Space of modular forms of level 50, weight 4, and trivial character

• Label: 50.4.a
• Level: $50$
• Weight: $4$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $30$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$50 = 2 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	50.a (trivial)
Character field:			$\Q$
Newform subspaces:			$5$
Sturm bound:			$30$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(50))$.

	Total	New	Old
Modular forms	29	5	24
Cusp forms	17	5	12
Eisenstein series	12	0	12

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

$2$	$5$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$8$	$2$	$6$		$5$	$2$	$3$		$3$	$0$	$3$
$+$	$-$	$-$		$6$	$1$	$5$		$3$	$1$	$2$		$3$	$0$	$3$
$-$	$+$	$-$		$7$	$0$	$7$		$4$	$0$	$4$		$3$	$0$	$3$
$-$	$-$	$+$		$8$	$2$	$6$		$5$	$2$	$3$		$3$	$0$	$3$
Plus space		$+$		$16$	$4$	$12$		$10$	$4$	$6$		$6$	$0$	$6$
Minus space		$-$		$13$	$1$	$12$		$7$	$1$	$6$		$6$	$0$	$6$

Trace form

5 * q - 2 * q^2 + 8 * q^3 + 20 * q^4 + 20 * q^6 + 4 * q^7 - 8 * q^8 + 35 * q^9 + 10 * q^11 + 32 * q^12 + 58 * q^13 - 40 * q^14 + 80 * q^16 - 66 * q^17 - 74 * q^18 - 150 * q^19 - 340 * q^21 - 24 * q^22 - 132 * q^23 + 80 * q^24 - 180 * q^26 + 80 * q^27 + 16 * q^28 - 150 * q^29 + 260 * q^31 - 32 * q^32 + 96 * q^33 - 40 * q^34 + 140 * q^36 + 34 * q^37 + 200 * q^38 + 120 * q^39 + 760 * q^41 - 64 * q^42 - 32 * q^43 + 40 * q^44 - 280 * q^46 + 204 * q^47 + 128 * q^48 + 1965 * q^49 - 490 * q^51 + 232 * q^52 - 222 * q^53 - 700 * q^54 - 160 * q^56 - 800 * q^57 + 180 * q^58 - 1300 * q^59 + 1510 * q^61 - 304 * q^62 + 148 * q^63 + 320 * q^64 + 340 * q^66 + 1024 * q^67 - 264 * q^68 - 2380 * q^69 - 3240 * q^71 - 296 * q^72 - 362 * q^73 + 1460 * q^74 - 600 * q^76 + 48 * q^77 - 928 * q^78 - 300 * q^79 - 1195 * q^81 + 876 * q^82 - 72 * q^83 - 1360 * q^84 + 920 * q^86 - 720 * q^87 - 96 * q^88 + 1300 * q^89 + 2760 * q^91 - 528 * q^92 + 1216 * q^93 + 160 * q^94 + 320 * q^96 - 1106 * q^97 + 654 * q^98 + 2920 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(50))$ 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	5
50.4.a.a	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	$\Q$	None	✓	✓			50.4.a.a	$1$	$0$	$-2$	$-7$	$0$	$34$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-7q^{3}+4q^{4}+14q^{6}+34q^{7}+\cdots$
50.4.a.b	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	$\Q$	None	✓		✓	✓	10.4.b.a	$1$	$1$	$-2$	$-2$	$0$	$-26$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-26q^{7}+\cdots$
50.4.a.c	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	$\Q$	None	✓				10.4.a.a	$1$	$0$	$-2$	$8$	$0$	$4$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+8q^{3}+4q^{4}-2^{4}q^{6}+4q^{7}+\cdots$
50.4.a.d	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	$\Q$	None	✓				10.4.b.a	$1$	$0$	$2$	$2$	$0$	$26$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+26q^{7}+\cdots$
50.4.a.e	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	$\Q$	None	✓	✓			50.4.a.a	$1$	$0$	$2$	$7$	$0$	$-34$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+7q^{3}+4q^{4}+14q^{6}-34q^{7}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(50))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(50)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(10))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(25))$$^{\oplus 2}$