Space of modular forms of level 425, weight 2, and trivial character

• Label: 425.2.a
• Level: $425$
• Weight: $2$
• Character orbit: 425.a
• Rep. character: $\chi_{425}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $10$
• Sturm bound: $90$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(90\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(425))\).

	Total	New	Old
Modular forms	50	26	24
Cusp forms	39	26	13
Eisenstein series	11	0	11

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

\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(3\)	\(6\)		\(7\)	\(3\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(15\)	\(9\)	\(6\)		\(12\)	\(9\)	\(3\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(16\)	\(9\)	\(7\)		\(13\)	\(9\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(10\)	\(5\)	\(5\)		\(7\)	\(5\)	\(2\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(19\)	\(8\)	\(11\)		\(14\)	\(8\)	\(6\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(31\)	\(18\)	\(13\)		\(25\)	\(18\)	\(7\)		\(6\)	\(0\)	\(6\)

Trace form

26 * q + 2 * q^2 + 30 * q^4 + 4 * q^7 + 6 * q^8 + 22 * q^9 - 4 * q^11 - 4 * q^12 + 8 * q^13 - 16 * q^14 + 38 * q^16 + 2 * q^17 - 2 * q^18 + 8 * q^19 + 4 * q^21 - 8 * q^22 - 36 * q^24 - 16 * q^26 + 12 * q^27 + 16 * q^28 - 8 * q^29 + 8 * q^31 - 6 * q^32 - 28 * q^33 - 2 * q^34 + 26 * q^36 + 12 * q^37 - 8 * q^38 + 8 * q^39 - 24 * q^41 - 4 * q^42 - 4 * q^43 + 20 * q^44 - 20 * q^47 + 24 * q^48 + 22 * q^49 - 4 * q^51 - 8 * q^52 - 20 * q^53 - 40 * q^54 - 44 * q^56 + 4 * q^58 + 8 * q^59 + 12 * q^61 - 4 * q^62 + 24 * q^63 + 54 * q^64 - 48 * q^66 + 20 * q^67 + 6 * q^68 + 8 * q^69 + 36 * q^71 + 2 * q^72 + 32 * q^73 - 48 * q^74 - 16 * q^76 - 4 * q^77 - 36 * q^78 + 40 * q^79 - 30 * q^81 - 20 * q^83 - 68 * q^84 - 24 * q^86 + 8 * q^87 - 28 * q^88 + 4 * q^89 + 48 * q^91 + 12 * q^92 + 28 * q^93 - 48 * q^94 - 152 * q^96 - 4 * q^97 + 38 * q^98 - 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(425))\) 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	17
425.2.a.a	$425$	$2$	425.a	1.a	$1$	$1$	$1$	$3.394$	\(\Q\)	None	✓				85.2.a.a	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}-q^{4}+2q^{6}+2q^{7}+3q^{8}+\cdots\)
425.2.a.b	$425$	$2$	425.a	1.a	$1$	$1$	$1$	$3.394$	\(\Q\)	None	✓	✓			425.2.a.b	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}-q^{7}+3q^{8}+\cdots\)
425.2.a.c	$425$	$2$	425.a	1.a	$1$	$1$	$1$	$3.394$	\(\Q\)	None	✓	✓			425.2.a.b	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
425.2.a.d	$425$	$2$	425.a	1.a	$1$	$1$	$1$	$3.394$	\(\Q\)	None	✓				17.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{7}-3q^{8}-3q^{9}+2q^{13}+\cdots\)
425.2.a.e	$425$	$2$	425.a	1.a	$1$	$2$	$2$	$3.394$	\(\Q(\sqrt{3}) \)	None	✓				85.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1-\beta )q^{3}+q^{4}+(-3-\beta )q^{6}+\cdots\)
425.2.a.f	$425$	$2$	425.a	1.a	$1$	$2$	$2$	$3.394$	\(\Q(\sqrt{2}) \)	None	✓				85.2.a.b	$1$	$0$	\(2\)	\(4\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(2-\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)
425.2.a.g	$425$	$2$	425.a	1.a	$1$	$4$	$4$	$3.394$	4.4.6224.1	None	✓		✓		85.2.b.a	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+(-1+\beta _{1})q^{3}+(1+\cdots)q^{4}+\cdots\)
425.2.a.h	$425$	$2$	425.a	1.a	$1$	$4$	$4$	$3.394$	4.4.6224.1	None	✓				85.2.b.a	$1$	$0$	\(2\)	\(4\)	\(0\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+(1-\beta _{1})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
425.2.a.i	$425$	$2$	425.a	1.a	$1$	$5$	$5$	$3.394$	5.5.1893456.1	None	✓	✓	✓		425.2.a.i	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(2+\beta _{1}+\beta _{2}+\beta _{4})q^{4}+\cdots\)
425.2.a.j	$425$	$2$	425.a	1.a	$1$	$5$	$5$	$3.394$	5.5.1893456.1	None	✓	✓	✓		425.2.a.i	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(2+\beta _{1}+\beta _{2}+\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(425))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(425)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)