Space of modular forms of level 399, weight 2, and trivial character

• Label: 399.2.a
• Level: $399$
• Weight: $2$
• Character orbit: 399.a
• Rep. character: $\chi_{399}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $7$
• Sturm bound: $106$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	399.a (trivial)
Character field:			$\Q$
Newform subspaces:			$7$
Sturm bound:			$106$
Trace bound:			$3$
Distinguishing $T_p$:			$2$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(399))$.

	Total	New	Old
Modular forms	56	19	37
Cusp forms	49	19	30
Eisenstein series	7	0	7

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

$3$	$7$	$19$	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$4$	$1$	$3$		$4$	$1$	$3$		$0$	$0$	$0$
$+$	$+$	$-$	$-$		$8$	$3$	$5$		$7$	$3$	$4$		$1$	$0$	$1$
$+$	$-$	$+$	$-$		$9$	$5$	$4$		$8$	$5$	$3$		$1$	$0$	$1$
$+$	$-$	$-$	$+$		$5$	$1$	$4$		$4$	$1$	$3$		$1$	$0$	$1$
$-$	$+$	$+$	$-$		$10$	$4$	$6$		$9$	$4$	$5$		$1$	$0$	$1$
$-$	$+$	$-$	$+$		$6$	$0$	$6$		$5$	$0$	$5$		$1$	$0$	$1$
$-$	$-$	$+$	$+$		$5$	$0$	$5$		$4$	$0$	$4$		$1$	$0$	$1$
$-$	$-$	$-$	$-$		$9$	$5$	$4$		$8$	$5$	$3$		$1$	$0$	$1$
Plus space			$+$		$20$	$2$	$18$		$17$	$2$	$15$		$3$	$0$	$3$
Minus space			$-$		$36$	$17$	$19$		$32$	$17$	$15$		$4$	$0$	$4$

Trace form

19 * q + 5 * q^2 - q^3 + 21 * q^4 + 10 * q^5 - 3 * q^6 + 3 * q^7 + 9 * q^8 + 19 * q^9 + 6 * q^10 + 4 * q^11 + 9 * q^12 + 10 * q^13 + q^14 - 6 * q^15 + 29 * q^16 + 22 * q^17 + 5 * q^18 - q^19 - 2 * q^20 - q^21 - 12 * q^22 - 8 * q^23 - 15 * q^24 + 29 * q^25 - 18 * q^26 - q^27 + 5 * q^28 + 18 * q^29 + 6 * q^30 - 7 * q^32 - 4 * q^33 - 14 * q^34 - 6 * q^35 + 21 * q^36 + 2 * q^37 - 3 * q^38 + 18 * q^39 - 18 * q^40 + 22 * q^41 + q^42 + 36 * q^43 - 4 * q^44 + 10 * q^45 - 16 * q^46 - 8 * q^47 + q^48 + 19 * q^49 - 37 * q^50 - 2 * q^51 - 10 * q^52 - 6 * q^53 - 3 * q^54 - 24 * q^55 + 21 * q^56 + 3 * q^57 - 50 * q^58 + 4 * q^59 - 42 * q^60 + 10 * q^61 - 56 * q^62 + 3 * q^63 + 29 * q^64 + 28 * q^65 + 4 * q^66 - 20 * q^67 + 2 * q^68 - 16 * q^69 - 18 * q^70 - 40 * q^71 + 9 * q^72 + 30 * q^73 - 18 * q^74 + 17 * q^75 - 7 * q^76 + 4 * q^77 - 34 * q^78 + 16 * q^79 - 98 * q^80 + 19 * q^81 - 70 * q^82 - 52 * q^83 - 7 * q^84 - 44 * q^85 - 68 * q^86 - 14 * q^87 - 108 * q^88 + 38 * q^89 + 6 * q^90 - 6 * q^91 - 88 * q^92 - 48 * q^94 - 6 * q^95 - 63 * q^96 - 26 * q^97 + 5 * q^98 + 4 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(399))$ 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	19
399.2.a.a	$399$	$2$	399.a	1.a	$1$	$1$	$1$	$3.186$	$\Q$	None	✓	✓	✓	✓	399.2.a.a	$1$	$1$	$-1$	$-1$	$0$	$1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}-q^{4}+q^{6}+q^{7}+3q^{8}+\cdots$
399.2.a.b	$399$	$2$	399.a	1.a	$1$	$1$	$1$	$3.186$	$\Q$	None	✓	✓			399.2.a.b	$1$	$0$	$-1$	$1$	$4$	$-1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}-q^{4}+4q^{5}-q^{6}-q^{7}+\cdots$
399.2.a.c	$399$	$2$	399.a	1.a	$1$	$1$	$1$	$3.186$	$\Q$	None	✓	✓	✓	✓	399.2.a.c	$1$	$1$	$1$	$-1$	$0$	$-1$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}-q^{4}-q^{6}-q^{7}-3q^{8}+\cdots$
399.2.a.d	$399$	$2$	399.a	1.a	$1$	$3$	$3$	$3.186$	3.3.148.1	None	✓	✓	✓	✓	399.2.a.d	$1$	$0$	$1$	$-3$	$4$	$-3$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots$
399.2.a.e	$399$	$2$	399.a	1.a	$1$	$3$	$3$	$3.186$	3.3.404.1	None	✓	✓	✓		399.2.a.e	$1$	$0$	$1$	$3$	$0$	$-3$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
399.2.a.f	$399$	$2$	399.a	1.a	$1$	$5$	$5$	$3.186$	5.5.1240016.1	None	✓	✓	✓	✓	399.2.a.f	$1$	$0$	$1$	$5$	$-2$	$5$		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{3}q^{2}+q^{3}+(1+\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots$
399.2.a.g	$399$	$2$	399.a	1.a	$1$	$5$	$5$	$3.186$	5.5.368464.1	None	✓	✓	✓	✓	399.2.a.g	$1$	$0$	$3$	$-5$	$4$	$5$		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(399))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(399)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(57))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(133))$$^{\oplus 2}$