Space of modular forms of level 272, weight 4, and trivial character

• Label: 272.4.a
• Level: $272$
• Weight: $4$
• Character orbit: 272.a
• Rep. character: $\chi_{272}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $11$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(272))\).

	Total	New	Old
Modular forms	114	24	90
Cusp forms	102	24	78
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.

\(2\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(10\)

Trace form

24 * q - 6 * q^3 + 50 * q^7 + 216 * q^9 - 86 * q^11 + 12 * q^15 + 180 * q^19 + 136 * q^21 - 26 * q^23 + 768 * q^25 + 360 * q^27 - 256 * q^29 - 54 * q^31 - 24 * q^33 + 228 * q^35 - 8 * q^37 + 432 * q^39 + 296 * q^41 + 364 * q^43 - 440 * q^45 + 960 * q^47 + 512 * q^49 - 306 * q^51 + 376 * q^53 - 612 * q^55 + 680 * q^57 - 1020 * q^59 + 1926 * q^63 + 184 * q^65 + 816 * q^67 - 496 * q^69 + 254 * q^71 - 768 * q^73 + 1670 * q^75 + 1280 * q^77 - 674 * q^79 + 1024 * q^81 + 1340 * q^83 - 700 * q^87 - 600 * q^89 + 792 * q^91 - 2192 * q^93 - 1688 * q^95 + 536 * q^97 - 26 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(272))\) 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	17
272.4.a.a	$272$	$4$	272.a	1.a	$1$	$1$	$1$	$16.049$	\(\Q\)	None	✓				34.4.a.a	$1$	$0$	\(0\)	\(2\)	\(-18\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-18q^{5}+10q^{7}-23q^{9}+6q^{11}+\cdots\)
272.4.a.b	$272$	$4$	272.a	1.a	$1$	$1$	$1$	$16.049$	\(\Q\)	None	✓				68.4.a.a	$1$	$1$	\(0\)	\(2\)	\(-8\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-8q^{5}+12q^{7}-23q^{9}+10q^{11}+\cdots\)
272.4.a.c	$272$	$4$	272.a	1.a	$1$	$1$	$1$	$16.049$	\(\Q\)	None	✓				34.4.a.b	$1$	$1$	\(0\)	\(2\)	\(16\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2^{4}q^{5}-24q^{7}-23q^{9}-62q^{11}+\cdots\)
272.4.a.d	$272$	$4$	272.a	1.a	$1$	$1$	$1$	$16.049$	\(\Q\)	None	✓				17.4.a.a	$1$	$0$	\(0\)	\(8\)	\(6\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+6q^{5}+28q^{7}+37q^{9}+24q^{11}+\cdots\)
272.4.a.e	$272$	$4$	272.a	1.a	$1$	$2$	$2$	$16.049$	\(\Q(\sqrt{13}) \)	None	✓				34.4.a.c	$1$	$0$	\(0\)	\(-6\)	\(-4\)	\(6\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(-2-4\beta )q^{5}+(3+\beta )q^{7}+\cdots\)
272.4.a.f	$272$	$4$	272.a	1.a	$1$	$2$	$2$	$16.049$	\(\Q(\sqrt{3}) \)	None	✓				136.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(-12\)	\(36\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-6+2\beta )q^{5}+(18+\cdots)q^{7}+\cdots\)
272.4.a.g	$272$	$4$	272.a	1.a	$1$	$3$	$3$	$16.049$	3.3.1556.1	None	✓		✓		136.4.a.c	$1$	$1$	\(0\)	\(-8\)	\(2\)	\(-12\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(-4+\cdots)q^{7}+\cdots\)
272.4.a.h	$272$	$4$	272.a	1.a	$1$	$3$	$3$	$16.049$	3.3.2636.1	None	✓		✓		17.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(-8\)	\(-22\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-3+\beta _{2})q^{5}+\cdots\)
272.4.a.i	$272$	$4$	272.a	1.a	$1$	$3$	$3$	$16.049$	3.3.1524.1	None	✓		✓		68.4.a.b	$1$	$0$	\(0\)	\(-4\)	\(26\)	\(-8\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(9-\beta _{1})q^{5}+(-2+\cdots)q^{7}+\cdots\)
272.4.a.j	$272$	$4$	272.a	1.a	$1$	$3$	$3$	$16.049$	3.3.8396.1	None	✓				136.4.a.b	$1$	$0$	\(0\)	\(4\)	\(-8\)	\(2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(-3+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
272.4.a.k	$272$	$4$	272.a	1.a	$1$	$4$	$4$	$16.049$	4.4.550476.1	None	✓		✓		136.4.a.d	$1$	$0$	\(0\)	\(2\)	\(8\)	\(22\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(272))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(272)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 2}\)