Space of modular forms of level 1225, weight 4, and trivial character

• Label: 1225.4.a
• Level: $1225$
• Weight: $4$
• Character orbit: 1225.a
• Rep. character: $\chi_{1225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $187$
• Newform subspaces: $46$
• Sturm bound: $560$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 46 \)
Sturm bound:			\(560\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1225))\).

	Total	New	Old
Modular forms	444	202	242
Cusp forms	396	187	209
Eisenstein series	48	15	33

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	Dim
\(+\)	\(+\)	\(+\)	\(45\)
\(+\)	\(-\)	\(-\)	\(45\)
\(-\)	\(+\)	\(-\)	\(46\)
\(-\)	\(-\)	\(+\)	\(51\)
Plus space		\(+\)	\(96\)
Minus space		\(-\)	\(91\)

Trace form

187 * q - 4 * q^3 + 714 * q^4 + 16 * q^6 + 48 * q^8 + 1571 * q^9 - 20 * q^11 - 174 * q^12 + 6 * q^13 + 2650 * q^16 - 116 * q^17 - 232 * q^18 + 80 * q^19 + 44 * q^22 + 196 * q^23 + 180 * q^24 - 264 * q^26 + 176 * q^27 + 50 * q^29 + 36 * q^31 + 1184 * q^32 + 376 * q^33 - 32 * q^34 + 5500 * q^36 + 488 * q^37 - 390 * q^38 - 112 * q^39 + 286 * q^41 + 454 * q^43 - 918 * q^44 + 672 * q^46 - 734 * q^47 - 2110 * q^48 - 292 * q^51 + 1876 * q^52 - 948 * q^53 + 2830 * q^54 - 1694 * q^57 + 1692 * q^58 - 330 * q^59 - 434 * q^61 - 564 * q^62 + 11238 * q^64 + 3542 * q^66 - 644 * q^67 + 2314 * q^68 - 88 * q^69 - 404 * q^71 - 1852 * q^72 - 512 * q^73 - 748 * q^74 + 3840 * q^76 + 3816 * q^78 - 1542 * q^79 + 12263 * q^81 - 690 * q^82 - 3504 * q^83 - 672 * q^86 - 1464 * q^87 + 344 * q^88 - 1550 * q^89 + 2780 * q^92 + 5686 * q^93 + 3488 * q^94 - 4 * q^96 + 4548 * q^97 - 1472 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\) 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
1225.4.a.a	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				35.4.e.a	$1$	$1$	\(-3\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-2q^{3}+q^{4}+6q^{6}+21q^{8}+\cdots\)
1225.4.a.b	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				35.4.e.a	$1$	$0$	\(-3\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+2q^{3}+q^{4}-6q^{6}+21q^{8}+\cdots\)
1225.4.a.c	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				7.4.c.a	$1$	$0$	\(-2\)	\(-7\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-7q^{3}-4q^{4}+14q^{6}+24q^{8}+\cdots\)
1225.4.a.d	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				7.4.c.a	$1$	$1$	\(-2\)	\(7\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+7q^{3}-4q^{4}-14q^{6}+24q^{8}+\cdots\)
1225.4.a.e	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				35.4.a.a	$1$	$1$	\(-1\)	\(-8\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-8q^{3}-7q^{4}+8q^{6}+15q^{8}+\cdots\)
1225.4.a.f	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				245.4.a.b	$1$	$1$	\(-1\)	\(-6\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-6q^{3}-7q^{4}+6q^{6}+15q^{8}+\cdots\)
1225.4.a.g	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				245.4.a.b	$1$	$1$	\(-1\)	\(6\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+6q^{3}-7q^{4}-6q^{6}+15q^{8}+\cdots\)
1225.4.a.h	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				25.4.a.a	$1$	$0$	\(-1\)	\(7\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+7q^{3}-7q^{4}-7q^{6}+15q^{8}+\cdots\)
1225.4.a.i	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				25.4.a.a	$1$	$1$	\(1\)	\(-7\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{3}-7q^{4}-7q^{6}-15q^{8}+\cdots\)
1225.4.a.j	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				7.4.a.a	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-7q^{4}-2q^{6}-15q^{8}+\cdots\)
1225.4.a.k	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	None	✓				5.4.a.a	$1$	$1$	\(4\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2q^{3}+8q^{4}+8q^{6}-23q^{9}+\cdots\)
