Space of modular forms of level 245, weight 6, and trivial character

• Label: 245.6.a
• Level: $245$
• Weight: $6$
• Character orbit: 245.a
• Rep. character: $\chi_{245}(1,\cdot)$
• Character field: $\Q$
• Dimension: $69$
• Newform subspaces: $13$
• Sturm bound: $168$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(168\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(245))\).

	Total	New	Old
Modular forms	148	69	79
Cusp forms	132	69	63
Eisenstein series	16	0	16

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
\(+\)	\(+\)	\(+\)		\(34\)	\(16\)	\(18\)		\(30\)	\(16\)	\(14\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(39\)	\(19\)	\(20\)		\(35\)	\(19\)	\(16\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(38\)	\(18\)	\(20\)		\(34\)	\(18\)	\(16\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(37\)	\(16\)	\(21\)		\(33\)	\(16\)	\(17\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(71\)	\(32\)	\(39\)		\(63\)	\(32\)	\(31\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(77\)	\(37\)	\(40\)		\(69\)	\(37\)	\(32\)		\(8\)	\(0\)	\(8\)

Trace form

69 * q + 6 * q^2 - 40 * q^3 + 1168 * q^4 - 25 * q^5 + 72 * q^6 - 348 * q^8 + 5765 * q^9 - 150 * q^10 + 1420 * q^11 - 3796 * q^12 - 690 * q^13 + 1000 * q^15 + 19124 * q^16 - 1822 * q^17 + 7202 * q^18 - 1916 * q^19 + 700 * q^20 - 1560 * q^22 + 1436 * q^23 + 15124 * q^24 + 43125 * q^25 + 8808 * q^26 - 1732 * q^27 + 994 * q^29 - 7800 * q^30 - 1692 * q^31 + 500 * q^32 + 26628 * q^33 + 23364 * q^34 + 94824 * q^36 + 30054 * q^37 - 700 * q^38 - 50040 * q^39 - 12600 * q^40 - 14246 * q^41 - 31096 * q^43 - 15132 * q^44 - 125 * q^45 + 17468 * q^46 - 35116 * q^47 - 13332 * q^48 + 3750 * q^50 + 10056 * q^51 - 11776 * q^52 - 8634 * q^53 + 21628 * q^54 - 24100 * q^55 - 108112 * q^57 - 45776 * q^58 - 67192 * q^59 + 80900 * q^60 - 7230 * q^61 + 97320 * q^62 + 238988 * q^64 + 26550 * q^65 + 108732 * q^66 - 66040 * q^67 - 37900 * q^68 - 129780 * q^69 + 40200 * q^71 + 95728 * q^72 + 88014 * q^73 - 114536 * q^74 - 25000 * q^75 + 101676 * q^76 + 146472 * q^78 - 118008 * q^79 + 48400 * q^80 + 280549 * q^81 - 360968 * q^82 + 105540 * q^83 - 61550 * q^85 + 227696 * q^86 + 241348 * q^87 - 47344 * q^88 + 306526 * q^89 - 163450 * q^90 + 65020 * q^92 + 155112 * q^93 - 309980 * q^94 + 82100 * q^95 + 256188 * q^96 - 72870 * q^97 + 1230636 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\) 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
245.6.a.a	$245$	$6$	245.a	1.a	$1$	$1$	$1$	$39.294$	\(\Q\)	None	✓				35.6.a.a	$1$	$0$	\(-8\)	\(-1\)	\(-25\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-q^{3}+2^{5}q^{4}-5^{2}q^{5}+8q^{6}+\cdots\)
245.6.a.b	$245$	$6$	245.a	1.a	$1$	$1$	$1$	$39.294$	\(\Q\)	None	✓				5.6.a.a	$1$	$0$	\(2\)	\(4\)	\(-25\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{3}-28q^{4}-5^{2}q^{5}+8q^{6}+\cdots\)
245.6.a.c	$245$	$6$	245.a	1.a	$1$	$2$	$2$	$39.294$	\(\Q(\sqrt{65}) \)	None	✓				35.6.a.b	$1$	$1$	\(1\)	\(-3\)	\(50\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-3+3\beta )q^{3}+(-2^{4}+\beta )q^{4}+\cdots\)
245.6.a.d	$245$	$6$	245.a	1.a	$1$	$3$	$3$	$39.294$	3.3.577880.1	None	✓				35.6.a.c	$1$	$1$	\(-6\)	\(-26\)	\(75\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-9-\beta _{2})q^{3}+(38+\cdots)q^{4}+\cdots\)
245.6.a.e	$245$	$6$	245.a	1.a	$1$	$4$	$4$	$39.294$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				35.6.a.d	$1$	$0$	\(7\)	\(-14\)	\(-100\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(-4+\beta _{1}-\beta _{3})q^{3}+\cdots\)
245.6.a.f	$245$	$6$	245.a	1.a	$1$	$5$	$5$	$39.294$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			245.6.a.f	$1$	$1$	\(-3\)	\(-7\)	\(125\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+\cdots\)
245.6.a.g	$245$	$6$	245.a	1.a	$1$	$5$	$5$	$39.294$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			245.6.a.f	$1$	$0$	\(-3\)	\(7\)	\(-125\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots\)
245.6.a.h	$245$	$6$	245.a	1.a	$1$	$6$	$6$	$39.294$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		35.6.e.a	$1$	$1$	\(-5\)	\(-20\)	\(150\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-3-\beta _{1}+\beta _{3})q^{3}+\cdots\)
245.6.a.i	$245$	$6$	245.a	1.a	$1$	$6$	$6$	$39.294$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				35.6.e.a	$1$	$1$	\(-5\)	\(20\)	\(-150\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3+\beta _{1}-\beta _{3})q^{3}+\cdots\)
245.6.a.j	$245$	$6$	245.a	1.a	$1$	$8$	$8$	$39.294$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				35.6.e.b	$1$	$0$	\(3\)	\(-2\)	\(200\)	\(0\)		$-$	$+$	$-$	$2^{5}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(5^{2}+\beta _{1}+\beta _{2})q^{4}+\cdots\)
245.6.a.k	$245$	$6$	245.a	1.a	$1$	$8$	$8$	$39.294$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		35.6.e.b	$1$	$0$	\(3\)	\(2\)	\(-200\)	\(0\)		$+$	$-$	$-$	$2^{5}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(5^{2}+\beta _{1}+\beta _{2})q^{4}+\cdots\)
245.6.a.l	$245$	$6$	245.a	1.a	$1$	$10$	$10$	$39.294$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		245.6.a.l	$1$	$1$	\(10\)	\(-58\)	\(-250\)	\(0\)		$+$	$+$	$+$	$2^{3}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-6+\beta _{3})q^{3}+(18-\beta _{1}+\cdots)q^{4}+\cdots\)
245.6.a.m	$245$	$6$	245.a	1.a	$1$	$10$	$10$	$39.294$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		245.6.a.l	$1$	$0$	\(10\)	\(58\)	\(250\)	\(0\)		$-$	$+$	$-$	$2^{3}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(6-\beta _{3})q^{3}+(18-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(245))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(245)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)