Space of modular forms of level 4225, weight 2, and trivial character

• Label: 4225.2.a
• Level: $4225$
• Weight: $2$
• Character orbit: 4225.a
• Rep. character: $\chi_{4225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $229$
• Newform subspaces: $54$
• Sturm bound: $910$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 54 \)
Sturm bound:			\(910\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4225))\).

	Total	New	Old
Modular forms	496	262	234
Cusp forms	413	229	184
Eisenstein series	83	33	50

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

\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(119\)	\(59\)	\(60\)		\(99\)	\(54\)	\(45\)		\(20\)	\(5\)	\(15\)
\(+\)	\(-\)	\(-\)		\(128\)	\(63\)	\(65\)		\(107\)	\(57\)	\(50\)		\(21\)	\(6\)	\(15\)
\(-\)	\(+\)	\(-\)		\(126\)	\(72\)	\(54\)		\(105\)	\(62\)	\(43\)		\(21\)	\(10\)	\(11\)
\(-\)	\(-\)	\(+\)		\(123\)	\(68\)	\(55\)		\(102\)	\(56\)	\(46\)		\(21\)	\(12\)	\(9\)
Plus space		\(+\)		\(242\)	\(127\)	\(115\)		\(201\)	\(110\)	\(91\)		\(41\)	\(17\)	\(24\)
Minus space		\(-\)		\(254\)	\(135\)	\(119\)		\(212\)	\(119\)	\(93\)		\(42\)	\(16\)	\(26\)

