Space of modular forms of level 320, weight 10, and trivial character

• Label: 320.10.a
• Level: $320$
• Weight: $10$
• Character orbit: 320.a
• Rep. character: $\chi_{320}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $30$
• Sturm bound: $480$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(480\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(320))\).

	Total	New	Old
Modular forms	444	72	372
Cusp forms	420	72	348
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.

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(17\)
\(+\)	\(-\)	$-$	\(19\)
\(-\)	\(+\)	$-$	\(19\)
\(-\)	\(-\)	$+$	\(17\)
Plus space		\(+\)	\(34\)
Minus space		\(-\)	\(38\)

Trace form

\( 72 q + 472392 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
320.10.a.a	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-204\)	\(-625\)	\(-5432\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-204q^{3}-5^{4}q^{5}-5432q^{7}+21933q^{9}+\cdots\)
320.10.a.b	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-174\)	\(625\)	\(4658\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-174q^{3}+5^{4}q^{5}+4658q^{7}+10593q^{9}+\cdots\)
320.10.a.c	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-114\)	\(625\)	\(-4242\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-114q^{3}+5^{4}q^{5}-4242q^{7}-6687q^{9}+\cdots\)
320.10.a.d	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-48\)	\(-625\)	\(532\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-48q^{3}-5^{4}q^{5}+532q^{7}-17379q^{9}+\cdots\)
320.10.a.e	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-46\)	\(625\)	\(-10318\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-46q^{3}+5^{4}q^{5}-10318q^{7}-17567q^{9}+\cdots\)
320.10.a.f	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(46\)	\(625\)	\(10318\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+46q^{3}+5^{4}q^{5}+10318q^{7}-17567q^{9}+\cdots\)
320.10.a.g	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(48\)	\(-625\)	\(-532\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+48q^{3}-5^{4}q^{5}-532q^{7}-17379q^{9}+\cdots\)
320.10.a.h	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(114\)	\(625\)	\(4242\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+114q^{3}+5^{4}q^{5}+4242q^{7}-6687q^{9}+\cdots\)
320.10.a.i	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(174\)	\(625\)	\(-4658\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+174q^{3}+5^{4}q^{5}-4658q^{7}+10593q^{9}+\cdots\)
320.10.a.j	$320$	$10$	320.a	1.a	$1$	$1$	$1$	$164.811$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(204\)	\(-625\)	\(5432\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+204q^{3}-5^{4}q^{5}+5432q^{7}+21933q^{9}+\cdots\)
320.10.a.k	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$1$	\(0\)	\(-260\)	\(-1250\)	\(1700\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-130-\beta )q^{3}-5^{4}q^{5}+(850+107\beta )q^{7}+\cdots\)
320.10.a.l	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{79}) \)	None	✓	$1$	$1$	\(0\)	\(-260\)	\(1250\)	\(380\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-130+\beta )q^{3}+5^{4}q^{5}+(190-69\beta )q^{7}+\cdots\)
320.10.a.m	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$1$	\(0\)	\(-116\)	\(1250\)	\(-11284\)		$-$	$-$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-58+\beta )q^{3}+5^{4}q^{5}+(-5642+\cdots)q^{7}+\cdots\)
320.10.a.n	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(-108\)	\(1250\)	\(-908\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-54+\beta )q^{3}+5^{4}q^{5}+(-454-13\beta )q^{7}+\cdots\)
320.10.a.o	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{6049}) \)	None	✓	$1$	$0$	\(0\)	\(-92\)	\(-1250\)	\(6908\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-46-\beta )q^{3}-5^{4}q^{5}+(3454-37\beta )q^{7}+\cdots\)
