Space of modular forms of level 525, weight 8, and trivial character

• Label: 525.8.a
• Level: $525$
• Weight: $8$
• Character orbit: 525.a
• Rep. character: $\chi_{525}(1,\cdot)$
• Character field: $\Q$
• Dimension: $132$
• Newform subspaces: $25$
• Sturm bound: $640$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(640\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(525))\).

	Total	New	Old
Modular forms	572	132	440
Cusp forms	548	132	416
Eisenstein series	24	0	24

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

\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(74\)	\(15\)	\(59\)		\(71\)	\(15\)	\(56\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(71\)	\(17\)	\(54\)		\(68\)	\(17\)	\(51\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(70\)	\(17\)	\(53\)		\(67\)	\(17\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(71\)	\(17\)	\(54\)		\(68\)	\(17\)	\(51\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(69\)	\(14\)	\(55\)		\(66\)	\(14\)	\(52\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(72\)	\(18\)	\(54\)		\(69\)	\(18\)	\(51\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(73\)	\(19\)	\(54\)		\(70\)	\(19\)	\(51\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(72\)	\(15\)	\(57\)		\(69\)	\(15\)	\(54\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(290\)	\(69\)	\(221\)		\(278\)	\(69\)	\(209\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(282\)	\(63\)	\(219\)		\(270\)	\(63\)	\(207\)		\(12\)	\(0\)	\(12\)

