Space of modular forms of level 350, weight 8, and trivial character

• Label: 350.8.a
• Level: $350$
• Weight: $8$
• Character orbit: 350.a
• Rep. character: $\chi_{350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $66$
• Newform subspaces: $27$
• Sturm bound: $480$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	350.a (trivial)
Character field:			$\Q$
Newform subspaces:			$27$
Sturm bound:			$480$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{8}(\Gamma_0(350))$.

	Total	New	Old
Modular forms	432	66	366
Cusp forms	408	66	342
Eisenstein series	24	0	24

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

$2$	$5$	$7$	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$57$	$8$	$49$		$54$	$8$	$46$		$3$	$0$	$3$
$+$	$+$	$-$	$-$		$51$	$7$	$44$		$48$	$7$	$41$		$3$	$0$	$3$
$+$	$-$	$+$	$-$		$52$	$8$	$44$		$49$	$8$	$41$		$3$	$0$	$3$
$+$	$-$	$-$	$+$		$56$	$10$	$46$		$53$	$10$	$43$		$3$	$0$	$3$
$-$	$+$	$+$	$-$		$54$	$8$	$46$		$51$	$8$	$43$		$3$	$0$	$3$
$-$	$+$	$-$	$+$		$54$	$9$	$45$		$51$	$9$	$42$		$3$	$0$	$3$
$-$	$-$	$+$	$+$		$53$	$9$	$44$		$50$	$9$	$41$		$3$	$0$	$3$
$-$	$-$	$-$	$-$		$55$	$7$	$48$		$52$	$7$	$45$		$3$	$0$	$3$
Plus space			$+$		$220$	$36$	$184$		$208$	$36$	$172$		$12$	$0$	$12$
Minus space			$-$		$212$	$30$	$182$		$200$	$30$	$170$		$12$	$0$	$12$

Trace form

66 * q + 26 * q^3 + 4224 * q^4 + 688 * q^6 + 40074 * q^9 + 5520 * q^11 + 1664 * q^12 - 15410 * q^13 - 5488 * q^14 + 270336 * q^16 + 42064 * q^17 - 10528 * q^18 + 3558 * q^19 - 74774 * q^21 - 52128 * q^22 + 71520 * q^23 + 44032 * q^24 + 66000 * q^26 - 15508 * q^27 - 590776 * q^29 + 616388 * q^31 - 126000 * q^33 - 344000 * q^34 + 2564736 * q^36 + 334176 * q^37 + 370256 * q^38 - 1126088 * q^39 + 222612 * q^41 + 148176 * q^42 - 652556 * q^43 + 353280 * q^44 - 1255360 * q^46 + 262468 * q^47 + 106496 * q^48 + 7764834 * q^49 + 5522240 * q^51 - 986240 * q^52 - 1106692 * q^53 - 182528 * q^54 - 351232 * q^56 + 3297948 * q^57 + 2873824 * q^58 + 796682 * q^59 + 7993390 * q^61 - 1095264 * q^62 - 2841412 * q^63 + 17301504 * q^64 - 19030944 * q^66 + 4167872 * q^67 + 2692096 * q^68 + 19739040 * q^69 + 10063536 * q^71 - 673792 * q^72 + 5821708 * q^73 + 9733920 * q^74 + 227712 * q^76 + 5664988 * q^77 - 6097856 * q^78 - 5836864 * q^79 - 1543122 * q^81 - 6424352 * q^82 + 6363814 * q^83 - 4785536 * q^84 + 142816 * q^86 - 14454764 * q^87 - 3336192 * q^88 - 20703952 * q^89 + 10827138 * q^91 + 4577280 * q^92 - 13065544 * q^93 - 8504352 * q^94 + 2818048 * q^96 + 5287456 * q^97 + 26161484 * q^99

