Space of modular forms of level 64, weight 22, and trivial character

• Label: 64.22.a
• Level: $64$
• Weight: $22$
• Character orbit: 64.a
• Rep. character: $\chi_{64}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $17$
• Sturm bound: $176$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(176\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(64))\).

	Total	New	Old
Modular forms	174	43	131
Cusp forms	162	41	121
Eisenstein series	12	2	10

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

\(2\)	Dim
\(+\)	\(20\)
\(-\)	\(21\)

Trace form

41 * q + 2 * q^5 + 135984591637 * q^9 - 1065395863654 * q^13 + 3481428994002 * q^17 + 132761067402368 * q^21 + 3807066340697127 * q^25 - 2433173450216598 * q^29 - 10718205122066112 * q^33 - 17830227570674174 * q^37 + 2550812783630522 * q^41 + 827402995178917914 * q^45 + 2792729320416420033 * q^49 + 2140956801021598770 * q^53 - 3378567006828484416 * q^57 - 7508615876911824022 * q^61 - 6125468142667127532 * q^65 - 35904133933074186880 * q^69 - 5771745784109926806 * q^73 + 214596272327840298368 * q^77 + 418629249161332174801 * q^81 - 65877839254108777212 * q^85 + 359902412185241718330 * q^89 - 1367800221219678191104 * q^93 - 200185457171576537182 * q^97

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(64))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
64.22.a.a	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	None	✓				1.22.a.a	$1$	$0$	\(0\)	\(-128844\)	\(-21640950\)	\(768078808\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-128844q^{3}-21640950q^{5}+768078808q^{7}+\cdots\)
64.22.a.b	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	None	✓				2.22.a.a	$1$	$1$	\(0\)	\(-71604\)	\(28693770\)	\(-853202392\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-71604q^{3}+28693770q^{5}-853202392q^{7}+\cdots\)
64.22.a.c	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	None	✓				2.22.a.b	$1$	$1$	\(0\)	\(-59316\)	\(-4975350\)	\(1427425832\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-59316q^{3}-4975350q^{5}+1427425832q^{7}+\cdots\)
64.22.a.d	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.22.a.a	$2$	$0$	\(0\)	\(0\)	\(-13398638\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-13398638q^{5}-3^{21}q^{9}-970515066006q^{13}+\cdots\)
64.22.a.e	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	None	✓				2.22.a.b	$1$	$0$	\(0\)	\(59316\)	\(-4975350\)	\(-1427425832\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+59316q^{3}-4975350q^{5}-1427425832q^{7}+\cdots\)
64.22.a.f	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	None	✓				2.22.a.a	$1$	$0$	\(0\)	\(71604\)	\(28693770\)	\(853202392\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+71604q^{3}+28693770q^{5}+853202392q^{7}+\cdots\)
64.22.a.g	$64$	$22$	64.a	1.a	$1$	$1$	$1$	$178.866$	\(\Q\)	None	✓				1.22.a.a	$1$	$1$	\(0\)	\(128844\)	\(-21640950\)	\(-768078808\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+128844q^{3}-21640950q^{5}-768078808q^{7}+\cdots\)
64.22.a.h	$64$	$22$	64.a	1.a	$1$	$2$	$2$	$178.866$	\(\Q(\sqrt{358549}) \)	None	✓				8.22.a.a	$1$	$0$	\(0\)	\(-105432\)	\(-2108140\)	\(-444771792\)		$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-52716-\beta )q^{3}+(-1054070+\cdots)q^{5}+\cdots\)
64.22.a.i	$64$	$22$	64.a	1.a	$1$	$2$	$2$	$178.866$	\(\Q(\sqrt{2161}) \)	None	✓				4.22.a.a	$1$	$1$	\(0\)	\(-65640\)	\(-13689324\)	\(-260508080\)		$+$	$+$	$2^{8}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+(-32820-\beta )q^{3}+(-6844662+\cdots)q^{5}+\cdots\)
64.22.a.j	$64$	$22$	64.a	1.a	$1$	$2$	$2$	$178.866$	\(\Q(\sqrt{2161}) \)	None	✓				4.22.a.a	$1$	$0$	\(0\)	\(65640\)	\(-13689324\)	\(260508080\)		$-$	$-$	$2^{8}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+(32820-\beta )q^{3}+(-6844662+204\beta )q^{5}+\cdots\)
64.22.a.k	$64$	$22$	64.a	1.a	$1$	$2$	$2$	$178.866$	\(\Q(\sqrt{358549}) \)	None	✓				8.22.a.a	$1$	$1$	\(0\)	\(105432\)	\(-2108140\)	\(444771792\)		$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(52716-\beta )q^{3}+(-1054070+20\beta )q^{5}+\cdots\)
64.22.a.l	$64$	$22$	64.a	1.a	$1$	$3$	$3$	$178.866$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				8.22.a.b	$1$	$1$	\(0\)	\(-96764\)	\(24111774\)	\(295988280\)		$+$	$+$	$2^{21}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-32255+\beta _{1})q^{3}+(8037261-9\beta _{1}+\cdots)q^{5}+\cdots\)
64.22.a.m	$64$	$22$	64.a	1.a	$1$	$3$	$3$	$178.866$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				8.22.a.b	$1$	$0$	\(0\)	\(96764\)	\(24111774\)	\(-295988280\)		$-$	$-$	$2^{21}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(32255-\beta _{1})q^{3}+(8037261-9\beta _{1}+\cdots)q^{5}+\cdots\)
64.22.a.n	$64$	$22$	64.a	1.a	$1$	$4$	$4$	$178.866$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				32.22.a.b	$2$	$0$	\(0\)	\(0\)	\(37124680\)	\(0\)		$-$	$-$	$2^{30}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(9281170+115\beta _{2})q^{5}+\cdots\)
64.22.a.o	$64$	$22$	64.a	1.a	$1$	$5$	$5$	$178.866$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				32.22.a.c	$1$	$1$	\(0\)	\(-23144\)	\(-18311174\)	\(63978640\)		$+$	$+$	$2^{48}\cdot 3^{2}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-4629+\beta _{1})q^{3}+(-3662238+\cdots)q^{5}+\cdots\)
64.22.a.p	$64$	$22$	64.a	1.a	$1$	$5$	$5$	$178.866$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				32.22.a.c	$1$	$1$	\(0\)	\(23144\)	\(-18311174\)	\(-63978640\)		$+$	$+$	$2^{48}\cdot 3^{2}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(4629-\beta _{1})q^{3}+(-3662238+14\beta _{1}+\cdots)q^{5}+\cdots\)
64.22.a.q	$64$	$22$	64.a	1.a	$1$	$6$	$6$	$178.866$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		32.22.a.e	$2$	$0$	\(0\)	\(0\)	\(-7887252\)	\(0\)		$-$	$-$	$2^{71}\cdot 3^{8}\cdot 5^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1314542+\beta _{2})q^{5}+(-1762\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(64))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(64)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 7}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)