Space of modular forms of level 350, weight 6, and trivial character

• Label: 350.6.a
• Level: $350$
• Weight: $6$
• Character orbit: 350.a
• Rep. character: $\chi_{350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $26$
• Sturm bound: $360$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(360\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(350))\).

	Total	New	Old
Modular forms	312	48	264
Cusp forms	288	48	240
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	\(7\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(8\)
Plus space			\(+\)	\(21\)
Minus space			\(-\)	\(27\)

Trace form

48 * q - 8 * q^2 + 26 * q^3 + 768 * q^4 - 8 * q^6 - 128 * q^8 + 4416 * q^9 - 952 * q^11 + 416 * q^12 + 518 * q^13 + 392 * q^14 + 12288 * q^16 + 2212 * q^17 - 5656 * q^18 - 4122 * q^19 - 98 * q^21 + 1872 * q^22 + 8224 * q^23 - 128 * q^24 - 1560 * q^26 - 8164 * q^27 + 12612 * q^29 - 6572 * q^31 - 2048 * q^32 + 12288 * q^33 + 28976 * q^34 + 70656 * q^36 - 8004 * q^37 - 24888 * q^38 + 39976 * q^39 + 32320 * q^41 - 3528 * q^42 + 9596 * q^43 - 15232 * q^44 - 13664 * q^46 + 86420 * q^47 + 6656 * q^48 + 115248 * q^49 - 94144 * q^51 + 8288 * q^52 + 5648 * q^53 - 83840 * q^54 + 6272 * q^56 - 12156 * q^57 + 27104 * q^58 + 47818 * q^59 + 59126 * q^61 + 1488 * q^62 + 44492 * q^63 + 196608 * q^64 + 113040 * q^66 - 146168 * q^67 + 35392 * q^68 - 240816 * q^69 - 211184 * q^71 - 90496 * q^72 + 110552 * q^73 + 50080 * q^74 - 65952 * q^76 - 47628 * q^77 + 132064 * q^78 + 41680 * q^79 + 454452 * q^81 + 49088 * q^82 - 190410 * q^83 - 1568 * q^84 + 335696 * q^86 - 211292 * q^87 + 29952 * q^88 + 187220 * q^89 + 11466 * q^91 + 131584 * q^92 + 339080 * q^93 + 40560 * q^94 - 2048 * q^96 - 112012 * q^97 - 19208 * q^98 + 579332 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\) 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
350.6.a.a	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.c.c	$1$	$1$	\(-4\)	\(-13\)	\(0\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-13q^{3}+2^{4}q^{4}+52q^{6}+7^{2}q^{7}+\cdots\)
350.6.a.b	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				14.6.a.b	$1$	$0$	\(-4\)	\(-8\)	\(0\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-8q^{3}+2^{4}q^{4}+2^{5}q^{6}+7^{2}q^{7}+\cdots\)
350.6.a.c	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.c.a	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+7^{2}q^{7}-2^{6}q^{8}-3^{5}q^{9}+\cdots\)
350.6.a.d	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.a.f	$1$	$1$	\(-4\)	\(11\)	\(0\)	\(-49\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+11q^{3}+2^{4}q^{4}-44q^{6}-7^{2}q^{7}+\cdots\)
350.6.a.e	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.a.e	$1$	$0$	\(-4\)	\(17\)	\(0\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+17q^{3}+2^{4}q^{4}-68q^{6}+7^{2}q^{7}+\cdots\)
350.6.a.f	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.c.b	$1$	$1$	\(-4\)	\(27\)	\(0\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+3^{3}q^{3}+2^{4}q^{4}-108q^{6}+\cdots\)
350.6.a.g	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.c.b	$1$	$1$	\(4\)	\(-27\)	\(0\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-3^{3}q^{3}+2^{4}q^{4}-108q^{6}+\cdots\)
350.6.a.h	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.a.d	$1$	$1$	\(4\)	\(-11\)	\(0\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-11q^{3}+2^{4}q^{4}-44q^{6}+7^{2}q^{7}+\cdots\)
350.6.a.i	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				14.6.a.a	$1$	$0$	\(4\)	\(-10\)	\(0\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-10q^{3}+2^{4}q^{4}-40q^{6}-7^{2}q^{7}+\cdots\)
350.6.a.j	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.c.a	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-7^{2}q^{7}+2^{6}q^{8}-3^{5}q^{9}+\cdots\)