320.10.a.p	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{6049}) \)	None	✓	$1$	$1$	\(0\)	\(92\)	\(-1250\)	\(-6908\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(46-\beta )q^{3}-5^{4}q^{5}+(-3454-37\beta )q^{7}+\cdots\)
320.10.a.q	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$1$	\(0\)	\(108\)	\(1250\)	\(908\)		$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(54+\beta )q^{3}+5^{4}q^{5}+(454-13\beta )q^{7}+\cdots\)
320.10.a.r	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$0$	\(0\)	\(116\)	\(1250\)	\(11284\)		$+$	$-$	$-$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(58+\beta )q^{3}+5^{4}q^{5}+(5642+35\beta )q^{7}+\cdots\)
320.10.a.s	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$0$	\(0\)	\(260\)	\(-1250\)	\(-1700\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(130-\beta )q^{3}-5^{4}q^{5}+(-850+107\beta )q^{7}+\cdots\)
320.10.a.t	$320$	$10$	320.a	1.a	$1$	$2$	$2$	$164.811$	\(\Q(\sqrt{79}) \)	None	✓	$1$	$0$	\(0\)	\(260\)	\(1250\)	\(-380\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(130+\beta )q^{3}+5^{4}q^{5}+(-190-69\beta )q^{7}+\cdots\)
320.10.a.u	$320$	$10$	320.a	1.a	$1$	$3$	$3$	$164.811$	3.3.7117.1	None	✓	$1$	$1$	\(0\)	\(-84\)	\(-1875\)	\(-5520\)		$+$	$+$	$+$	$2^{9}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-28-\beta _{1})q^{3}-5^{4}q^{5}+(-1840+\cdots)q^{7}+\cdots\)
320.10.a.v	$320$	$10$	320.a	1.a	$1$	$3$	$3$	$164.811$	3.3.7117.1	None	✓	$1$	$0$	\(0\)	\(84\)	\(-1875\)	\(5520\)		$-$	$+$	$-$	$2^{9}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(28+\beta _{1})q^{3}-5^{4}q^{5}+(1840-17\beta _{1}+\cdots)q^{7}+\cdots\)
320.10.a.w	$320$	$10$	320.a	1.a	$1$	$4$	$4$	$164.811$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-176\)	\(-2500\)	\(1392\)		$+$	$+$	$+$	$2^{13}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-44-\beta _{1})q^{3}-5^{4}q^{5}+(348-14\beta _{1}+\cdots)q^{7}+\cdots\)
320.10.a.x	$320$	$10$	320.a	1.a	$1$	$4$	$4$	$164.811$	\(\Q(\sqrt{7}, \sqrt{418})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-2500\)	\(0\)		$-$	$+$	$-$	$2^{11}\cdot 5\cdot 13$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5^{4}q^{5}+(8\beta _{1}-9\beta _{2})q^{7}+\cdots\)
320.10.a.y	$320$	$10$	320.a	1.a	$1$	$4$	$4$	$164.811$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(2500\)	\(0\)		$-$	$-$	$+$	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5^{4}q^{5}+(14\beta _{1}+7\beta _{2})q^{7}+\cdots\)
320.10.a.z	$320$	$10$	320.a	1.a	$1$	$4$	$4$	$164.811$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(2500\)	\(0\)		$-$	$-$	$+$	$2^{12}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5^{4}q^{5}+(-34\beta _{1}+\beta _{2})q^{7}+\cdots\)
320.10.a.ba	$320$	$10$	320.a	1.a	$1$	$4$	$4$	$164.811$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(176\)	\(-2500\)	\(-1392\)		$+$	$+$	$+$	$2^{13}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(44+\beta _{1})q^{3}-5^{4}q^{5}+(-348+14\beta _{1}+\cdots)q^{7}+\cdots\)
320.10.a.bb	$320$	$10$	320.a	1.a	$1$	$5$	$5$	$164.811$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-14\)	\(3125\)	\(-3410\)		$+$	$-$	$-$	$2^{23}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+5^{4}q^{5}+(-683+3\beta _{1}+\cdots)q^{7}+\cdots\)
320.10.a.bc	$320$	$10$	320.a	1.a	$1$	$5$	$5$	$164.811$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(14\)	\(3125\)	\(3410\)		$+$	$-$	$-$	$2^{23}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{3}+5^{4}q^{5}+(683-3\beta _{1}+\cdots)q^{7}+\cdots\)
320.10.a.bd	$320$	$10$	320.a	1.a	$1$	$6$	$6$	$164.811$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-3750\)	\(0\)		$-$	$+$	$-$	$2^{33}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5^{4}q^{5}+(-3\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(320))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(320)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 7}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)