Space of modular forms of level 6069, weight 2, and trivial character

• Label: 6069.2.a
• Level: $6069$
• Weight: $2$
• Character orbit: 6069.a
• Rep. character: $\chi_{6069}(1,\cdot)$
• Character field: $\Q$
• Dimension: $272$
• Newform subspaces: $39$
• Sturm bound: $1632$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$6069 = 3 \cdot 7 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6069.a (trivial)
Character field:			$\Q$
Newform subspaces:			$39$
Sturm bound:			$1632$
Trace bound:			$7$
Distinguishing $T_p$:			$2$, $5$, $11$, $23$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(6069))$.

	Total	New	Old
Modular forms	852	272	580
Cusp forms	781	272	509
Eisenstein series	71	0	71

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

$3$	$7$	$17$	Fricke	Dim
$+$	$+$	$+$	$+$	$33$
$+$	$+$	$-$	$-$	$36$
$+$	$-$	$+$	$-$	$31$
$+$	$-$	$-$	$+$	$36$
$-$	$+$	$+$	$-$	$40$
$-$	$+$	$-$	$+$	$28$
$-$	$-$	$+$	$+$	$24$
$-$	$-$	$-$	$-$	$44$
Plus space			$+$	$121$
Minus space			$-$	$151$

Trace form

272 * q + 276 * q^4 + 8 * q^5 + 4 * q^6 - 2 * q^7 + 272 * q^9 - 8 * q^12 + 16 * q^13 - 2 * q^14 + 8 * q^15 + 276 * q^16 + 16 * q^19 + 2 * q^21 - 32 * q^22 - 8 * q^23 + 12 * q^24 + 296 * q^25 - 16 * q^26 - 6 * q^28 + 8 * q^29 + 16 * q^31 + 8 * q^33 - 4 * q^35 + 276 * q^36 + 8 * q^37 + 32 * q^38 + 8 * q^40 + 16 * q^41 + 2 * q^42 - 8 * q^43 + 40 * q^44 + 8 * q^45 + 8 * q^46 - 8 * q^47 + 272 * q^49 - 8 * q^50 + 80 * q^52 - 24 * q^53 + 4 * q^54 - 8 * q^55 - 18 * q^56 + 8 * q^57 + 16 * q^58 - 8 * q^59 + 16 * q^60 + 24 * q^61 - 8 * q^62 - 2 * q^63 + 284 * q^64 + 32 * q^65 + 8 * q^66 + 32 * q^67 + 16 * q^69 - 4 * q^70 - 8 * q^71 + 16 * q^73 + 8 * q^74 - 32 * q^78 + 40 * q^80 + 272 * q^81 + 8 * q^82 - 32 * q^83 + 6 * q^84 + 8 * q^87 - 72 * q^88 + 40 * q^89 - 4 * q^91 + 24 * q^92 + 16 * q^93 + 32 * q^94 - 72 * q^95 + 28 * q^96 - 8 * q^97

