 * q^53 + 300 * q^55 + 16 * q^57 - 652 * q^59 + 714 * q^61 - 2052 * q^63 + 672 * q^65 + 1566 * q^67 - 1168 * q^71 + 332 * q^73 - 828 * q^75 + 728 * q^77 - 820 * q^79 + 9150 * q^81 + 950 * q^83 + 732 * q^85 - 480 * q^87 + 276 * q^89 - 696 * q^91 + 4044 * q^93 + 172 * q^95 - 2020 * q^97 + 4674 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(2116))$ 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	23
2116.4.a.a	$2116$	$4$	2116.a	1.a	$1$	$3$	$3$	$124.848$	3.3.1229.1	None	✓				92.4.a.a	$1$	$0$	$0$	$-4$	$10$	$46$		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(4+\beta _{1}-3\beta _{2})q^{5}+\cdots$
2116.4.a.b	$2116$	$4$	2116.a	1.a	$1$	$3$	$3$	$124.848$	3.3.28669.1	None	✓				92.4.a.b	$1$	$0$	$0$	$8$	$0$	$-42$		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+(3+\beta _{2})q^{3}-\beta _{1}q^{5}+(-14+\beta _{1}+\cdots)q^{7}+\cdots$
2116.4.a.c	$2116$	$4$	2116.a	1.a	$1$	$6$	$6$	$124.848$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	✓			2116.4.a.c	$1$	$0$	$0$	$1$	$-5$	$-20$		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{3})q^{5}+(-3+\cdots)q^{7}+\cdots$
2116.4.a.d	$2116$	$4$	2116.a	1.a	$1$	$6$	$6$	$124.848$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	✓			2116.4.a.c	$1$	$0$	$0$	$1$	$5$	$20$		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(1-\beta _{1}-\beta _{3})q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots$
2116.4.a.e	$2116$	$4$	2116.a	1.a	$1$	$6$	$6$	$124.848$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	✓			2116.4.a.e	$2$	$0$	$0$	$8$	$0$	$0$		$-$	$-$	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	$q+(1-\beta _{3})q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$
2116.4.a.f	$2116$	$4$	2116.a	1.a	$1$	$10$	$10$	$124.848$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓	✓			2116.4.a.f	$2$	$1$	$0$	$-12$	$0$	$0$		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots$
2116.4.a.g	$2116$	$4$	2116.a	1.a	$1$	$12$	$12$	$124.848$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓			2116.4.a.g	$2$	$0$	$0$	$-2$	$0$	$0$		$-$	$-$	$+$	$2^{14}\cdot 3^{4}$	$\mathrm{SU}(2)$	$q-\beta _{6}q^{3}+(\beta _{1}+\beta _{8})q^{5}+(-\beta _{1}+\beta _{7}+\cdots)q^{7}+\cdots$
2116.4.a.h	$2116$	$4$	2116.a	1.a	$1$	$20$	$20$	$124.848$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	✓	✓			2116.4.a.h	$2$	$1$	$0$	$0$	$0$	$0$		$-$	$+$	$-$	$2^{23}\cdot 3^{2}\cdot 23^{2}$	$\mathrm{SU}(2)$	$q+\beta _{5}q^{3}+\beta _{11}q^{5}+(\beta _{12}+\beta _{15})q^{7}+\cdots$
2116.4.a.i	$2116$	$4$	2116.a	1.a	$1$	$30$	$30$	$124.848$		None	✓		✓		92.4.e.a	$1$	$1$	$0$	$-2$	$-50$	$2$		$-$	$+$	$-$		$\mathrm{SU}(2)$
2116.4.a.j	$2116$	$4$	2116.a	1.a	$1$	$30$	$30$	$124.848$		None	✓		✓		92.4.e.a	$1$	$0$	$0$	$-2$	$50$	$-2$		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(2116))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(2116)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(46))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(92))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(529))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(1058))$$^{\oplus 2}$