Space of modular forms of level 368, weight 4, and trivial character

• Label: 368.4.a
• Level: $368$
• Weight: $4$
• Character orbit: 368.a
• Rep. character: $\chi_{368}(1,\cdot)$
• Character field: $\Q$
• Dimension: $33$
• Newform subspaces: $14$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(368))\).

	Total	New	Old
Modular forms	150	33	117
Cusp forms	138	33	105
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.

\(2\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(8\)
\(+\)	\(-\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(7\)
\(-\)	\(-\)	\(+\)	\(10\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(15\)

Trace form

33 * q - 6 * q^3 + 2 * q^5 + 36 * q^7 + 317 * q^9 - 46 * q^13 - 102 * q^17 + 180 * q^19 + 69 * q^23 + 1015 * q^25 - 150 * q^27 - 142 * q^29 - 54 * q^31 - 408 * q^33 - 192 * q^35 + 330 * q^37 + 30 * q^39 + 98 * q^41 + 768 * q^43 + 90 * q^45 + 266 * q^47 + 849 * q^49 + 1012 * q^51 + 106 * q^53 + 296 * q^55 - 552 * q^57 - 884 * q^59 - 190 * q^61 - 416 * q^63 - 348 * q^65 - 168 * q^67 + 2858 * q^71 - 310 * q^73 - 898 * q^75 - 184 * q^77 - 1764 * q^79 + 3673 * q^81 - 1220 * q^83 - 1588 * q^85 + 674 * q^87 - 830 * q^89 + 2836 * q^91 - 1068 * q^93 + 1352 * q^95 + 466 * q^97 + 6424 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(368))\) 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	23
368.4.a.a	$368$	$4$	368.a	1.a	$1$	$1$	$1$	$21.713$	\(\Q\)	None	✓				184.4.a.b	$1$	$1$	\(0\)	\(-8\)	\(-4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-4q^{5}+4q^{7}+37q^{9}-26q^{11}+\cdots\)
368.4.a.b	$368$	$4$	368.a	1.a	$1$	$1$	$1$	$21.713$	\(\Q\)	None	✓				46.4.a.a	$1$	$1$	\(0\)	\(1\)	\(-10\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-10q^{5}+12q^{7}-26q^{9}+42q^{11}+\cdots\)
368.4.a.c	$368$	$4$	368.a	1.a	$1$	$1$	$1$	$21.713$	\(\Q\)	None	✓				184.4.a.a	$1$	$0$	\(0\)	\(4\)	\(22\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+22q^{5}-8q^{7}-11q^{9}+20q^{11}+\cdots\)
368.4.a.d	$368$	$4$	368.a	1.a	$1$	$1$	$1$	$21.713$	\(\Q\)	None	✓				23.4.a.a	$1$	$1$	\(0\)	\(5\)	\(-6\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-6q^{5}+8q^{7}-2q^{9}-34q^{11}+\cdots\)
368.4.a.e	$368$	$4$	368.a	1.a	$1$	$1$	$1$	$21.713$	\(\Q\)	None	✓				46.4.a.b	$1$	$0$	\(0\)	\(9\)	\(-20\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-20q^{5}-2q^{7}+54q^{9}+52q^{11}+\cdots\)
368.4.a.f	$368$	$4$	368.a	1.a	$1$	$2$	$2$	$21.713$	\(\Q(\sqrt{73}) \)	None	✓				46.4.a.d	$1$	$1$	\(0\)	\(-3\)	\(10\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(6-2\beta )q^{5}+(-8+4\beta )q^{7}+\cdots\)
368.4.a.g	$368$	$4$	368.a	1.a	$1$	$2$	$2$	$21.713$	\(\Q(\sqrt{41}) \)	None	✓				46.4.a.c	$1$	$0$	\(0\)	\(1\)	\(10\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+3\beta )q^{3}+(4+2\beta )q^{5}+(-2+\cdots)q^{7}+\cdots\)
368.4.a.h	$368$	$4$	368.a	1.a	$1$	$3$	$3$	$21.713$	3.3.28669.1	None	✓		✓		92.4.a.b	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-42\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{3}+\beta _{1}q^{5}+(-14+\beta _{1}+\cdots)q^{7}+\cdots\)
368.4.a.i	$368$	$4$	368.a	1.a	$1$	$3$	$3$	$21.713$	3.3.761.1	None	✓				184.4.a.d	$1$	$1$	\(0\)	\(-1\)	\(16\)	\(-18\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(5+2\beta _{1}+\beta _{2})q^{5}+(-5+\cdots)q^{7}+\cdots\)
368.4.a.j	$368$	$4$	368.a	1.a	$1$	$3$	$3$	$21.713$	3.3.761.1	None	✓				184.4.a.c	$1$	$0$	\(0\)	\(3\)	\(2\)	\(28\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(1-\beta _{1}-\beta _{2})q^{5}+(10+\cdots)q^{7}+\cdots\)
368.4.a.k	$368$	$4$	368.a	1.a	$1$	$3$	$3$	$21.713$	3.3.1229.1	None	✓				92.4.a.a	$1$	$0$	\(0\)	\(4\)	\(-10\)	\(46\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-4-\beta _{1}+3\beta _{2})q^{5}+\cdots\)
368.4.a.l	$368$	$4$	368.a	1.a	$1$	$4$	$4$	$21.713$	4.4.334189.1	None	✓		✓		23.4.a.b	$1$	$0$	\(0\)	\(-7\)	\(14\)	\(-16\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{3}+(3+\beta _{1}-\beta _{3})q^{5}+\cdots\)
368.4.a.m	$368$	$4$	368.a	1.a	$1$	$4$	$4$	$21.713$	4.4.2822449.1	None	✓		✓		184.4.a.f	$1$	$0$	\(0\)	\(-5\)	\(-2\)	\(32\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{1}+\beta _{3})q^{5}+\cdots\)
368.4.a.n	$368$	$4$	368.a	1.a	$1$	$4$	$4$	$21.713$	4.4.167313.1	None	✓		✓		184.4.a.e	$1$	$1$	\(0\)	\(-1\)	\(-20\)	\(10\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-5+\beta _{2}+\beta _{3})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(368))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(368)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 2}\)