Space of modular forms of level 245, weight 10, and trivial character

• Label: 245.10.a
• Level: $245$
• Weight: $10$
• Character orbit: 245.a
• Rep. character: $\chi_{245}(1,\cdot)$
• Character field: $\Q$
• Dimension: $123$
• Newform subspaces: $15$
• Sturm bound: $280$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(280\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(245))\).

	Total	New	Old
Modular forms	260	123	137
Cusp forms	244	123	121
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
\(+\)	\(+\)	\(+\)		\(62\)	\(29\)	\(33\)		\(58\)	\(29\)	\(29\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(67\)	\(33\)	\(34\)		\(63\)	\(33\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(66\)	\(31\)	\(35\)		\(62\)	\(31\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(65\)	\(30\)	\(35\)		\(61\)	\(30\)	\(31\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(127\)	\(59\)	\(68\)		\(119\)	\(59\)	\(60\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(133\)	\(64\)	\(69\)		\(125\)	\(64\)	\(61\)		\(8\)	\(0\)	\(8\)

Trace form

123 * q + 50 * q^2 + 146 * q^3 + 30976 * q^4 - 625 * q^5 + 120 * q^6 + 55332 * q^8 + 747647 * q^9 - 21250 * q^10 - 66196 * q^11 + 199436 * q^12 - 93534 * q^13 - 31250 * q^15 + 7517236 * q^16 + 356990 * q^17 + 28958 * q^18 - 518568 * q^19 - 932500 * q^20 - 5260264 * q^22 + 5283346 * q^23 + 1570132 * q^24 + 48046875 * q^25 - 19131552 * q^26 + 1722308 * q^27 - 4108130 * q^29 + 855000 * q^30 + 12667708 * q^31 + 42764404 * q^32 - 46451880 * q^33 - 14559780 * q^34 + 128215896 * q^36 + 73439126 * q^37 + 14270500 * q^38 + 30064716 * q^39 - 32835000 * q^40 - 61003334 * q^41 - 107787142 * q^43 + 39890212 * q^44 - 39565625 * q^45 + 124732556 * q^46 - 28362334 * q^47 + 201892524 * q^48 + 19531250 * q^50 + 168701244 * q^51 - 88413024 * q^52 + 63368254 * q^53 + 118759324 * q^54 + 48335000 * q^55 + 486450704 * q^57 - 291910792 * q^58 + 130890704 * q^59 + 183702500 * q^60 + 315200674 * q^61 + 206213544 * q^62 + 1740196620 * q^64 - 73201250 * q^65 + 896660604 * q^66 + 325989198 * q^67 + 663170644 * q^68 - 439827468 * q^69 - 286870420 * q^71 - 1133150864 * q^72 - 370085770 * q^73 - 1334085264 * q^74 + 57031250 * q^75 + 1523192108 * q^76 + 1039087272 * q^78 + 732597960 * q^79 - 1336790000 * q^80 + 4424658199 * q^81 - 372376144 * q^82 - 1331291174 * q^83 + 689276250 * q^85 + 2620304736 * q^86 - 3282442652 * q^87 - 1109567152 * q^88 + 2769543798 * q^89 + 452386250 * q^90 - 44664676 * q^92 + 1352429664 * q^93 + 4915128468 * q^94 + 808802500 * q^95 - 599218692 * q^96 - 678170858 * q^97 + 485561676 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(245))\) 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
245.10.a.a	$245$	$10$	245.a	1.a	$1$	$1$	$1$	$126.184$	\(\Q\)	None	✓				5.10.a.a	$1$	$1$	\(-8\)	\(114\)	\(625\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+114q^{3}-448q^{4}+5^{4}q^{5}+\cdots\)
245.10.a.b	$245$	$10$	245.a	1.a	$1$	$1$	$1$	$126.184$	\(\Q\)	None	✓				35.10.a.a	$1$	$0$	\(28\)	\(116\)	\(-625\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+28q^{2}+116q^{3}+272q^{4}-5^{4}q^{5}+\cdots\)
245.10.a.c	$245$	$10$	245.a	1.a	$1$	$2$	$2$	$126.184$	\(\Q(\sqrt{2}) \)	None	✓				35.10.a.b	$1$	$0$	\(-24\)	\(174\)	\(-1250\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-12+\beta )q^{2}+(87-54\beta )q^{3}+(-360+\cdots)q^{4}+\cdots\)
245.10.a.d	$245$	$10$	245.a	1.a	$1$	$2$	$2$	$126.184$	\(\Q(\sqrt{1009}) \)	None	✓				5.10.a.b	$1$	$0$	\(-10\)	\(-260\)	\(-1250\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{2}+(-130-2\beta )q^{3}+(522+\cdots)q^{4}+\cdots\)
