Space of modular forms of level 425, weight 6, and trivial character

• Label: 425.6.a
• Level: $425$
• Weight: $6$
• Character orbit: 425.a
• Rep. character: $\chi_{425}(1,\cdot)$
• Character field: $\Q$
• Dimension: $126$
• Newform subspaces: $13$
• Sturm bound: $270$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(270\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(425))\).

	Total	New	Old
Modular forms	230	126	104
Cusp forms	218	126	92
Eisenstein series	12	0	12

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
\(+\)	\(+\)	\(+\)		\(54\)	\(27\)	\(27\)		\(51\)	\(27\)	\(24\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(60\)	\(33\)	\(27\)		\(57\)	\(33\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(61\)	\(35\)	\(26\)		\(58\)	\(35\)	\(23\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(55\)	\(31\)	\(24\)		\(52\)	\(31\)	\(21\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(109\)	\(58\)	\(51\)		\(103\)	\(58\)	\(45\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(121\)	\(68\)	\(53\)		\(115\)	\(68\)	\(47\)		\(6\)	\(0\)	\(6\)

Trace form

126 * q - 2 * q^2 + 16 * q^3 + 1962 * q^4 + 122 * q^6 + 268 * q^7 - 558 * q^8 + 10362 * q^9 - 1108 * q^11 + 742 * q^12 - 112 * q^13 + 2628 * q^14 + 34658 * q^16 + 578 * q^17 - 1362 * q^18 + 7176 * q^19 - 8388 * q^21 + 614 * q^22 + 5512 * q^23 + 4338 * q^24 + 7396 * q^26 + 8380 * q^27 + 2228 * q^28 + 14636 * q^29 - 22688 * q^31 + 622 * q^32 - 8828 * q^33 - 2312 * q^34 + 230002 * q^36 + 13944 * q^37 + 27296 * q^38 - 31368 * q^39 - 39120 * q^41 - 14476 * q^42 + 6892 * q^43 + 77018 * q^44 - 5968 * q^46 - 8708 * q^47 + 11426 * q^48 + 374994 * q^49 - 10404 * q^51 + 19860 * q^52 + 17260 * q^53 - 77180 * q^54 - 101216 * q^56 - 9600 * q^57 - 17214 * q^58 - 41800 * q^59 + 99312 * q^61 + 25648 * q^62 + 140288 * q^63 + 291010 * q^64 - 66372 * q^66 + 141908 * q^67 + 27744 * q^68 + 129576 * q^69 + 154156 * q^71 - 241606 * q^72 - 94848 * q^73 - 203098 * q^74 + 396208 * q^76 + 54164 * q^77 - 137568 * q^78 + 128304 * q^79 + 917294 * q^81 - 21532 * q^82 + 30300 * q^83 - 440076 * q^84 - 227404 * q^86 + 41240 * q^87 + 183430 * q^88 - 12132 * q^89 - 354480 * q^91 + 374316 * q^92 + 97572 * q^93 + 19480 * q^94 + 545238 * q^96 - 218260 * q^97 - 325182 * q^98 - 831864 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(425))\) 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
425.6.a.a	$425$	$6$	425.a	1.a	$1$	$1$	$1$	$68.163$	\(\Q\)	None	✓				17.6.a.b	$1$	$0$	\(-1\)	\(18\)	\(0\)	\(-28\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+18q^{3}-31q^{4}-18q^{6}-28q^{7}+\cdots\)
425.6.a.b	$425$	$6$	425.a	1.a	$1$	$1$	$1$	$68.163$	\(\Q\)	None	✓				17.6.a.a	$1$	$0$	\(6\)	\(-10\)	\(0\)	\(196\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}-10q^{3}+4q^{4}-60q^{6}+14^{2}q^{7}+\cdots\)
425.6.a.c	$425$	$6$	425.a	1.a	$1$	$4$	$4$	$68.163$	4.4.5416116.1	None	✓				17.6.a.c	$1$	$1$	\(-3\)	\(-28\)	\(0\)	\(-284\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-7-\beta _{1}+\beta _{3})q^{3}+\cdots\)
425.6.a.d	$425$	$6$	425.a	1.a	$1$	$5$	$5$	$68.163$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				85.6.a.a	$1$	$1$	\(7\)	\(36\)	\(0\)	\(204\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(8-\beta _{1}+\beta _{3})q^{3}+(8+\cdots)q^{4}+\cdots\)
425.6.a.e	$425$	$6$	425.a	1.a	$1$	$7$	$7$	$68.163$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				85.6.a.b	$1$	$1$	\(-9\)	\(-18\)	\(0\)	\(-50\)		$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-3-\beta _{2})q^{3}+(20+\cdots)q^{4}+\cdots\)
425.6.a.f	$425$	$6$	425.a	1.a	$1$	$8$	$8$	$68.163$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				85.6.a.d	$1$	$0$	\(-5\)	\(-18\)	\(0\)	\(46\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(24+\cdots)q^{4}+\cdots\)
425.6.a.g	$425$	$6$	425.a	1.a	$1$	$8$	$8$	$68.163$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				85.6.a.c	$1$	$0$	\(3\)	\(36\)	\(0\)	\(184\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{3})q^{3}+(23+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
425.6.a.h	$425$	$6$	425.a	1.a	$1$	$11$	$11$	$68.163$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓			425.6.a.h	$1$	$1$	\(-1\)	\(-9\)	\(0\)	\(-219\)		$-$	$-$	$+$	$2^{12}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(8+\beta _{2})q^{4}+\cdots\)
425.6.a.i	$425$	$6$	425.a	1.a	$1$	$11$	$11$	$68.163$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓		425.6.a.h	$1$	$1$	\(1\)	\(9\)	\(0\)	\(219\)		$+$	$+$	$+$	$2^{12}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(8+\beta _{2})q^{4}+\cdots\)
425.6.a.j	$425$	$6$	425.a	1.a	$1$	$15$	$15$	$68.163$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓			425.6.a.j	$1$	$0$	\(-1\)	\(-9\)	\(0\)	\(-181\)		$-$	$+$	$-$	$2^{9}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{6})q^{3}+(18+\beta _{1}+\cdots)q^{4}+\cdots\)
425.6.a.k	$425$	$6$	425.a	1.a	$1$	$15$	$15$	$68.163$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		425.6.a.j	$1$	$0$	\(1\)	\(9\)	\(0\)	\(181\)		$+$	$-$	$-$	$2^{9}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{6})q^{3}+(18+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
425.6.a.l	$425$	$6$	425.a	1.a	$1$	$20$	$20$	$68.163$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		85.6.b.a	$1$	$1$	\(-8\)	\(-36\)	\(0\)	\(-490\)		$-$	$-$	$+$	$2^{10}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2-\beta _{4})q^{3}+(2^{4}+\beta _{1}+\cdots)q^{4}+\cdots\)
425.6.a.m	$425$	$6$	425.a	1.a	$1$	$20$	$20$	$68.163$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		85.6.b.a	$1$	$0$	\(8\)	\(36\)	\(0\)	\(490\)		$-$	$+$	$-$	$2^{10}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{4})q^{3}+(2^{4}+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(425))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(425)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)