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(6069))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	17
6069.2.a.a	$6069$	$2$	6069.a	1.a	$1$	$1$	$1$	$48.461$	$\Q$	None	✓				357.2.a.a	$1$	$1$	$-2$	$-1$	$3$	$-1$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-q^{3}+2q^{4}+3q^{5}+2q^{6}+\cdots$
6069.2.a.b	$6069$	$2$	6069.a	1.a	$1$	$1$	$1$	$48.461$	$\Q$	None	✓				21.2.a.a	$1$	$0$	$-1$	$-1$	$2$	$1$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+q^{7}+\cdots$
6069.2.a.c	$6069$	$2$	6069.a	1.a	$1$	$1$	$1$	$48.461$	$\Q$	None	✓				357.2.a.c	$1$	$0$	$0$	$1$	$-1$	$-1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-2q^{4}-q^{5}-q^{7}+q^{9}+5q^{11}+\cdots$
6069.2.a.d	$6069$	$2$	6069.a	1.a	$1$	$1$	$1$	$48.461$	$\Q$	None	✓				357.2.a.b	$1$	$1$	$0$	$1$	$-1$	$1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}-3q^{11}+\cdots$
6069.2.a.e	$6069$	$2$	6069.a	1.a	$1$	$1$	$1$	$48.461$	$\Q$	None	✓				357.2.a.d	$1$	$0$	$2$	$-1$	$-1$	$1$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-q^{3}+2q^{4}-q^{5}-2q^{6}+q^{7}+\cdots$
6069.2.a.f	$6069$	$2$	6069.a	1.a	$1$	$2$	$2$	$48.461$	$\Q(\sqrt{3})$	None	✓				357.2.a.e	$1$	$0$	$-2$	$-2$	$4$	$2$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots$
6069.2.a.g	$6069$	$2$	6069.a	1.a	$1$	$2$	$2$	$48.461$	$\Q(\sqrt{2})$	None	✓				357.2.a.f	$1$	$1$	$0$	$2$	$2$	$2$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{3}+(1+\beta )q^{5}+\beta q^{6}+q^{7}+\cdots$
6069.2.a.h	$6069$	$2$	6069.a	1.a	$1$	$2$	$2$	$48.461$	$\Q(\sqrt{13})$	None	✓	✓			6069.2.a.h	$1$	$1$	$1$	$-2$	$0$	$-2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}-q^{7}+\cdots$
6069.2.a.i	$6069$	$2$	6069.a	1.a	$1$	$2$	$2$	$48.461$	$\Q(\sqrt{13})$	None	✓	✓			6069.2.a.h	$1$	$0$	$1$	$2$	$0$	$2$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+q^{7}+\cdots$
6069.2.a.j	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	$\Q(\zeta_{14})^+$	None	✓	✓			6069.2.a.j	$1$	$1$	$-2$	$-3$	$-2$	$-3$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$
6069.2.a.k	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	3.3.148.1	None	✓				357.2.f.a	$1$	$1$	$-2$	$-3$	$3$	$-3$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}-q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots$
6069.2.a.l	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	$\Q(\zeta_{14})^+$	None	✓	✓			6069.2.a.j	$1$	$0$	$-2$	$3$	$2$	$3$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$
6069.2.a.m	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	3.3.148.1	None	✓				357.2.f.a	$1$	$1$	$-2$	$3$	$-3$	$3$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots$
6069.2.a.n	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	$\Q(\zeta_{18})^+$	None	✓	✓			6069.2.a.n	$1$	$0$	$0$	$-3$	$0$	$3$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots$
6069.2.a.o	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	3.3.257.1	None	✓	✓			6069.2.a.o	$1$	$0$	$0$	$-3$	$2$	$-3$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}-q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots$
6069.2.a.p	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	$\Q(\zeta_{18})^+$	None	✓	✓			6069.2.a.n	$1$	$1$	$0$	$3$	$0$	$-3$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots$
6069.2.a.q	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	3.3.257.1	None	✓	✓			6069.2.a.o	$1$	$1$	$0$	$3$	$-2$	$3$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots$
6069.2.a.r	$6069$	$2$	6069.a	1.a	$1$	$3$	$3$	$48.461$	3.3.316.1	None	✓				357.2.a.g	$1$	$1$	$1$	$-3$	$-2$	$-3$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$
6069.2.a.s	$6069$	$2$	6069.a	1.a	$1$	$4$	$4$	$48.461$	4.4.7232.1	None	✓				357.2.a.h	$1$	$0$	$2$	$4$	$2$	$-4$		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+q^{3}+(1-\beta _{2}-\beta _{3})q^{4}-\beta _{1}q^{5}+\cdots$
6069.2.a.t	$6069$	$2$	6069.a	1.a	$1$	$5$	$5$	$48.461$	5.5.1669781.1	None	✓	✓			6069.2.a.t	$1$	$1$	$-1$	$-5$	$-4$	$5$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$
6069.2.a.u	$6069$	$2$	6069.a	1.a	$1$	$5$	$5$	$48.461$	5.5.1669781.1	None	✓	✓			6069.2.a.t	$1$	$0$	$-1$	$5$	$4$	$-5$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots$
6069.2.a.v	$6069$	$2$	6069.a	1.a	$1$	$5$	$5$	$48.461$	5.5.1502576.1	None	✓				357.2.f.b	$1$	$0$	$0$	$-5$	$3$	$5$		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots$