245.10.a.e	$245$	$10$	245.a	1.a	$1$	$4$	$4$	$126.184$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				35.10.a.c	$1$	$1$	\(-19\)	\(18\)	\(2500\)	\(0\)		$-$	$-$	$+$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{1})q^{2}+(4-\beta _{2})q^{3}+(435+\cdots)q^{4}+\cdots\)
245.10.a.f	$245$	$10$	245.a	1.a	$1$	$5$	$5$	$126.184$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.10.a.d	$1$	$1$	\(2\)	\(-140\)	\(3125\)	\(0\)		$-$	$-$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-28-\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots\)
245.10.a.g	$245$	$10$	245.a	1.a	$1$	$6$	$6$	$126.184$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				35.10.a.e	$1$	$0$	\(15\)	\(124\)	\(-3750\)	\(0\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(20+\beta _{1}-\beta _{2})q^{3}+(503+\cdots)q^{4}+\cdots\)
245.10.a.h	$245$	$10$	245.a	1.a	$1$	$9$	$9$	$126.184$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			245.10.a.h	$1$	$1$	\(1\)	\(-268\)	\(5625\)	\(0\)		$-$	$-$	$+$	$2^{7}\cdot 3^{4}\cdot 5^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-30-\beta _{1}+\beta _{3})q^{3}+(208+\cdots)q^{4}+\cdots\)
245.10.a.i	$245$	$10$	245.a	1.a	$1$	$9$	$9$	$126.184$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			245.10.a.h	$1$	$0$	\(1\)	\(268\)	\(-5625\)	\(0\)		$+$	$-$	$-$	$2^{7}\cdot 3^{4}\cdot 5^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(30+\beta _{1}-\beta _{3})q^{3}+(208+\cdots)q^{4}+\cdots\)
245.10.a.j	$245$	$10$	245.a	1.a	$1$	$11$	$11$	$126.184$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		35.10.e.a	$1$	$1$	\(-33\)	\(-56\)	\(6875\)	\(0\)		$-$	$-$	$+$	$2^{10}\cdot 3^{2}\cdot 5^{3}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(-5-\beta _{3})q^{3}+(250+\cdots)q^{4}+\cdots\)
245.10.a.k	$245$	$10$	245.a	1.a	$1$	$11$	$11$	$126.184$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				35.10.e.a	$1$	$1$	\(-33\)	\(56\)	\(-6875\)	\(0\)		$+$	$+$	$+$	$2^{10}\cdot 3^{2}\cdot 5^{3}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(5+\beta _{3})q^{3}+(250+\cdots)q^{4}+\cdots\)
245.10.a.l	$245$	$10$	245.a	1.a	$1$	$13$	$13$	$126.184$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		35.10.e.b	$1$	$0$	\(-1\)	\(-268\)	\(-8125\)	\(0\)		$+$	$-$	$-$	$2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-21-\beta _{3})q^{3}+(274+\beta _{1}+\cdots)q^{4}+\cdots\)
245.10.a.m	$245$	$10$	245.a	1.a	$1$	$13$	$13$	$126.184$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				35.10.e.b	$1$	$0$	\(-1\)	\(268\)	\(8125\)	\(0\)		$-$	$+$	$-$	$2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(21+\beta _{3})q^{3}+(274+\beta _{1}+\cdots)q^{4}+\cdots\)
245.10.a.n	$245$	$10$	245.a	1.a	$1$	$18$	$18$	$126.184$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		245.10.a.n	$1$	$1$	\(66\)	\(-112\)	\(-11250\)	\(0\)		$+$	$+$	$+$	$2^{18}\cdot 3^{4}\cdot 5^{4}\cdot 7^{16}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(-6+\beta _{4})q^{3}+(249+\cdots)q^{4}+\cdots\)
245.10.a.o	$245$	$10$	245.a	1.a	$1$	$18$	$18$	$126.184$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		245.10.a.n	$1$	$0$	\(66\)	\(112\)	\(11250\)	\(0\)		$-$	$+$	$-$	$2^{18}\cdot 3^{4}\cdot 5^{4}\cdot 7^{16}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(6-\beta _{4})q^{3}+(249-6\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(245))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(245)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)