Space of modular forms of level 35, weight 4, and trivial character

• Label: 35.4.a
• Level: $35$
• Weight: $4$
• Character orbit: 35.a
• Rep. character: $\chi_{35}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $3$
• Sturm bound: $16$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$35 = 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	35.a (trivial)
Character field:			$\Q$
Newform subspaces:			$3$
Sturm bound:			$16$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(35))$.

	Total	New	Old
Modular forms	14	6	8
Cusp forms	10	6	4
Eisenstein series	4	0	4

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

$5$	$7$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$5$	$2$	$3$		$4$	$2$	$2$		$1$	$0$	$1$
$+$	$-$	$-$		$3$	$1$	$2$		$2$	$1$	$1$		$1$	$0$	$1$
$-$	$+$	$-$		$2$	$0$	$2$		$1$	$0$	$1$		$1$	$0$	$1$
$-$	$-$	$+$		$4$	$3$	$1$		$3$	$3$	$0$		$1$	$0$	$1$
Plus space		$+$		$9$	$5$	$4$		$7$	$5$	$2$		$2$	$0$	$2$
Minus space		$-$		$5$	$1$	$4$		$3$	$1$	$2$		$2$	$0$	$2$

Trace form

6 * q + 6 * q^2 - 4 * q^3 + 26 * q^4 + 8 * q^6 + 14 * q^7 + 18 * q^8 + 130 * q^9 - 60 * q^10 - 76 * q^11 - 204 * q^12 + 16 * q^13 - 70 * q^14 + 40 * q^15 + 130 * q^16 - 196 * q^17 - 358 * q^18 + 244 * q^19 - 56 * q^21 + 12 * q^22 + 152 * q^23 + 44 * q^24 + 150 * q^25 - 308 * q^26 + 176 * q^27 - 98 * q^28 + 256 * q^29 + 200 * q^30 - 48 * q^31 + 650 * q^32 + 296 * q^33 + 464 * q^34 + 140 * q^35 - 214 * q^36 + 76 * q^37 - 100 * q^38 - 372 * q^39 - 240 * q^40 + 436 * q^41 + 168 * q^42 - 344 * q^43 - 1512 * q^44 + 160 * q^45 + 400 * q^46 - 544 * q^47 - 1900 * q^48 + 294 * q^49 + 150 * q^50 - 1180 * q^51 + 2376 * q^52 + 76 * q^53 + 388 * q^54 - 360 * q^55 - 546 * q^56 - 1944 * q^57 + 392 * q^58 - 1972 * q^59 - 500 * q^60 + 1280 * q^61 - 984 * q^62 + 742 * q^63 + 1570 * q^64 + 360 * q^65 + 1348 * q^66 - 112 * q^67 + 2484 * q^68 + 360 * q^69 + 140 * q^70 - 1368 * q^71 - 590 * q^72 + 1068 * q^73 + 2084 * q^74 - 100 * q^75 + 3284 * q^76 - 336 * q^77 + 3972 * q^78 - 1020 * q^79 - 1440 * q^80 + 470 * q^81 - 420 * q^82 - 2484 * q^83 + 84 * q^84 + 460 * q^85 - 72 * q^86 - 1584 * q^87 - 4888 * q^88 - 844 * q^89 - 2320 * q^90 - 588 * q^91 + 136 * q^92 + 6120 * q^93 - 2500 * q^94 + 460 * q^95 - 6908 * q^96 + 3868 * q^97 + 294 * q^98 - 2104 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(35))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	7
35.4.a.a	$35$	$4$	35.a	1.a	$1$	$1$	$1$	$2.065$	$\Q$	None	✓	✓	✓	✓	35.4.a.a	$1$	$1$	$1$	$-8$	$-5$	$7$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-8q^{3}-7q^{4}-5q^{5}-8q^{6}+\cdots$
35.4.a.b	$35$	$4$	35.a	1.a	$1$	$2$	$2$	$2.065$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	35.4.a.b	$1$	$0$	$8$	$2$	$-10$	$-14$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(4+\beta )q^{2}+(1-4\beta )q^{3}+(10+8\beta )q^{4}+\cdots$
35.4.a.c	$35$	$4$	35.a	1.a	$1$	$3$	$3$	$2.065$	3.3.14360.1	None	✓	✓	✓	✓	35.4.a.c	$1$	$0$	$-3$	$2$	$15$	$21$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(35))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(35)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$^{\oplus 2}$