Space of modular forms of level 350, weight 10, and trivial character

• Label: 350.10.a
• Level: $350$
• Weight: $10$
• Character orbit: 350.a
• Rep. character: $\chi_{350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $86$
• Newform subspaces: $24$
• Sturm bound: $600$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(600\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(350))\).

	Total	New	Old
Modular forms	552	86	466
Cusp forms	528	86	442
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
\(+\)	\(+\)	\(+\)	\(+\)		\(66\)	\(10\)	\(56\)		\(63\)	\(10\)	\(53\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(72\)	\(11\)	\(61\)		\(69\)	\(11\)	\(58\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(71\)	\(12\)	\(59\)		\(68\)	\(12\)	\(56\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(67\)	\(10\)	\(57\)		\(64\)	\(10\)	\(54\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(69\)	\(10\)	\(59\)		\(66\)	\(10\)	\(56\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(69\)	\(9\)	\(60\)		\(66\)	\(9\)	\(57\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(70\)	\(11\)	\(59\)		\(67\)	\(11\)	\(56\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(68\)	\(13\)	\(55\)		\(65\)	\(13\)	\(52\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(272\)	\(40\)	\(232\)		\(260\)	\(40\)	\(220\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(280\)	\(46\)	\(234\)		\(268\)	\(46\)	\(222\)		\(12\)	\(0\)	\(12\)