Trace form

229 * q - 3 * q^2 + 213 * q^4 + 8 * q^6 + 4 * q^7 - 3 * q^8 + 193 * q^9 - 8 * q^12 + 10 * q^14 + 173 * q^16 - 11 * q^18 + 4 * q^19 + 20 * q^21 - 6 * q^22 - 2 * q^23 + 28 * q^24 + 18 * q^27 + 28 * q^28 + 6 * q^29 + 24 * q^31 + q^32 - 20 * q^33 + 26 * q^34 + 171 * q^36 - 10 * q^37 + 36 * q^38 - 10 * q^41 + 2 * q^43 - 32 * q^44 + 4 * q^46 + 12 * q^47 + 30 * q^48 + 147 * q^49 - 58 * q^51 + 6 * q^53 + 4 * q^54 + 12 * q^56 - 4 * q^57 + 2 * q^58 - 20 * q^59 + 24 * q^61 - 6 * q^62 - 4 * q^63 + 119 * q^64 + 26 * q^66 - 16 * q^67 + 70 * q^68 - 12 * q^71 + 21 * q^72 - 14 * q^73 + 32 * q^74 + 60 * q^76 - 4 * q^77 + 22 * q^79 + 53 * q^81 + 28 * q^82 - 40 * q^83 + 64 * q^84 - 16 * q^86 + 44 * q^87 - 34 * q^88 - 2 * q^89 + 40 * q^92 + 60 * q^93 - 30 * q^94 + 36 * q^96 - 2 * q^97 - 51 * q^98 - 80 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\) 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	13
4225.2.a.a	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.c	$1$	$0$	\(-2\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
4225.2.a.b	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.a.a	$1$	$0$	\(-2\)	\(1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
4225.2.a.c	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.c	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
4225.2.a.d	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.a	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}-q^{4}+2q^{6}-5q^{7}+3q^{8}+\cdots\)
4225.2.a.e	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				65.2.d.a	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}-q^{4}+2q^{6}+3q^{8}+q^{9}+\cdots\)
4225.2.a.f	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.a	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}-5q^{7}+3q^{8}+\cdots\)
4225.2.a.g	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				65.2.a.a	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}-4q^{7}+3q^{8}+\cdots\)
4225.2.a.h	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				65.2.d.a	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}+3q^{8}+q^{9}+\cdots\)
4225.2.a.i	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-4q^{7}-2q^{9}+6q^{11}+\cdots\)
4225.2.a.j	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+4q^{7}-2q^{9}+6q^{11}+\cdots\)
4225.2.a.k	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				65.2.d.a	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}-2q^{6}-3q^{8}+q^{9}+\cdots\)
4225.2.a.l	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.a	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}-2q^{6}+5q^{7}-3q^{8}+\cdots\)
4225.2.a.m	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				65.2.d.a	$1$	$1$	\(1\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}+2q^{6}-3q^{8}+q^{9}+\cdots\)
4225.2.a.n	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.a	$1$	$0$	\(1\)	\(2\)	\(0\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}+2q^{6}+5q^{7}-3q^{8}+\cdots\)
4225.2.a.o	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.c	$1$	$0$	\(2\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
4225.2.a.p	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.a.a	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
4225.2.a.q	$4225$	$2$	4225.a	1.a	$1$	$1$	$1$	$33.737$	\(\Q\)	None	✓				325.2.c.c	$1$	$1$	\(2\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-2q^{7}+\cdots\)
4225.2.a.r	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{2}) \)	None	✓				65.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
4225.2.a.s	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{2}) \)	None	✓				325.2.a.f	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
4225.2.a.t	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{13}) \)	None	✓				65.2.e.b	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
4225.2.a.u	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{5}) \)	None	✓				65.2.e.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
4225.2.a.v	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{3}) \)	None	✓				13.2.e.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-2q^{3}+q^{4}-2\beta q^{6}-\beta q^{8}+\cdots\)
4225.2.a.w	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{3}) \)	None	✓				65.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{3}+q^{4}+(3-\beta )q^{6}+\cdots\)
4225.2.a.x	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{13}) \)	None	✓				65.2.e.b	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+q^{7}+\cdots\)
4225.2.a.y	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{5}) \)	None	✓				65.2.e.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
4225.2.a.z	$4225$	$2$	4225.a	1.a	$1$	$2$	$2$	$33.737$	\(\Q(\sqrt{2}) \)	None	✓				325.2.a.f	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+2\beta q^{3}+(1+2\beta )q^{4}+(4+\cdots)q^{6}+\cdots\)
4225.2.a.ba	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	3.3.148.1	None	✓				65.2.b.a	$1$	$0$	\(-3\)	\(4\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bb	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	\(\Q(\zeta_{14})^+\)	None	✓				169.2.a.b	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bc	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	3.3.564.1	None	✓				65.2.c.a	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4225.2.a.bd	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	\(\Q(\zeta_{14})^+\)	None	✓				845.2.a.h	$1$	$1$	\(-1\)	\(5\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
4225.2.a.be	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	3.3.564.1	None	✓				65.2.c.a	$1$	$0$	\(1\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4225.2.a.bf	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	\(\Q(\zeta_{14})^+\)	None	✓				845.2.a.h	$1$	$0$	\(1\)	\(5\)	\(0\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
4225.2.a.bg	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	\(\Q(\zeta_{14})^+\)	None	✓				169.2.a.b	$1$	$1$	\(2\)	\(2\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bh	$4225$	$2$	4225.a	1.a	$1$	$3$	$3$	$33.737$	3.3.148.1	None	✓				65.2.b.a	$1$	$0$	\(3\)	\(-4\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bi	$4225$	$2$	4225.a	1.a	$1$	$4$	$4$	$33.737$	4.4.4752.1	None	✓				65.2.m.a	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
4225.2.a.bj	$4225$	$2$	4225.a	1.a	$1$	$4$	$4$	$33.737$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓				65.2.l.a	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4225.2.a.bk	$4225$	$2$	4225.a	1.a	$1$	$4$	$4$	$33.737$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓				65.2.l.a	$2$	$1$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4225.2.a.bl	$4225$	$2$	4225.a	1.a	$1$	$4$	$4$	$33.737$	4.4.4752.1	None	✓				65.2.m.a	$1$	$0$	\(2\)	\(2\)	\(0\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
4225.2.a.bm	$4225$	$2$	4225.a	1.a	$1$	$5$	$5$	$33.737$	5.5.1068321.1	None	✓				325.2.e.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bn	$4225$	$2$	4225.a	1.a	$1$	$5$	$5$	$33.737$	5.5.1068321.1	None	✓				325.2.e.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bo	$4225$	$2$	4225.a	1.a	$1$	$5$	$5$	$33.737$	5.5.1068321.1	None	✓				325.2.e.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bp	$4225$	$2$	4225.a	1.a	$1$	$5$	$5$	$33.737$	5.5.1068321.1	None	✓				325.2.e.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bq	$4225$	$2$	4225.a	1.a	$1$	$6$	$6$	$33.737$	6.6.199374400.1	None	✓				65.2.n.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.br	$4225$	$2$	4225.a	1.a	$1$	$6$	$6$	$33.737$	6.6.199374400.1	None	✓				65.2.n.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{5})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bs	$4225$	$2$	4225.a	1.a	$1$	$9$	$9$	$33.737$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				845.2.a.n	$1$	$1$	\(-3\)	\(-7\)	\(0\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots\)
4225.2.a.bt	$4225$	$2$	4225.a	1.a	$1$	$9$	$9$	$33.737$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				845.2.a.n	$1$	$0$	\(3\)	\(-7\)	\(0\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)
4225.2.a.bu	$4225$	$2$	4225.a	1.a	$1$	$10$	$10$	$33.737$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				325.2.n.e	$2$	$1$	\(0\)	\(-6\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bv	$4225$	$2$	4225.a	1.a	$1$	$10$	$10$	$33.737$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				325.2.n.e	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bw	$4225$	$2$	4225.a	1.a	$1$	$12$	$12$	$33.737$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			4225.2.a.bw	$1$	$1$	\(-7\)	\(-1\)	\(0\)	\(-12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.bx	$4225$	$2$	4225.a	1.a	$1$	$12$	$12$	$33.737$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		4225.2.a.bw	$1$	$1$	\(-7\)	\(1\)	\(0\)	\(-12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.by	$4225$	$2$	4225.a	1.a	$1$	$12$	$12$	$33.737$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			4225.2.a.bw	$1$	$0$	\(7\)	\(-1\)	\(0\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.bz	$4225$	$2$	4225.a	1.a	$1$	$12$	$12$	$33.737$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		4225.2.a.bw	$1$	$0$	\(7\)	\(1\)	\(0\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.ca	$4225$	$2$	4225.a	1.a	$1$	$18$	$18$	$33.737$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓		✓		845.2.b.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
4225.2.a.cb	$4225$	$2$	4225.a	1.a	$1$	$18$	$18$	$33.737$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓		✓		845.2.b.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4225)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 2}\)