Space of modular forms of level 350, weight 4, and trivial character

• Label: 350.4.a
• Level: $350$
• Weight: $4$
• Character orbit: 350.a
• Rep. character: $\chi_{350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $24$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(350))\).

	Total	New	Old
Modular forms	192	28	164
Cusp forms	168	28	140
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
\(+\)	\(+\)	\(+\)	\(+\)		\(27\)	\(3\)	\(24\)		\(24\)	\(3\)	\(21\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(21\)	\(2\)	\(19\)		\(18\)	\(2\)	\(16\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(22\)	\(3\)	\(19\)		\(19\)	\(3\)	\(16\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(26\)	\(5\)	\(21\)		\(23\)	\(5\)	\(18\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(24\)	\(4\)	\(20\)		\(21\)	\(4\)	\(17\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(24\)	\(5\)	\(19\)		\(21\)	\(5\)	\(16\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(23\)	\(4\)	\(19\)		\(20\)	\(4\)	\(16\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(25\)	\(2\)	\(23\)		\(22\)	\(2\)	\(20\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(100\)	\(17\)	\(83\)		\(88\)	\(17\)	\(71\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(92\)	\(11\)	\(81\)		\(80\)	\(11\)	\(69\)		\(12\)	\(0\)	\(12\)

Trace form

28 * q + 4 * q^2 - 10 * q^3 + 112 * q^4 - 20 * q^6 + 16 * q^8 + 276 * q^9 - 40 * q^11 - 40 * q^12 - 42 * q^13 - 28 * q^14 + 448 * q^16 + 28 * q^17 + 44 * q^18 + 178 * q^19 + 70 * q^21 + 8 * q^22 + 160 * q^23 - 80 * q^24 + 444 * q^26 + 44 * q^27 + 444 * q^29 - 268 * q^31 + 64 * q^32 + 744 * q^33 - 24 * q^34 + 1104 * q^36 + 188 * q^37 - 204 * q^38 - 272 * q^39 - 848 * q^41 + 84 * q^42 + 764 * q^43 - 160 * q^44 + 336 * q^46 - 1420 * q^47 - 160 * q^48 + 1372 * q^49 + 920 * q^51 - 168 * q^52 - 136 * q^53 + 1600 * q^54 - 112 * q^56 - 948 * q^57 + 848 * q^58 + 502 * q^59 - 2242 * q^61 + 360 * q^62 + 644 * q^63 + 1792 * q^64 - 456 * q^66 - 304 * q^67 + 112 * q^68 + 4776 * q^69 + 2872 * q^71 + 176 * q^72 - 968 * q^73 - 3344 * q^74 + 712 * q^76 + 84 * q^77 + 1264 * q^78 + 144 * q^79 + 7968 * q^81 - 1248 * q^82 - 3630 * q^83 + 280 * q^84 - 3832 * q^86 + 2500 * q^87 + 32 * q^88 - 3100 * q^89 - 1106 * q^91 + 640 * q^92 + 3200 * q^93 + 1592 * q^94 - 320 * q^96 + 4156 * q^97 + 196 * q^98 - 4780 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(350))\) 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
350.4.a.a	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.a	$1$	$1$	\(-2\)	\(-8\)	\(0\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-8q^{3}+4q^{4}+2^{4}q^{6}-7q^{7}+\cdots\)
350.4.a.b	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.a.f	$1$	$0$	\(-2\)	\(-7\)	\(0\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-7q^{3}+4q^{4}+14q^{6}-7q^{7}+\cdots\)
350.4.a.c	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.a.e	$1$	$1$	\(-2\)	\(-5\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-5q^{3}+4q^{4}+10q^{6}+7q^{7}+\cdots\)
350.4.a.d	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.d	$1$	$0$	\(-2\)	\(-1\)	\(0\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+4q^{4}+2q^{6}+7q^{7}+\cdots\)
350.4.a.e	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.e	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-7q^{7}+\cdots\)
350.4.a.f	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				14.4.a.b	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-7q^{7}+\cdots\)
350.4.a.g	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.g	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
350.4.a.h	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.h	$1$	$1$	\(-2\)	\(4\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{3}+4q^{4}-8q^{6}+7q^{7}+\cdots\)
350.4.a.i	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.c.a	$1$	$1$	\(-2\)	\(7\)	\(0\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+7q^{3}+4q^{4}-14q^{6}-7q^{7}+\cdots\)
350.4.a.j	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.j	$1$	$0$	\(-2\)	\(10\)	\(0\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+10q^{3}+4q^{4}-20q^{6}+7q^{7}+\cdots\)
350.4.a.k	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.j	$1$	$1$	\(2\)	\(-10\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-10q^{3}+4q^{4}-20q^{6}-7q^{7}+\cdots\)
350.4.a.l	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				14.4.a.a	$1$	$0$	\(2\)	\(-8\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-8q^{3}+4q^{4}-2^{4}q^{6}+7q^{7}+\cdots\)
350.4.a.m	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.c.a	$1$	$1$	\(2\)	\(-7\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-7q^{3}+4q^{4}-14q^{6}+7q^{7}+\cdots\)
350.4.a.n	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.h	$1$	$0$	\(2\)	\(-4\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{3}+4q^{4}-8q^{6}-7q^{7}+\cdots\)
350.4.a.o	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.a.d	$1$	$1$	\(2\)	\(-4\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{3}+4q^{4}-8q^{6}-7q^{7}+\cdots\)
350.4.a.p	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.g	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
350.4.a.q	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.e	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-2q^{3}+4q^{4}-4q^{6}+7q^{7}+\cdots\)
350.4.a.r	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.a.c	$1$	$1$	\(2\)	\(1\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}+2q^{6}-7q^{7}+\cdots\)
350.4.a.s	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.d	$1$	$1$	\(2\)	\(1\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}+2q^{6}-7q^{7}+\cdots\)
350.4.a.t	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.a.b	$1$	$0$	\(2\)	\(3\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
350.4.a.u	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓	✓			350.4.a.a	$1$	$0$	\(2\)	\(8\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+8q^{3}+4q^{4}+2^{4}q^{6}+7q^{7}+\cdots\)
350.4.a.v	$350$	$4$	350.a	1.a	$1$	$1$	$1$	$20.651$	\(\Q\)	None	✓				70.4.a.a	$1$	$0$	\(2\)	\(8\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+8q^{3}+4q^{4}+2^{4}q^{6}+7q^{7}+\cdots\)
350.4.a.w	$350$	$4$	350.a	1.a	$1$	$3$	$3$	$20.651$	3.3.51960.1	None	✓		✓		70.4.c.b	$1$	$0$	\(-6\)	\(-7\)	\(0\)	\(21\)		$+$	$-$	$-$	$+$	$5$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}+(4+2\beta _{1}+\cdots)q^{6}+\cdots\)
350.4.a.x	$350$	$4$	350.a	1.a	$1$	$3$	$3$	$20.651$	3.3.51960.1	None	✓		✓		70.4.c.b	$1$	$0$	\(6\)	\(7\)	\(0\)	\(-21\)		$-$	$-$	$+$	$+$	$5$	$\mathrm{SU}(2)$	\(q+2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+(4+2\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

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