Decomposition of $S_{8}^{\mathrm{new}}(\Gamma_0(350))$ 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	7
350.8.a.a	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				70.8.a.b	$1$	$1$	$-8$	$-9$	$0$	$343$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}-9q^{3}+2^{6}q^{4}+72q^{6}+7^{3}q^{7}+\cdots$
350.8.a.b	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				70.8.c.b	$1$	$1$	$-8$	$3$	$0$	$-343$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+3q^{3}+2^{6}q^{4}-24q^{6}-7^{3}q^{7}+\cdots$
350.8.a.c	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				70.8.c.a	$1$	$1$	$-8$	$63$	$0$	$-343$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+63q^{3}+2^{6}q^{4}-504q^{6}+\cdots$
350.8.a.d	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				14.8.a.b	$1$	$1$	$-8$	$66$	$0$	$343$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+66q^{3}+2^{6}q^{4}-528q^{6}+\cdots$
350.8.a.e	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				70.8.a.a	$1$	$0$	$-8$	$93$	$0$	$-343$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+93q^{3}+2^{6}q^{4}-744q^{6}+\cdots$
350.8.a.f	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				70.8.c.a	$1$	$1$	$8$	$-63$	$0$	$343$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}-63q^{3}+2^{6}q^{4}-504q^{6}+\cdots$
350.8.a.g	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				70.8.c.b	$1$	$1$	$8$	$-3$	$0$	$343$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}-3q^{3}+2^{6}q^{4}-24q^{6}+7^{3}q^{7}+\cdots$
350.8.a.h	$350$	$8$	350.a	1.a	$1$	$1$	$1$	$109.335$	$\Q$	None	✓				14.8.a.a	$1$	$0$	$8$	$82$	$0$	$343$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+82q^{3}+2^{6}q^{4}+656q^{6}+\cdots$
350.8.a.i	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{214})$	None	✓				70.8.c.c	$1$	$1$	$-16$	$-80$	$0$	$-686$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+(-40+\beta )q^{3}+2^{6}q^{4}+(320+\cdots)q^{6}+\cdots$
350.8.a.j	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{1969})$	None	✓				14.8.a.c	$1$	$0$	$-16$	$-70$	$0$	$-686$		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-8q^{2}+(-35-\beta )q^{3}+2^{6}q^{4}+(280+\cdots)q^{6}+\cdots$
350.8.a.k	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{12121})$	None	✓				70.8.a.h	$1$	$1$	$-16$	$-29$	$0$	$686$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+(-14-\beta )q^{3}+2^{6}q^{4}+(112+\cdots)q^{6}+\cdots$
350.8.a.l	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{8761})$	None	✓				70.8.a.g	$1$	$0$	$-16$	$5$	$0$	$-686$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+(3-\beta )q^{3}+2^{6}q^{4}+(-24+\cdots)q^{6}+\cdots$
350.8.a.m	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{1401})$	None	✓				70.8.a.f	$1$	$0$	$16$	$-45$	$0$	$686$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+(-22-\beta )q^{3}+2^{6}q^{4}+(-176+\cdots)q^{6}+\cdots$
350.8.a.n	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{18481})$	None	✓				70.8.a.e	$1$	$1$	$16$	$-31$	$0$	$-686$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+(-15-\beta )q^{3}+2^{6}q^{4}+(-120+\cdots)q^{6}+\cdots$
350.8.a.o	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{11761})$	None	✓				70.8.a.d	$1$	$0$	$16$	$-25$	$0$	$686$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+(-12-\beta )q^{3}+2^{6}q^{4}+(-96+\cdots)q^{6}+\cdots$
350.8.a.p	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{9241})$	None	✓				70.8.a.c	$1$	$1$	$16$	$-11$	$0$	$-686$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+(-5-\beta )q^{3}+2^{6}q^{4}+(-40+\cdots)q^{6}+\cdots$
350.8.a.q	$350$	$8$	350.a	1.a	$1$	$2$	$2$	$109.335$	$\Q(\sqrt{214})$	None	✓				70.8.c.c	$1$	$1$	$16$	$80$	$0$	$686$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+(40+\beta )q^{3}+2^{6}q^{4}+(320+\cdots)q^{6}+\cdots$