1225.4.a.l	$1225$	$4$	1225.a	1.a	$1$	$1$	$1$	$72.277$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.4.a.a	$2$	$1$	\(5\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+5q^{2}+17q^{4}+45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
1225.4.a.m	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{2}) \)	None	✓				35.4.a.b	$1$	$1$	\(-8\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{2}+(1+4\beta )q^{3}+(10-8\beta )q^{4}+\cdots\)
1225.4.a.n	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{2}) \)	None	✓				245.4.b.b	$2$	$1$	\(-6\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-6\beta q^{3}+q^{4}+18\beta q^{6}+21q^{8}+\cdots\)
1225.4.a.o	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			1225.4.a.o	$2$	$1$	\(-5\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-2-\beta )q^{2}+(1+5\beta )q^{4}+(-11+\cdots)q^{8}+\cdots\)
1225.4.a.p	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{11}) \)	None	✓				245.4.a.i	$1$	$1$	\(-2\)	\(-10\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-5q^{3}+(4-2\beta )q^{4}+\cdots\)
1225.4.a.q	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{11}) \)	None	✓				245.4.a.i	$1$	$1$	\(-2\)	\(10\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+5q^{3}+(4-2\beta )q^{4}+\cdots\)
1225.4.a.r	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{41}) \)	None	✓				175.4.a.d	$1$	$0$	\(-1\)	\(-5\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-2-\beta )q^{3}+(2+\beta )q^{4}+(10+\cdots)q^{6}+\cdots\)
1225.4.a.s	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-35}) \)	✓				245.4.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$N(\mathrm{U}(1))$	\(q+2\beta q^{3}-8q^{4}+53q^{9}+72q^{11}+\cdots\)
1225.4.a.t	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{41}) \)	None	✓				175.4.a.d	$1$	$1$	\(1\)	\(5\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{3}+(2+\beta )q^{4}+(10+\cdots)q^{6}+\cdots\)
1225.4.a.u	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			1225.4.a.o	$2$	$0$	\(5\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(3-\beta )q^{2}+(6-5\beta )q^{4}+(19-8\beta )q^{8}+\cdots\)
1225.4.a.v	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{2}) \)	None	✓				35.4.e.b	$1$	$1$	\(6\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(-1+3\beta )q^{3}+(3+6\beta )q^{4}+\cdots\)
1225.4.a.w	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{2}) \)	None	✓				245.4.b.b	$2$	$1$	\(6\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+6\beta q^{3}+q^{4}+18\beta q^{6}-21q^{8}+\cdots\)
1225.4.a.x	$1225$	$4$	1225.a	1.a	$1$	$2$	$2$	$72.277$	\(\Q(\sqrt{2}) \)	None	✓				35.4.e.b	$1$	$0$	\(6\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(1-3\beta )q^{3}+(3+6\beta )q^{4}+\cdots\)
1225.4.a.y	$1225$	$4$	1225.a	1.a	$1$	$3$	$3$	$72.277$	3.3.14360.1	None	✓				35.4.a.c	$1$	$1$	\(3\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+(4+\cdots)q^{4}+\cdots\)
1225.4.a.z	$1225$	$4$	1225.a	1.a	$1$	$4$	$4$	$72.277$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				175.4.a.g	$1$	$1$	\(-4\)	\(-3\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(9+\cdots)q^{4}+\cdots\)
1225.4.a.ba	$1225$	$4$	1225.a	1.a	$1$	$4$	$4$	$72.277$	4.4.23265040.2	None	✓	✓			1225.4.a.ba	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(2-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
1225.4.a.bb	$1225$	$4$	1225.a	1.a	$1$	$4$	$4$	$72.277$	\(\Q(\sqrt{2}, \sqrt{65})\)	None	✓				49.4.a.e	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)
