Space of modular forms of level 1098, weight 2, and trivial character

• Label: 1098.2.a
• Level: $1098$
• Weight: $2$
• Character orbit: 1098.a
• Rep. character: $\chi_{1098}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $17$
• Sturm bound: $372$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$1098 = 2 \cdot 3^{2} \cdot 61$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1098.a (trivial)
Character field:			$\Q$
Newform subspaces:			$17$
Sturm bound:			$372$
Trace bound:			$7$
Distinguishing $T_p$:			$5$, $7$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(1098))$.

	Total	New	Old
Modular forms	194	25	169
Cusp forms	179	25	154
Eisenstein series	15	0	15

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

$2$	$3$	$61$	Fricke	Dim
$+$	$+$	$+$	$+$	$4$
$+$	$+$	$-$	$-$	$1$
$+$	$-$	$+$	$-$	$4$
$+$	$-$	$-$	$+$	$3$
$-$	$+$	$+$	$-$	$4$
$-$	$+$	$-$	$+$	$1$
$-$	$-$	$+$	$+$	$2$
$-$	$-$	$-$	$-$	$6$
Plus space			$+$	$10$
Minus space			$-$	$15$

Trace form

25 * q + q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + q^8 + 2 * q^10 + 4 * q^11 + 4 * q^13 - 4 * q^14 + 25 * q^16 + 10 * q^17 - 6 * q^19 + 4 * q^20 + 6 * q^22 - 12 * q^23 + 25 * q^25 + 10 * q^26 + 4 * q^28 - 6 * q^29 - 12 * q^31 + q^32 + 10 * q^34 - 4 * q^35 - 14 * q^37 + 16 * q^38 + 2 * q^40 - 14 * q^41 - 8 * q^43 + 4 * q^44 + 8 * q^46 - 4 * q^47 + 25 * q^49 + 7 * q^50 + 4 * q^52 - 10 * q^53 - 16 * q^55 - 4 * q^56 + 28 * q^59 - 3 * q^61 + 14 * q^62 + 25 * q^64 + 6 * q^65 - 8 * q^67 + 10 * q^68 + 14 * q^70 + 28 * q^71 + 2 * q^73 + 8 * q^74 - 6 * q^76 + 30 * q^77 - 32 * q^79 + 4 * q^80 + 6 * q^82 - 6 * q^83 - 8 * q^85 - 12 * q^86 + 6 * q^88 - 10 * q^89 - 72 * q^91 - 12 * q^92 + 20 * q^94 - 12 * q^95 - 12 * q^97 + 17 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(1098))$ 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	3	61
1098.2.a.a	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.f	$1$	$1$	$-1$	$0$	$-1$	$-2$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots$
1098.2.a.b	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.g	$1$	$1$	$-1$	$0$	$-1$	$1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots$
1098.2.a.c	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.e	$1$	$0$	$-1$	$0$	$1$	$2$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots$
1098.2.a.d	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓	✓			1098.2.a.d	$1$	$1$	$-1$	$0$	$1$	$4$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots$
1098.2.a.e	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.d	$1$	$1$	$-1$	$0$	$3$	$-3$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+3q^{5}-3q^{7}-q^{8}-3q^{10}+\cdots$
1098.2.a.f	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓	✓	✓	✓	1098.2.a.f	$1$	$0$	$-1$	$0$	$3$	$0$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+3q^{5}-q^{8}-3q^{10}+2q^{11}+\cdots$
1098.2.a.g	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓	✓	✓	✓	1098.2.a.f	$1$	$1$	$1$	$0$	$-3$	$0$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-3q^{5}+q^{8}-3q^{10}-2q^{11}+\cdots$
1098.2.a.h	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				122.2.a.a	$1$	$1$	$1$	$0$	$-1$	$-5$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}-5q^{7}+q^{8}-q^{10}+\cdots$
1098.2.a.i	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.c	$1$	$1$	$1$	$0$	$-1$	$-2$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots$
1098.2.a.j	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓	✓			1098.2.a.d	$1$	$0$	$1$	$0$	$-1$	$4$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-q^{10}+\cdots$
1098.2.a.k	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.a	$1$	$0$	$1$	$0$	$2$	$4$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{5}+4q^{7}+q^{8}+2q^{10}+\cdots$
1098.2.a.l	$1098$	$2$	1098.a	1.a	$1$	$1$	$1$	$8.768$	$\Q$	None	✓				366.2.a.b	$1$	$0$	$1$	$0$	$3$	$-1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+3q^{5}-q^{7}+q^{8}+3q^{10}+\cdots$
1098.2.a.m	$1098$	$2$	1098.a	1.a	$1$	$2$	$2$	$8.768$	$\Q(\sqrt{17})$	None	✓				366.2.a.h	$1$	$0$	$2$	$0$	$0$	$-3$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1-2\beta )q^{5}+(-2+\beta )q^{7}+\cdots$
1098.2.a.n	$1098$	$2$	1098.a	1.a	$1$	$2$	$2$	$8.768$	$\Q(\sqrt{13})$	None	✓				122.2.a.b	$1$	$0$	$2$	$0$	$0$	$5$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(2+\beta )q^{7}+q^{8}-2\beta q^{11}+\cdots$
1098.2.a.o	$1098$	$2$	1098.a	1.a	$1$	$3$	$3$	$8.768$	3.3.892.1	None	✓	✓	✓		1098.2.a.o	$1$	$1$	$-3$	$0$	$-4$	$-2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1-\beta _{1})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots$
1098.2.a.p	$1098$	$2$	1098.a	1.a	$1$	$3$	$3$	$8.768$	3.3.229.1	None	✓		✓		122.2.a.c	$1$	$0$	$-3$	$0$	$-1$	$4$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(\beta _{1}+\beta _{2})q^{5}+(1+2\beta _{1}+\cdots)q^{7}+\cdots$
1098.2.a.q	$1098$	$2$	1098.a	1.a	$1$	$3$	$3$	$8.768$	3.3.892.1	None	✓	✓	✓		1098.2.a.o	$1$	$0$	$3$	$0$	$4$	$-2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1+\beta _{1})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(1098))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(1098)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(122))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(183))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(366))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(549))$$^{\oplus 2}$