Trace form

132 * q - 30 * q^2 + 8554 * q^4 + 108 * q^6 + 686 * q^7 - 4290 * q^8 + 96228 * q^9 + 1432 * q^11 + 15336 * q^12 - 31256 * q^13 - 8918 * q^14 + 537706 * q^16 + 59576 * q^17 - 21870 * q^18 + 52072 * q^19 - 18522 * q^21 - 18952 * q^22 + 61952 * q^23 + 54756 * q^24 + 495980 * q^26 + 107702 * q^28 - 329168 * q^29 - 239904 * q^31 - 322770 * q^32 - 272160 * q^33 - 936820 * q^34 + 6235866 * q^36 + 196424 * q^37 - 2153864 * q^38 + 30672 * q^39 - 1318880 * q^41 + 148176 * q^42 + 389280 * q^43 + 2461516 * q^44 + 1204308 * q^46 - 387952 * q^47 + 2041200 * q^48 + 15529668 * q^49 - 3098520 * q^51 - 7948716 * q^52 + 270240 * q^53 + 78732 * q^54 - 6126666 * q^56 - 723384 * q^57 - 978020 * q^58 - 555176 * q^59 + 8799848 * q^61 + 5610552 * q^62 + 500094 * q^63 + 32321982 * q^64 - 6138936 * q^66 + 7113120 * q^67 + 2514644 * q^68 + 5280336 * q^69 + 5161496 * q^71 - 3127410 * q^72 + 14782632 * q^73 + 3370648 * q^74 + 4949464 * q^76 + 9796080 * q^77 + 12170952 * q^78 - 6972640 * q^79 + 70150212 * q^81 - 19477284 * q^82 - 19789984 * q^83 - 3556224 * q^84 + 56359252 * q^86 - 4626720 * q^87 + 16803784 * q^88 + 62384552 * q^89 - 2169132 * q^91 + 7863008 * q^92 + 16251624 * q^93 + 49785552 * q^94 + 11293236 * q^96 - 5169016 * q^97 - 3529470 * q^98 + 1043928 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(525))\) 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	7
525.8.a.a	$525$	$8$	525.a	1.a	$1$	$1$	$1$	$164.002$	\(\Q\)	None	✓				105.8.a.b	$1$	$1$	\(-18\)	\(27\)	\(0\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-18q^{2}+3^{3}q^{3}+14^{2}q^{4}-486q^{6}+\cdots\)
525.8.a.b	$525$	$8$	525.a	1.a	$1$	$1$	$1$	$164.002$	\(\Q\)	None	✓				21.8.a.a	$1$	$1$	\(-2\)	\(-27\)	\(0\)	\(343\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3^{3}q^{3}-124q^{4}+54q^{6}+\cdots\)
525.8.a.c	$525$	$8$	525.a	1.a	$1$	$1$	$1$	$164.002$	\(\Q\)	None	✓				105.8.a.a	$1$	$1$	\(2\)	\(27\)	\(0\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3^{3}q^{3}-124q^{4}+54q^{6}+\cdots\)
525.8.a.d	$525$	$8$	525.a	1.a	$1$	$2$	$2$	$164.002$	\(\Q(\sqrt{67}) \)	None	✓				21.8.a.c	$1$	$0$	\(-12\)	\(54\)	\(0\)	\(686\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-6+\beta )q^{2}+3^{3}q^{3}+(176-12\beta )q^{4}+\cdots\)
525.8.a.e	$525$	$8$	525.a	1.a	$1$	$2$	$2$	$164.002$	\(\Q(\sqrt{1065}) \)	None	✓				21.8.a.b	$1$	$1$	\(9\)	\(54\)	\(0\)	\(-686\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}+3^{3}q^{3}+(163-9\beta )q^{4}+\cdots\)
525.8.a.f	$525$	$8$	525.a	1.a	$1$	$2$	$2$	$164.002$	\(\Q(\sqrt{2}) \)	None	✓				105.8.a.c	$1$	$0$	\(12\)	\(-54\)	\(0\)	\(-686\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(6+2\beta )q^{2}-3^{3}q^{3}+(6^{2}+24\beta )q^{4}+\cdots\)
525.8.a.g	$525$	$8$	525.a	1.a	$1$	$3$	$3$	$164.002$	3.3.2910828.1	None	✓				21.8.a.d	$1$	$0$	\(3\)	\(-81\)	\(0\)	\(-1029\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3^{3}q^{3}+(75+\beta _{2})q^{4}+\cdots\)
525.8.a.h	$525$	$8$	525.a	1.a	$1$	$4$	$4$	$164.002$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.i	$1$	$0$	\(-25\)	\(-108\)	\(0\)	\(-1372\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-6-\beta _{1})q^{2}-3^{3}q^{3}+(2^{5}+12\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.i	$525$	$8$	525.a	1.a	$1$	$4$	$4$	$164.002$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.h	$1$	$1$	\(-11\)	\(108\)	\(0\)	\(-1372\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+3^{3}q^{3}+(6^{2}+5\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.j	$525$	$8$	525.a	1.a	$1$	$4$	$4$	$164.002$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.g	$1$	$1$	\(-4\)	\(-108\)	\(0\)	\(1372\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}-3^{3}q^{3}+(26+\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.k	$525$	$8$	525.a	1.a	$1$	$4$	$4$	$164.002$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.f	$1$	$1$	\(1\)	\(-108\)	\(0\)	\(1372\)		$+$	$+$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3^{3}q^{3}+(105+\beta _{2})q^{4}-3^{3}\beta _{1}q^{6}+\cdots\)
525.8.a.l	$525$	$8$	525.a	1.a	$1$	$4$	$4$	$164.002$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.e	$1$	$0$	\(5\)	\(108\)	\(0\)	\(1372\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+3^{3}q^{3}+(105+2\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.m	$525$	$8$	525.a	1.a	$1$	$4$	$4$	$164.002$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.d	$1$	$0$	\(10\)	\(108\)	\(0\)	\(1372\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+3^{3}q^{3}+(29-4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.n	$525$	$8$	525.a	1.a	$1$	$6$	$6$	$164.002$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			525.8.a.n	$1$	$1$	\(-17\)	\(162\)	\(0\)	\(2058\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+3^{3}q^{3}+(66-3\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.o	$525$	$8$	525.a	1.a	$1$	$6$	$6$	$164.002$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			525.8.a.o	$1$	$0$	\(-3\)	\(-162\)	\(0\)	\(2058\)		$+$	$-$	$-$	$+$	$2^{4}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3^{3}q^{3}+(66+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.p	$525$	$8$	525.a	1.a	$1$	$6$	$6$	$164.002$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		525.8.a.o	$1$	$1$	\(3\)	\(162\)	\(0\)	\(-2058\)		$-$	$+$	$+$	$-$	$2^{4}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3^{3}q^{3}+(66+\beta _{2})q^{4}+\cdots\)
525.8.a.q	$525$	$8$	525.a	1.a	$1$	$6$	$6$	$164.002$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		525.8.a.n	$1$	$0$	\(17\)	\(-162\)	\(0\)	\(-2058\)		$+$	$+$	$+$	$+$	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}-3^{3}q^{3}+(66-3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.r	$525$	$8$	525.a	1.a	$1$	$8$	$8$	$164.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		525.8.a.r	$1$	$1$	\(-12\)	\(-216\)	\(0\)	\(2744\)		$+$	$+$	$-$	$-$	$2^{4}\cdot 3^{4}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}-3^{3}q^{3}+(78-\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.s	$525$	$8$	525.a	1.a	$1$	$8$	$8$	$164.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			525.8.a.s	$1$	$1$	\(-6\)	\(-216\)	\(0\)	\(-2744\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3^{3}q^{3}+(78-2\beta _{1}+\cdots)q^{4}+\cdots\)
525.8.a.t	$525$	$8$	525.a	1.a	$1$	$8$	$8$	$164.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		525.8.a.s	$1$	$0$	\(6\)	\(216\)	\(0\)	\(2744\)		$-$	$+$	$-$	$+$	$2^{4}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3^{3}q^{3}+(78-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.u	$525$	$8$	525.a	1.a	$1$	$8$	$8$	$164.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			525.8.a.r	$1$	$0$	\(12\)	\(216\)	\(0\)	\(-2744\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 3^{4}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+3^{3}q^{3}+(78-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.v	$525$	$8$	525.a	1.a	$1$	$9$	$9$	$164.002$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		105.8.d.a	$1$	$1$	\(-1\)	\(243\)	\(0\)	\(3087\)		$-$	$-$	$-$	$-$	$2^{10}\cdot 3^{3}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3^{3}q^{3}+(41+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.w	$525$	$8$	525.a	1.a	$1$	$9$	$9$	$164.002$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		105.8.d.a	$1$	$1$	\(1\)	\(-243\)	\(0\)	\(-3087\)		$+$	$-$	$+$	$-$	$2^{10}\cdot 3^{3}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3^{3}q^{3}+(41+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.8.a.x	$525$	$8$	525.a	1.a	$1$	$11$	$11$	$164.002$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		105.8.d.b	$1$	$0$	\(-1\)	\(297\)	\(0\)	\(-3773\)		$-$	$-$	$+$	$+$	$2^{20}\cdot 3^{2}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3^{3}q^{3}+(69-\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.8.a.y	$525$	$8$	525.a	1.a	$1$	$11$	$11$	$164.002$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		105.8.d.b	$1$	$0$	\(1\)	\(-297\)	\(0\)	\(3773\)		$+$	$-$	$-$	$+$	$2^{20}\cdot 3^{2}\cdot 5^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3^{3}q^{3}+(69-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(525)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)