Space of modular forms of level 225, weight 10, and trivial character

• Label: 225.10.a
• Level: $225$
• Weight: $10$
• Character orbit: 225.a
• Rep. character: $\chi_{225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $24$
• Sturm bound: $300$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(300\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(225))\).

	Total	New	Old
Modular forms	282	73	209
Cusp forms	258	70	188
Eisenstein series	24	3	21

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(69\)	\(13\)	\(56\)		\(63\)	\(13\)	\(50\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(71\)	\(16\)	\(55\)		\(65\)	\(16\)	\(49\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(72\)	\(21\)	\(51\)		\(66\)	\(20\)	\(46\)		\(6\)	\(1\)	\(5\)
\(-\)	\(-\)	\(+\)		\(70\)	\(23\)	\(47\)		\(64\)	\(21\)	\(43\)		\(6\)	\(2\)	\(4\)
Plus space		\(+\)		\(139\)	\(36\)	\(103\)		\(127\)	\(34\)	\(93\)		\(12\)	\(2\)	\(10\)
Minus space		\(-\)		\(143\)	\(37\)	\(106\)		\(131\)	\(36\)	\(95\)		\(12\)	\(1\)	\(11\)

Trace form

70 * q + 32 * q^2 + 17922 * q^4 + 5638 * q^7 + 32940 * q^8 - 99738 * q^11 - 148516 * q^13 + 120072 * q^14 + 4164554 * q^16 + 315962 * q^17 + 1059982 * q^19 - 420944 * q^22 - 1192254 * q^23 + 3928908 * q^26 - 122144 * q^28 - 826002 * q^29 + 11154716 * q^31 + 24666772 * q^32 + 24557570 * q^34 - 18572232 * q^37 - 17323240 * q^38 - 25410504 * q^41 + 40819214 * q^43 - 228405234 * q^44 + 20818204 * q^46 + 125262422 * q^47 + 314859818 * q^49 - 196109592 * q^52 - 18769894 * q^53 - 27361620 * q^56 - 65147920 * q^58 - 391612404 * q^59 - 8832016 * q^61 + 38022984 * q^62 + 886502810 * q^64 + 564691858 * q^67 + 589124584 * q^68 - 465147732 * q^71 + 166031624 * q^73 + 176521572 * q^74 + 1142520042 * q^76 - 737802096 * q^77 + 82833184 * q^79 + 656687216 * q^82 + 1448251266 * q^83 - 2178393948 * q^86 - 654530880 * q^88 + 2699989944 * q^89 - 306057636 * q^91 - 3936201888 * q^92 - 2820398824 * q^94 - 1304850912 * q^97 + 6818382544 * q^98