350.8.a.r	$350$	$8$	350.a	1.a	$1$	$3$	$3$	$109.335$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓		350.8.a.r	$1$	$1$	$-24$	$-83$	$0$	$1029$		$+$	$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	$q-8q^{2}+(-28+\beta _{1})q^{3}+2^{6}q^{4}+(224+\cdots)q^{6}+\cdots$
350.8.a.s	$350$	$8$	350.a	1.a	$1$	$3$	$3$	$109.335$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓		350.8.a.s	$1$	$0$	$-24$	$53$	$0$	$-1029$		$+$	$+$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	$q-8q^{2}+(18+\beta _{1})q^{3}+2^{6}q^{4}+(-12^{2}+\cdots)q^{6}+\cdots$
350.8.a.t	$350$	$8$	350.a	1.a	$1$	$3$	$3$	$109.335$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓		350.8.a.s	$1$	$1$	$24$	$-53$	$0$	$1029$		$-$	$-$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	$q+8q^{2}+(-18-\beta _{1})q^{3}+2^{6}q^{4}+(-12^{2}+\cdots)q^{6}+\cdots$
350.8.a.u	$350$	$8$	350.a	1.a	$1$	$3$	$3$	$109.335$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓			350.8.a.r	$1$	$0$	$24$	$83$	$0$	$-1029$		$-$	$-$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	$q+8q^{2}+(28-\beta _{1})q^{3}+2^{6}q^{4}+(224+\cdots)q^{6}+\cdots$
350.8.a.v	$350$	$8$	350.a	1.a	$1$	$4$	$4$	$109.335$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓			350.8.a.v	$1$	$0$	$-32$	$-42$	$0$	$1372$		$+$	$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	$q-8q^{2}+(-10+\beta _{1})q^{3}+2^{6}q^{4}+(80+\cdots)q^{6}+\cdots$
350.8.a.w	$350$	$8$	350.a	1.a	$1$	$4$	$4$	$109.335$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓	✓		350.8.a.w	$1$	$1$	$-32$	$-14$	$0$	$-1372$		$+$	$-$	$+$	$-$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q-8q^{2}+(-4+\beta _{1})q^{3}+2^{6}q^{4}+(2^{5}+\cdots)q^{6}+\cdots$
350.8.a.x	$350$	$8$	350.a	1.a	$1$	$4$	$4$	$109.335$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓	✓		350.8.a.w	$1$	$0$	$32$	$14$	$0$	$1372$		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+8q^{2}+(4-\beta _{1})q^{3}+2^{6}q^{4}+(2^{5}-8\beta _{1}+\cdots)q^{6}+\cdots$
350.8.a.y	$350$	$8$	350.a	1.a	$1$	$4$	$4$	$109.335$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓	✓		350.8.a.v	$1$	$1$	$32$	$42$	$0$	$-1372$		$-$	$+$	$+$	$-$	$2^{2}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+8q^{2}+(10-\beta _{1})q^{3}+2^{6}q^{4}+(80+\cdots)q^{6}+\cdots$
350.8.a.z	$350$	$8$	350.a	1.a	$1$	$6$	$6$	$109.335$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓		✓		70.8.c.d	$1$	$0$	$-48$	$14$	$0$	$2058$		$+$	$-$	$-$	$+$	$2^{4}\cdot 3\cdot 5^{5}$	$\mathrm{SU}(2)$	$q-8q^{2}+(2+\beta _{1})q^{3}+2^{6}q^{4}+(-2^{4}+\cdots)q^{6}+\cdots$
350.8.a.ba	$350$	$8$	350.a	1.a	$1$	$6$	$6$	$109.335$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓		✓		70.8.c.d	$1$	$0$	$48$	$-14$	$0$	$-2058$		$-$	$-$	$+$	$+$	$2^{4}\cdot 3\cdot 5^{5}$	$\mathrm{SU}(2)$	$q+8q^{2}+(-2-\beta _{1})q^{3}+2^{6}q^{4}+(-2^{4}+\cdots)q^{6}+\cdots$

Decomposition of $S_{8}^{\mathrm{old}}(\Gamma_0(350))$ into lower level spaces

$S_{8}^{\mathrm{old}}(\Gamma_0(350)) \simeq$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))$$^{\oplus 6}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 8}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(7))$$^{\oplus 6}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(10))$$^{\oplus 4}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(14))$$^{\oplus 3}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(25))$$^{\oplus 4}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(35))$$^{\oplus 4}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(50))$$^{\oplus 2}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(70))$$^{\oplus 2}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(175))$$^{\oplus 2}$