350.6.a.k	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.a.c	$1$	$0$	\(4\)	\(3\)	\(0\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+3q^{3}+2^{4}q^{4}+12q^{6}-7^{2}q^{7}+\cdots\)
350.6.a.l	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.a.b	$1$	$1$	\(4\)	\(9\)	\(0\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
350.6.a.m	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.c.c	$1$	$1$	\(4\)	\(13\)	\(0\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+13q^{3}+2^{4}q^{4}+52q^{6}-7^{2}q^{7}+\cdots\)
350.6.a.n	$350$	$6$	350.a	1.a	$1$	$1$	$1$	$56.134$	\(\Q\)	None	✓				70.6.a.a	$1$	$0$	\(4\)	\(23\)	\(0\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+23q^{3}+2^{4}q^{4}+92q^{6}-7^{2}q^{7}+\cdots\)
350.6.a.o	$350$	$6$	350.a	1.a	$1$	$2$	$2$	$56.134$	\(\Q(\sqrt{79}) \)	None	✓	✓	✓		350.6.a.o	$1$	$1$	\(-8\)	\(-8\)	\(0\)	\(98\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-4+\beta )q^{3}+2^{4}q^{4}+(2^{4}+\cdots)q^{6}+\cdots\)
350.6.a.p	$350$	$6$	350.a	1.a	$1$	$2$	$2$	$56.134$	\(\Q(\sqrt{1129}) \)	None	✓				70.6.a.h	$1$	$1$	\(-8\)	\(-5\)	\(0\)	\(-98\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-2-\beta )q^{3}+2^{4}q^{4}+(8+4\beta )q^{6}+\cdots\)
350.6.a.q	$350$	$6$	350.a	1.a	$1$	$2$	$2$	$56.134$	\(\Q(\sqrt{3369}) \)	None	✓				70.6.a.g	$1$	$0$	\(-8\)	\(-3\)	\(0\)	\(98\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-1-\beta )q^{3}+2^{4}q^{4}+(4+4\beta )q^{6}+\cdots\)
350.6.a.r	$350$	$6$	350.a	1.a	$1$	$2$	$2$	$56.134$	\(\Q(\sqrt{79}) \)	None	✓	✓			350.6.a.r	$1$	$0$	\(-8\)	\(20\)	\(0\)	\(-98\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+(10+\beta )q^{3}+2^{4}q^{4}+(-40+\cdots)q^{6}+\cdots\)
350.6.a.s	$350$	$6$	350.a	1.a	$1$	$2$	$2$	$56.134$	\(\Q(\sqrt{79}) \)	None	✓	✓	✓		350.6.a.r	$1$	$1$	\(8\)	\(-20\)	\(0\)	\(98\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-10+\beta )q^{3}+2^{4}q^{4}+(-40+\cdots)q^{6}+\cdots\)
350.6.a.t	$350$	$6$	350.a	1.a	$1$	$2$	$2$	$56.134$	\(\Q(\sqrt{79}) \)	None	✓	✓	✓		350.6.a.o	$1$	$0$	\(8\)	\(8\)	\(0\)	\(-98\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}+(4+\beta )q^{3}+2^{4}q^{4}+(2^{4}+4\beta )q^{6}+\cdots\)
350.6.a.u	$350$	$6$	350.a	1.a	$1$	$3$	$3$	$56.134$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓		350.6.a.u	$1$	$1$	\(-12\)	\(-9\)	\(0\)	\(-147\)		$+$	$+$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-3+\beta _{1})q^{3}+2^{4}q^{4}+(12+\cdots)q^{6}+\cdots\)
350.6.a.v	$350$	$6$	350.a	1.a	$1$	$3$	$3$	$56.134$	3.3.1378776.1	None	✓	✓	✓		350.6.a.v	$1$	$0$	\(-12\)	\(-1\)	\(0\)	\(147\)		$+$	$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+4\beta _{1}q^{6}+\cdots\)
350.6.a.w	$350$	$6$	350.a	1.a	$1$	$3$	$3$	$56.134$	3.3.1378776.1	None	✓	✓	✓		350.6.a.v	$1$	$1$	\(12\)	\(1\)	\(0\)	\(-147\)		$-$	$-$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+4\beta _{1}q^{6}+\cdots\)
350.6.a.x	$350$	$6$	350.a	1.a	$1$	$3$	$3$	$56.134$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			350.6.a.u	$1$	$0$	\(12\)	\(9\)	\(0\)	\(147\)		$-$	$-$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(12-4\beta _{1}+\cdots)q^{6}+\cdots\)
350.6.a.y	$350$	$6$	350.a	1.a	$1$	$5$	$5$	$56.134$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		70.6.c.d	$1$	$0$	\(-20\)	\(-14\)	\(0\)	\(-245\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-3+\beta _{1})q^{3}+2^{4}q^{4}+(12+\cdots)q^{6}+\cdots\)
350.6.a.z	$350$	$6$	350.a	1.a	$1$	$5$	$5$	$56.134$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		70.6.c.d	$1$	$0$	\(20\)	\(14\)	\(0\)	\(245\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(12-4\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

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