Trace form

86 * q - 442 * q^3 + 22016 * q^4 + 3040 * q^6 + 552678 * q^9 + 36528 * q^11 - 113152 * q^12 + 217662 * q^13 + 76832 * q^14 + 5636096 * q^16 - 432680 * q^17 + 19520 * q^18 + 1848578 * q^19 + 427378 * q^21 + 247744 * q^22 + 1249824 * q^23 + 778240 * q^24 + 5466336 * q^26 - 15416740 * q^27 + 12648992 * q^29 - 9823676 * q^31 + 10689000 * q^33 - 37304448 * q^34 + 141485568 * q^36 + 47012080 * q^37 - 31595488 * q^38 - 32233568 * q^39 - 51910140 * q^41 - 6223392 * q^42 - 43983644 * q^43 + 9351168 * q^44 + 92511360 * q^46 - 105407996 * q^47 - 28966912 * q^48 + 495772886 * q^49 - 363754504 * q^51 + 55721472 * q^52 + 88933796 * q^53 + 331929088 * q^54 + 19668992 * q^56 - 93824748 * q^57 - 128310464 * q^58 + 687082598 * q^59 - 232329722 * q^61 + 154825536 * q^62 - 48356140 * q^63 + 1442840576 * q^64 + 203034432 * q^66 - 283265000 * q^67 - 110766080 * q^68 - 958235160 * q^69 + 48422040 * q^71 + 4997120 * q^72 + 735090764 * q^73 + 300447936 * q^74 + 473235968 * q^76 - 401917796 * q^77 - 1710429056 * q^78 - 763111392 * q^79 + 5567528802 * q^81 + 1710835776 * q^82 + 160701298 * q^83 + 109408768 * q^84 - 388510016 * q^86 + 3205091476 * q^87 + 63422464 * q^88 + 1438826432 * q^89 - 893772250 * q^91 + 319954944 * q^92 + 2031921632 * q^93 - 1704789056 * q^94 + 199229440 * q^96 - 3140619112 * q^97 - 2399054932 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\) 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
350.10.a.a	$350$	$10$	350.a	1.a	$1$	$1$	$1$	$180.263$	\(\Q\)	None	✓				14.10.a.b	$1$	$0$	\(-16\)	\(-170\)	\(0\)	\(2401\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}-170q^{3}+2^{8}q^{4}+2720q^{6}+\cdots\)
350.10.a.b	$350$	$10$	350.a	1.a	$1$	$1$	$1$	$180.263$	\(\Q\)	None	✓				70.10.a.b	$1$	$0$	\(16\)	\(-120\)	\(0\)	\(-2401\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-120q^{3}+2^{8}q^{4}-1920q^{6}+\cdots\)
350.10.a.c	$350$	$10$	350.a	1.a	$1$	$1$	$1$	$180.263$	\(\Q\)	None	✓				14.10.a.a	$1$	$1$	\(16\)	\(6\)	\(0\)	\(2401\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+6q^{3}+2^{8}q^{4}+96q^{6}+7^{4}q^{7}+\cdots\)
350.10.a.d	$350$	$10$	350.a	1.a	$1$	$1$	$1$	$180.263$	\(\Q\)	None	✓				70.10.a.a	$1$	$0$	\(16\)	\(87\)	\(0\)	\(-2401\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+87q^{3}+2^{8}q^{4}+1392q^{6}+\cdots\)
350.10.a.e	$350$	$10$	350.a	1.a	$1$	$2$	$2$	$180.263$	\(\Q(\sqrt{457}) \)	None	✓				70.10.a.g	$1$	$0$	\(-32\)	\(41\)	\(0\)	\(4802\)		$+$	$+$	$-$	$-$	$5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(20-\beta )q^{3}+2^{8}q^{4}+(-320+\cdots)q^{6}+\cdots\)
350.10.a.f	$350$	$10$	350.a	1.a	$1$	$2$	$2$	$180.263$	\(\Q(\sqrt{2473}) \)	None	✓				70.10.a.f	$1$	$1$	\(-32\)	\(143\)	\(0\)	\(-4802\)		$+$	$+$	$+$	$+$	$5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(71-\beta )q^{3}+2^{8}q^{4}+(-1136+\cdots)q^{6}+\cdots\)
350.10.a.g	$350$	$10$	350.a	1.a	$1$	$2$	$2$	$180.263$	\(\Q(\sqrt{5881}) \)	None	✓				70.10.a.e	$1$	$1$	\(32\)	\(-135\)	\(0\)	\(4802\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-68-\beta )q^{3}+2^{8}q^{4}+(-1088+\cdots)q^{6}+\cdots\)
350.10.a.h	$350$	$10$	350.a	1.a	$1$	$2$	$2$	$180.263$	\(\Q(\sqrt{3061}) \)	None	✓				70.10.a.d	$1$	$1$	\(32\)	\(-110\)	\(0\)	\(4802\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-55-\beta )q^{3}+2^{8}q^{4}+(-880+\cdots)q^{6}+\cdots\)
350.10.a.i	$350$	$10$	350.a	1.a	$1$	$2$	$2$	$180.263$	\(\Q(\sqrt{541}) \)	None	✓				70.10.a.c	$1$	$0$	\(32\)	\(-58\)	\(0\)	\(-4802\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-29-\beta )q^{3}+2^{8}q^{4}+(-464+\cdots)q^{6}+\cdots\)
350.10.a.j	$350$	$10$	350.a	1.a	$1$	$2$	$2$	$180.263$	\(\Q(\sqrt{2305}) \)	None	✓				14.10.a.c	$1$	$0$	\(32\)	\(14\)	\(0\)	\(-4802\)		$-$	$+$	$+$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(7+\beta )q^{3}+2^{8}q^{4}+(112+\cdots)q^{6}+\cdots\)
350.10.a.k	$350$	$10$	350.a	1.a	$1$	$3$	$3$	$180.263$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				70.10.a.i	$1$	$1$	\(-48\)	\(-206\)	\(0\)	\(-7203\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-69+\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.l	$350$	$10$	350.a	1.a	$1$	$3$	$3$	$180.263$	3.3.2997373.1	None	✓				70.10.a.h	$1$	$0$	\(-48\)	\(66\)	\(0\)	\(7203\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(22+\beta _{2})q^{3}+2^{8}q^{4}+(-352+\cdots)q^{6}+\cdots\)
350.10.a.m	$350$	$10$	350.a	1.a	$1$	$4$	$4$	$180.263$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			350.10.a.m	$1$	$1$	\(-64\)	\(-161\)	\(0\)	\(9604\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{2}\cdot 37$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-40-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.n	$350$	$10$	350.a	1.a	$1$	$4$	$4$	$180.263$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			350.10.a.n	$1$	$0$	\(-64\)	\(-7\)	\(0\)	\(-9604\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-2+\beta _{1})q^{3}+2^{8}q^{4}+(2^{5}+\cdots)q^{6}+\cdots\)
350.10.a.o	$350$	$10$	350.a	1.a	$1$	$4$	$4$	$180.263$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		350.10.a.n	$1$	$1$	\(64\)	\(7\)	\(0\)	\(9604\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(2-\beta _{1})q^{3}+2^{8}q^{4}+(2^{5}+\cdots)q^{6}+\cdots\)
350.10.a.p	$350$	$10$	350.a	1.a	$1$	$4$	$4$	$180.263$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		350.10.a.m	$1$	$0$	\(64\)	\(161\)	\(0\)	\(-9604\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3\cdot 5^{2}\cdot 37$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(40+\beta _{1})q^{3}+2^{8}q^{4}+(640+\cdots)q^{6}+\cdots\)
350.10.a.q	$350$	$10$	350.a	1.a	$1$	$5$	$5$	$180.263$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		350.10.a.q	$1$	$1$	\(-80\)	\(-96\)	\(0\)	\(-12005\)		$+$	$+$	$+$	$+$	$2^{4}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-19-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.r	$350$	$10$	350.a	1.a	$1$	$5$	$5$	$180.263$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		350.10.a.r	$1$	$0$	\(-80\)	\(74\)	\(0\)	\(12005\)		$+$	$+$	$-$	$-$	$2^{4}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(15-\beta _{1})q^{3}+2^{8}q^{4}+(-240+\cdots)q^{6}+\cdots\)
350.10.a.s	$350$	$10$	350.a	1.a	$1$	$5$	$5$	$180.263$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			350.10.a.r	$1$	$1$	\(80\)	\(-74\)	\(0\)	\(-12005\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-15+\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.t	$350$	$10$	350.a	1.a	$1$	$5$	$5$	$180.263$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			350.10.a.q	$1$	$0$	\(80\)	\(96\)	\(0\)	\(12005\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(19+\beta _{1})q^{3}+2^{8}q^{4}+(304+\cdots)q^{6}+\cdots\)
350.10.a.u	$350$	$10$	350.a	1.a	$1$	$6$	$6$	$180.263$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		70.10.c.a	$1$	$1$	\(-96\)	\(77\)	\(0\)	\(14406\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 3^{2}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(13+\beta _{1})q^{3}+2^{8}q^{4}+(-208+\cdots)q^{6}+\cdots\)
350.10.a.v	$350$	$10$	350.a	1.a	$1$	$6$	$6$	$180.263$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		70.10.c.a	$1$	$1$	\(96\)	\(-77\)	\(0\)	\(-14406\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 3^{2}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-13-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.w	$350$	$10$	350.a	1.a	$1$	$8$	$8$	$180.263$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		70.10.c.b	$1$	$0$	\(-128\)	\(-77\)	\(0\)	\(-19208\)		$+$	$-$	$+$	$-$	$2^{8}\cdot 3^{4}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-10+\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.x	$350$	$10$	350.a	1.a	$1$	$8$	$8$	$180.263$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		70.10.c.b	$1$	$0$	\(128\)	\(77\)	\(0\)	\(19208\)		$-$	$-$	$-$	$-$	$2^{8}\cdot 3^{4}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(10-\beta _{1})q^{3}+2^{8}q^{4}+(160+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

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