Space of modular forms of level 35, weight 10, and trivial character

• Label: 35.10.a
• Level: $35$
• Weight: $10$
• Character orbit: 35.a
• Rep. character: $\chi_{35}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $5$
• Sturm bound: $40$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$35 = 5 \cdot 7$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	35.a (trivial)
Character field:			$\Q$
Newform subspaces:			$5$
Sturm bound:			$40$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{10}(\Gamma_0(35))$.

	Total	New	Old
Modular forms	38	18	20
Cusp forms	34	18	16
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
$+$	$+$	$+$		$8$	$4$	$4$		$7$	$4$	$3$		$1$	$0$	$1$
$+$	$-$	$-$		$10$	$5$	$5$		$9$	$5$	$4$		$1$	$0$	$1$
$-$	$+$	$-$		$11$	$6$	$5$		$10$	$6$	$4$		$1$	$0$	$1$
$-$	$-$	$+$		$9$	$3$	$6$		$8$	$3$	$5$		$1$	$0$	$1$
Plus space		$+$		$17$	$7$	$10$		$15$	$7$	$8$		$2$	$0$	$2$
Minus space		$-$		$21$	$11$	$10$		$19$	$11$	$8$		$2$	$0$	$2$

Trace form

18 * q + 2 * q^2 - 292 * q^3 + 5122 * q^4 + 4304 * q^6 - 4802 * q^7 - 8766 * q^8 + 200470 * q^9 + 22500 * q^10 + 37972 * q^11 - 426588 * q^12 + 94448 * q^13 + 24010 * q^14 - 335000 * q^15 + 1693970 * q^16 - 1264652 * q^17 + 2094806 * q^18 + 1823828 * q^19 - 19208 * q^21 - 659076 * q^22 - 1968584 * q^23 + 2183948 * q^24 + 7031250 * q^25 + 21507516 * q^26 - 2035408 * q^27 - 10453954 * q^28 + 6445312 * q^29 + 3050000 * q^30 - 10915824 * q^31 - 28934550 * q^32 + 45734648 * q^33 - 5360792 * q^34 - 6002500 * q^35 + 92406098 * q^36 + 2690372 * q^37 - 18091900 * q^38 - 73514484 * q^39 + 50310000 * q^40 + 76904348 * q^41 - 12446784 * q^42 - 21190792 * q^43 - 95101704 * q^44 + 33820000 * q^45 - 88935040 * q^46 + 5509312 * q^47 - 163653100 * q^48 + 103766418 * q^49 + 781250 * q^50 - 132500620 * q^51 - 50849208 * q^52 - 120104252 * q^53 - 46141964 * q^54 - 92205000 * q^55 + 19548942 * q^56 - 168899208 * q^57 + 576773296 * q^58 + 9290476 * q^59 - 397512500 * q^60 - 86369600 * q^61 - 537252648 * q^62 - 77970074 * q^63 + 674436530 * q^64 - 48065000 * q^65 - 1327317596 * q^66 + 497623456 * q^67 - 397539388 * q^68 + 593805240 * q^69 - 48020000 * q^70 + 73818616 * q^71 + 1737012370 * q^72 - 162943356 * q^73 + 302347548 * q^74 - 114062500 * q^75 - 1877635388 * q^76 - 75180112 * q^77 - 4816044 * q^78 + 279703380 * q^79 + 1636780000 * q^80 + 2109004130 * q^81 + 640444260 * q^82 - 332763892 * q^83 + 1378337268 * q^84 - 382997500 * q^85 + 2463754824 * q^86 + 3805847952 * q^87 + 2168058136 * q^88 - 3579693828 * q^89 + 244730000 * q^90 + 86522436 * q^91 - 3018039768 * q^92 - 1624114440 * q^93 - 3851452820 * q^94 - 1305242500 * q^95 - 805565564 * q^96 + 59279636 * q^97 + 11529602 * q^98 + 4488271832 * q^99

Decomposition of $S_{10}^{\mathrm{new}}(\Gamma_0(35))$ 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
35.10.a.a	$35$	$10$	35.a	1.a	$1$	$1$	$1$	$18.026$	$\Q$	None	✓	✓			35.10.a.a	$1$	$1$	$28$	$-116$	$625$	$2401$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+28q^{2}-116q^{3}+272q^{4}+5^{4}q^{5}+\cdots$
35.10.a.b	$35$	$10$	35.a	1.a	$1$	$2$	$2$	$18.026$	$\Q(\sqrt{2})$	None	✓	✓	✓		35.10.a.b	$1$	$1$	$-24$	$-174$	$1250$	$4802$		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-12+\beta )q^{2}+(-87+54\beta )q^{3}+\cdots$
35.10.a.c	$35$	$10$	35.a	1.a	$1$	$4$	$4$	$18.026$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓	✓	✓	35.10.a.c	$1$	$1$	$-19$	$-18$	$-2500$	$-9604$		$+$	$+$	$+$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	$q+(-5-\beta _{1})q^{2}+(-4+\beta _{2})q^{3}+(435+\cdots)q^{4}+\cdots$
35.10.a.d	$35$	$10$	35.a	1.a	$1$	$5$	$5$	$18.026$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	✓	✓	✓	35.10.a.d	$1$	$0$	$2$	$140$	$-3125$	$12005$		$+$	$-$	$-$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(28+\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots$
35.10.a.e	$35$	$10$	35.a	1.a	$1$	$6$	$6$	$18.026$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	✓	✓	✓	35.10.a.e	$1$	$0$	$15$	$-124$	$3750$	$-14406$		$-$	$+$	$-$	$2^{3}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	$q+(3-\beta _{1})q^{2}+(-20-\beta _{1}+\beta _{2})q^{3}+\cdots$

Decomposition of $S_{10}^{\mathrm{old}}(\Gamma_0(35))$ into lower level spaces

$S_{10}^{\mathrm{old}}(\Gamma_0(35)) \simeq$ $S_{10}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 2}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(7))$$^{\oplus 2}$