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
225.10.a.a	$225$	$10$	225.a	1.a	$1$	$1$	$1$	$115.883$	\(\Q\)	None	✓				3.10.a.a	$1$	$0$	\(-36\)	\(0\)	\(0\)	\(4480\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6^{2}q^{2}+28^{2}q^{4}+4480q^{7}-9792q^{8}+\cdots\)
225.10.a.b	$225$	$10$	225.a	1.a	$1$	$1$	$1$	$115.883$	\(\Q\)	None	✓				5.10.a.a	$1$	$0$	\(-8\)	\(0\)	\(0\)	\(-4242\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-448q^{4}-4242q^{7}+7680q^{8}+\cdots\)
225.10.a.c	$225$	$10$	225.a	1.a	$1$	$1$	$1$	$115.883$	\(\Q\)	None	✓				15.10.a.a	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(7680\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-496q^{4}+7680q^{7}+4032q^{8}+\cdots\)
225.10.a.d	$225$	$10$	225.a	1.a	$1$	$1$	$1$	$115.883$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				9.10.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(12580\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{9}q^{4}+12580q^{7}-118370q^{13}+\cdots\)
225.10.a.e	$225$	$10$	225.a	1.a	$1$	$1$	$1$	$115.883$	\(\Q\)	None	✓				3.10.a.b	$1$	$0$	\(18\)	\(0\)	\(0\)	\(-9128\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+18q^{2}-188q^{4}-9128q^{7}-12600q^{8}+\cdots\)
225.10.a.f	$225$	$10$	225.a	1.a	$1$	$1$	$1$	$115.883$	\(\Q\)	None	✓				15.10.a.b	$1$	$0$	\(22\)	\(0\)	\(0\)	\(5988\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+22q^{2}-28q^{4}+5988q^{7}-11880q^{8}+\cdots\)
225.10.a.g	$225$	$10$	225.a	1.a	$1$	$2$	$2$	$115.883$	\(\Q(\sqrt{79}) \)	None	✓				75.10.a.e	$1$	$0$	\(-36\)	\(0\)	\(0\)	\(-3318\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-18+\beta )q^{2}+(2^{7}-6^{2}\beta )q^{4}+(-1659+\cdots)q^{7}+\cdots\)
225.10.a.h	$225$	$10$	225.a	1.a	$1$	$2$	$2$	$115.883$	\(\Q(\sqrt{1009}) \)	None	✓				5.10.a.b	$1$	$0$	\(-10\)	\(0\)	\(0\)	\(-1700\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{2}+(522+10\beta )q^{4}+(-850+\cdots)q^{7}+\cdots\)
225.10.a.i	$225$	$10$	225.a	1.a	$1$	$2$	$2$	$115.883$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓				45.10.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q-19\beta q^{2}+1293q^{4}-14839\beta q^{8}+\cdots\)
225.10.a.j	$225$	$10$	225.a	1.a	$1$	$2$	$2$	$115.883$	\(\Q(\sqrt{4729}) \)	None	✓				15.10.a.c	$1$	$0$	\(19\)	\(0\)	\(0\)	\(11872\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{2}+(770-19\beta )q^{4}+(5964+\cdots)q^{7}+\cdots\)
225.10.a.k	$225$	$10$	225.a	1.a	$1$	$2$	$2$	$115.883$	\(\Q(\sqrt{241}) \)	None	✓				15.10.a.d	$1$	$0$	\(31\)	\(0\)	\(0\)	\(-14112\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(15-\beta )q^{2}+(255-31\beta )q^{4}+(-6944+\cdots)q^{7}+\cdots\)
225.10.a.l	$225$	$10$	225.a	1.a	$1$	$2$	$2$	$115.883$	\(\Q(\sqrt{79}) \)	None	✓				75.10.a.e	$1$	$1$	\(36\)	\(0\)	\(0\)	\(3318\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(18+\beta )q^{2}+(2^{7}+6^{2}\beta )q^{4}+(1659+\cdots)q^{7}+\cdots\)
225.10.a.m	$225$	$10$	225.a	1.a	$1$	$3$	$3$	$115.883$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				25.10.a.c	$1$	$1$	\(-33\)	\(0\)	\(0\)	\(5258\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{2}+(114+9\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.n	$225$	$10$	225.a	1.a	$1$	$3$	$3$	$115.883$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				45.10.a.g	$1$	$1$	\(-25\)	\(0\)	\(0\)	\(-3890\)		$+$	$+$	$+$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-8+\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.o	$225$	$10$	225.a	1.a	$1$	$3$	$3$	$115.883$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				45.10.a.g	$1$	$1$	\(25\)	\(0\)	\(0\)	\(-3890\)		$+$	$+$	$+$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(8-\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.p	$225$	$10$	225.a	1.a	$1$	$3$	$3$	$115.883$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				25.10.a.c	$1$	$0$	\(33\)	\(0\)	\(0\)	\(-5258\)		$-$	$+$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{2}+(114+9\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.q	$225$	$10$	225.a	1.a	$1$	$4$	$4$	$115.883$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				15.10.b.a	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-9834\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(149-2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
225.10.a.r	$225$	$10$	225.a	1.a	$1$	$4$	$4$	$115.883$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				75.10.a.j	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-13036\)		$-$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{2}\cdot 23$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(445-6\beta _{1}+\beta _{2})q^{4}+(-3237+\cdots)q^{7}+\cdots\)
225.10.a.s	$225$	$10$	225.a	1.a	$1$	$4$	$4$	$115.883$	4.4.49740556.1	None	✓				5.10.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(342-\beta _{3})q^{4}+(133\beta _{1}-7^{2}\beta _{2}+\cdots)q^{7}+\cdots\)
225.10.a.t	$225$	$10$	225.a	1.a	$1$	$4$	$4$	$115.883$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		75.10.a.j	$1$	$0$	\(2\)	\(0\)	\(0\)	\(13036\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5^{2}\cdot 23$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(445-6\beta _{1}+\beta _{2})q^{4}+(3237+\cdots)q^{7}+\cdots\)
225.10.a.u	$225$	$10$	225.a	1.a	$1$	$4$	$4$	$115.883$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				15.10.b.a	$1$	$1$	\(3\)	\(0\)	\(0\)	\(9834\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(149-2\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.v	$225$	$10$	225.a	1.a	$1$	$6$	$6$	$115.883$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		225.10.a.v	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-9490\)		$+$	$+$	$+$	$2^{6}\cdot 3^{4}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(285+\beta _{2})q^{4}+(-1583+\cdots)q^{7}+\cdots\)
225.10.a.w	$225$	$10$	225.a	1.a	$1$	$6$	$6$	$115.883$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			225.10.a.v	$2$	$0$	\(0\)	\(0\)	\(0\)	\(9490\)		$+$	$-$	$-$	$2^{6}\cdot 3^{4}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(285+\beta _{2})q^{4}+(1583+5\beta _{2}+\cdots)q^{7}+\cdots\)
225.10.a.x	$225$	$10$	225.a	1.a	$1$	$8$	$8$	$115.883$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		45.10.b.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{22}\cdot 3^{8}\cdot 5^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(18-\beta _{1})q^{4}+\beta _{2}q^{7}+(-220\beta _{3}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(225)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)