Space of modular forms of level 136, weight 4, and trivial character

• Label: 136.4.a
• Level: $136$
• Weight: $4$
• Character orbit: 136.a
• Rep. character: $\chi_{136}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$136 = 2^{3} \cdot 17$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	136.a (trivial)
Character field:			$\Q$
Newform subspaces:			$4$
Sturm bound:			$72$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(136))$.

	Total	New	Old
Modular forms	58	12	46
Cusp forms	50	12	38
Eisenstein series	8	0	8

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$16$	$4$	$12$		$14$	$4$	$10$		$2$	$0$	$2$
$+$	$-$	$-$		$14$	$2$	$12$		$12$	$2$	$10$		$2$	$0$	$2$
$-$	$+$	$-$		$13$	$3$	$10$		$11$	$3$	$8$		$2$	$0$	$2$
$-$	$-$	$+$		$15$	$3$	$12$		$13$	$3$	$10$		$2$	$0$	$2$
Plus space		$+$		$31$	$7$	$24$		$27$	$7$	$20$		$4$	$0$	$4$
Minus space		$-$		$27$	$5$	$22$		$23$	$5$	$18$		$4$	$0$	$4$

Trace form

12 * q + 6 * q^3 - 10 * q^5 - 48 * q^7 + 136 * q^9 + 38 * q^11 + 36 * q^13 - 16 * q^15 - 34 * q^17 - 112 * q^19 + 296 * q^21 + 28 * q^23 + 552 * q^25 - 228 * q^27 - 258 * q^29 + 536 * q^31 - 116 * q^33 + 136 * q^35 - 326 * q^37 + 68 * q^39 + 428 * q^41 - 364 * q^43 - 674 * q^45 - 1296 * q^47 - 84 * q^49 + 306 * q^51 + 612 * q^53 + 64 * q^55 + 1240 * q^57 + 492 * q^59 - 1230 * q^61 - 1608 * q^63 + 116 * q^65 - 844 * q^67 - 1216 * q^69 + 172 * q^71 - 1664 * q^73 - 2542 * q^75 + 920 * q^77 + 1500 * q^79 + 2624 * q^81 - 1052 * q^83 - 170 * q^85 + 64 * q^87 - 120 * q^89 - 2040 * q^91 + 120 * q^93 + 4536 * q^95 + 1628 * q^97 + 542 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(136))$ 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
136.4.a.a	$136$	$4$	136.a	1.a	$1$	$2$	$2$	$8.024$	$\Q(\sqrt{3})$	None	✓	✓	✓	✓	136.4.a.a	$1$	$1$	$0$	$4$	$-12$	$-36$		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+(2+\beta )q^{3}+(-6-2\beta )q^{5}+(-18+\cdots)q^{7}+\cdots$
136.4.a.b	$136$	$4$	136.a	1.a	$1$	$3$	$3$	$8.024$	3.3.8396.1	None	✓	✓	✓	✓	136.4.a.b	$1$	$1$	$0$	$-4$	$-8$	$-2$		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(-1+\beta _{2})q^{3}+(-3+\beta _{1}-2\beta _{2})q^{5}+\cdots$
136.4.a.c	$136$	$4$	136.a	1.a	$1$	$3$	$3$	$8.024$	3.3.1556.1	None	✓	✓	✓	✓	136.4.a.c	$1$	$0$	$0$	$8$	$2$	$12$		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+(3-\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(4-\beta _{1}+\cdots)q^{7}+\cdots$
136.4.a.d	$136$	$4$	136.a	1.a	$1$	$4$	$4$	$8.024$	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 $S_{4}^{\mathrm{old}}(\Gamma_0(136))$ into lower level spaces

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