1225.4.a.bc	$1225$	$4$	1225.a	1.a	$1$	$4$	$4$	$72.277$	4.4.23265040.2	None	✓	✓			1225.4.a.ba	$2$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(2-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1225.4.a.bd	$1225$	$4$	1225.a	1.a	$1$	$4$	$4$	$72.277$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				175.4.a.g	$1$	$0$	\(4\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.be	$1225$	$4$	1225.a	1.a	$1$	$5$	$5$	$72.277$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.4.b.a	$1$	$0$	\(-4\)	\(10\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bf	$1225$	$4$	1225.a	1.a	$1$	$5$	$5$	$72.277$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.4.e.c	$1$	$1$	\(-1\)	\(-8\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2+\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots\)
1225.4.a.bg	$1225$	$4$	1225.a	1.a	$1$	$5$	$5$	$72.277$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.4.e.c	$1$	$0$	\(-1\)	\(8\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots\)
1225.4.a.bh	$1225$	$4$	1225.a	1.a	$1$	$5$	$5$	$72.277$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.4.b.a	$1$	$0$	\(4\)	\(-10\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bi	$1225$	$4$	1225.a	1.a	$1$	$6$	$6$	$72.277$	6.6.1163891200.1	None	✓				245.4.a.o	$1$	$0$	\(2\)	\(-16\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-3+\beta _{5})q^{3}+(2+3\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bj	$1225$	$4$	1225.a	1.a	$1$	$6$	$6$	$72.277$	6.6.1163891200.1	None	✓				245.4.a.o	$1$	$0$	\(2\)	\(16\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3-\beta _{5})q^{3}+(2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bk	$1225$	$4$	1225.a	1.a	$1$	$8$	$8$	$72.277$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		175.4.e.e	$1$	$1$	\(-1\)	\(-6\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bl	$1225$	$4$	1225.a	1.a	$1$	$8$	$8$	$72.277$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				175.4.e.e	$1$	$0$	\(-1\)	\(6\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bm	$1225$	$4$	1225.a	1.a	$1$	$8$	$8$	$72.277$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				245.4.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{6}q^{2}-\beta _{4}q^{3}+(9-\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1225.4.a.bn	$1225$	$4$	1225.a	1.a	$1$	$8$	$8$	$72.277$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				175.4.e.e	$1$	$1$	\(1\)	\(-6\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bo	$1225$	$4$	1225.a	1.a	$1$	$8$	$8$	$72.277$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				175.4.e.e	$1$	$0$	\(1\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bp	$1225$	$4$	1225.a	1.a	$1$	$10$	$10$	$72.277$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				35.4.j.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(-5+\cdots)q^{6}+\cdots\)
1225.4.a.bq	$1225$	$4$	1225.a	1.a	$1$	$10$	$10$	$72.277$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		35.4.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(5+\beta _{2}+\cdots)q^{6}+\cdots\)
1225.4.a.br	$1225$	$4$	1225.a	1.a	$1$	$12$	$12$	$72.277$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			1225.4.a.br	$2$	$1$	\(-10\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{2}+\beta _{3}q^{3}+(5-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bs	$1225$	$4$	1225.a	1.a	$1$	$12$	$12$	$72.277$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				245.4.b.g	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{2}+\beta _{2}q^{3}+(4+\beta _{3})q^{4}+(\beta _{8}+\cdots)q^{6}+\cdots\)
1225.4.a.bt	$1225$	$4$	1225.a	1.a	$1$	$12$	$12$	$72.277$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		1225.4.a.br	$2$	$0$	\(10\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{2}+\beta _{3}q^{3}+(5-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1225))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1225)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)