6069.2.a.w	$6069$	$2$	6069.a	1.a	$1$	$5$	$5$	$48.461$	5.5.1502576.1	None	✓				357.2.f.b	$1$	$0$	$0$	$5$	$-3$	$-5$		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots$
6069.2.a.x	$6069$	$2$	6069.a	1.a	$1$	$6$	$6$	$48.461$	6.6.2803712.1	None	✓				357.2.k.a	$1$	$1$	$0$	$-6$	$6$	$-6$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$
6069.2.a.y	$6069$	$2$	6069.a	1.a	$1$	$6$	$6$	$48.461$	6.6.2803712.1	None	✓				357.2.k.a	$1$	$1$	$0$	$6$	$-6$	$6$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots$
6069.2.a.z	$6069$	$2$	6069.a	1.a	$1$	$9$	$9$	$48.461$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓			6069.2.a.z	$1$	$0$	$0$	$-9$	$3$	$-9$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1-\beta _{3}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$
6069.2.a.ba	$6069$	$2$	6069.a	1.a	$1$	$9$	$9$	$48.461$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓			6069.2.a.ba	$1$	$0$	$0$	$-9$	$3$	$9$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1-\beta _{4}+\beta _{5})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$
6069.2.a.bb	$6069$	$2$	6069.a	1.a	$1$	$9$	$9$	$48.461$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓			6069.2.a.ba	$1$	$1$	$0$	$9$	$-3$	$-9$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(1-\beta _{4}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
6069.2.a.bc	$6069$	$2$	6069.a	1.a	$1$	$9$	$9$	$48.461$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓	✓		6069.2.a.z	$1$	$1$	$0$	$9$	$-3$	$9$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(1-\beta _{3}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$
6069.2.a.bd	$6069$	$2$	6069.a	1.a	$1$	$10$	$10$	$48.461$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓		✓		357.2.k.b	$1$	$0$	$4$	$-10$	$6$	$10$		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots$
6069.2.a.be	$6069$	$2$	6069.a	1.a	$1$	$10$	$10$	$48.461$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓				357.2.k.b	$1$	$0$	$4$	$10$	$-6$	$-10$		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots$
6069.2.a.bf	$6069$	$2$	6069.a	1.a	$1$	$15$	$15$	$48.461$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓	✓		6069.2.a.bf	$1$	$1$	$0$	$-15$	$-9$	$-15$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{12}+\cdots)q^{5}+\cdots$
6069.2.a.bg	$6069$	$2$	6069.a	1.a	$1$	$15$	$15$	$48.461$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓			6069.2.a.bg	$1$	$1$	$0$	$-15$	$-9$	$15$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{13}+\cdots)q^{5}+\cdots$
6069.2.a.bh	$6069$	$2$	6069.a	1.a	$1$	$15$	$15$	$48.461$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓	✓		6069.2.a.bg	$1$	$0$	$0$	$15$	$9$	$-15$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{13}+\cdots)q^{5}+\cdots$
6069.2.a.bi	$6069$	$2$	6069.a	1.a	$1$	$15$	$15$	$48.461$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓			6069.2.a.bf	$1$	$0$	$0$	$15$	$9$	$15$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{12}+\cdots)q^{5}+\cdots$
6069.2.a.bj	$6069$	$2$	6069.a	1.a	$1$	$16$	$16$	$48.461$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	✓		✓		357.2.u.a	$1$	$1$	$-4$	$-16$	$-4$	$16$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{14}q^{5}+\cdots$
6069.2.a.bk	$6069$	$2$	6069.a	1.a	$1$	$16$	$16$	$48.461$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	✓		✓		357.2.u.a	$1$	$1$	$-4$	$16$	$4$	$-16$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{14}q^{5}+\cdots$
6069.2.a.bl	$6069$	$2$	6069.a	1.a	$1$	$24$	$24$	$48.461$		None	✓		✓		357.2.u.b	$1$	$0$	$4$	$-24$	$-4$	$-24$		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
6069.2.a.bm	$6069$	$2$	6069.a	1.a	$1$	$24$	$24$	$48.461$		None	✓		✓		357.2.u.b	$1$	$0$	$4$	$24$	$4$	$24$		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(6069))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(6069)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(51))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(119))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(289))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(357))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(867))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(2023))$$^{\oplus 2}$