Space of modular forms of level 464, weight 4, and trivial character

• Label: 464.4.a
• Level: $464$
• Weight: $4$
• Character orbit: 464.a
• Rep. character: $\chi_{464}(1,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $14$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(240\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(464))\).

	Total	New	Old
Modular forms	186	42	144
Cusp forms	174	42	132
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.

\(2\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(12\)
\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(12\)
Plus space		\(+\)	\(24\)
Minus space		\(-\)	\(18\)

Trace form

42 * q - 6 * q^3 + 36 * q^7 + 398 * q^9 - 66 * q^11 + 72 * q^15 - 76 * q^17 + 90 * q^19 - 84 * q^23 + 1050 * q^25 + 228 * q^27 - 318 * q^31 - 384 * q^33 + 576 * q^35 + 732 * q^39 + 276 * q^41 + 510 * q^43 + 1058 * q^47 + 1802 * q^49 - 44 * q^51 + 1364 * q^55 + 296 * q^57 + 748 * q^59 - 912 * q^61 + 1276 * q^63 + 184 * q^65 - 168 * q^67 + 512 * q^69 + 112 * q^71 + 1044 * q^73 + 18 * q^75 + 656 * q^77 - 1610 * q^79 + 4098 * q^81 + 2196 * q^83 + 1536 * q^85 + 522 * q^87 - 1796 * q^89 + 1936 * q^91 + 800 * q^93 + 208 * q^95 + 756 * q^97 + 874 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(464))\) 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	29
464.4.a.a	$464$	$4$	464.a	1.a	$1$	$1$	$1$	$27.377$	\(\Q\)	None	✓				58.4.a.a	$1$	$1$	\(0\)	\(-7\)	\(5\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+5q^{5}+2q^{7}+22q^{9}-37q^{11}+\cdots\)
464.4.a.b	$464$	$4$	464.a	1.a	$1$	$1$	$1$	$27.377$	\(\Q\)	None	✓				58.4.a.b	$1$	$1$	\(0\)	\(7\)	\(-15\)	\(18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-15q^{5}+18q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
464.4.a.c	$464$	$4$	464.a	1.a	$1$	$2$	$2$	$27.377$	\(\Q(\sqrt{22}) \)	None	✓				116.4.a.b	$1$	$0$	\(0\)	\(-10\)	\(30\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-5+\beta )q^{3}+15q^{5}-2\beta q^{7}+(20+\cdots)q^{9}+\cdots\)
464.4.a.d	$464$	$4$	464.a	1.a	$1$	$2$	$2$	$27.377$	\(\Q(\sqrt{13}) \)	None	✓				116.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(20\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-5-2\beta )q^{5}+(10+4\beta )q^{7}+\cdots\)
464.4.a.e	$464$	$4$	464.a	1.a	$1$	$2$	$2$	$27.377$	\(\Q(\sqrt{6}) \)	None	✓				58.4.a.c	$1$	$0$	\(0\)	\(2\)	\(-10\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-5+6\beta )q^{5}+(8+8\beta )q^{7}+\cdots\)
464.4.a.f	$464$	$4$	464.a	1.a	$1$	$2$	$2$	$27.377$	\(\Q(\sqrt{2}) \)	None	✓				29.4.a.a	$1$	$0$	\(0\)	\(10\)	\(-10\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5+3\beta )q^{3}+(-5+4\beta )q^{5}+(8-10\beta )q^{7}+\cdots\)
464.4.a.g	$464$	$4$	464.a	1.a	$1$	$3$	$3$	$27.377$	3.3.229.1	None	✓				232.4.a.b	$1$	$0$	\(0\)	\(-6\)	\(4\)	\(-16\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+3\beta _{2})q^{3}+(1+3\beta _{1}-4\beta _{2})q^{5}+\cdots\)
464.4.a.h	$464$	$4$	464.a	1.a	$1$	$3$	$3$	$27.377$	3.3.4481.1	None	✓				232.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(11\)	\(38\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(4+\beta _{1}+\beta _{2})q^{5}+\cdots\)
464.4.a.i	$464$	$4$	464.a	1.a	$1$	$3$	$3$	$27.377$	3.3.19816.1	None	✓				58.4.a.d	$1$	$0$	\(0\)	\(-2\)	\(20\)	\(-24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(6+2\beta _{1}+\beta _{2})q^{5}+\cdots\)
464.4.a.j	$464$	$4$	464.a	1.a	$1$	$3$	$3$	$27.377$	3.3.148344.1	None	✓				116.4.a.c	$1$	$0$	\(0\)	\(10\)	\(-20\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-7+\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
464.4.a.k	$464$	$4$	464.a	1.a	$1$	$4$	$4$	$27.377$	4.4.225792.1	None	✓				232.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(8\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-5-\beta _{2})q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots\)
464.4.a.l	$464$	$4$	464.a	1.a	$1$	$5$	$5$	$27.377$	5.5.13458092.1	None	✓		✓		29.4.a.b	$1$	$1$	\(0\)	\(-8\)	\(10\)	\(-40\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{3})q^{3}+(2-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
464.4.a.m	$464$	$4$	464.a	1.a	$1$	$5$	$5$	$27.377$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		232.4.a.d	$1$	$1$	\(0\)	\(-4\)	\(10\)	\(-32\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(2+\beta _{3})q^{5}+(-6+\cdots)q^{7}+\cdots\)
464.4.a.n	$464$	$4$	464.a	1.a	$1$	$6$	$6$	$27.377$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		232.4.a.e	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(38\)		$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{3}+(-1-\beta _{3}-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(464))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(464)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 2}\)