Space of modular forms of level 176, weight 4, and trivial character

• Label: 176.4.a
• Level: $176$
• Weight: $4$
• Character orbit: 176.a
• Rep. character: $\chi_{176}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $10$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$176 = 2^{4} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	176.a (trivial)
Character field:			$\Q$
Newform subspaces:			$10$
Sturm bound:			$96$
Trace bound:			$5$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(176))$.

	Total	New	Old
Modular forms	78	15	63
Cusp forms	66	15	51
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.

$2$	$11$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$21$	$5$	$16$		$18$	$5$	$13$		$3$	$0$	$3$
$+$	$-$	$-$		$18$	$2$	$16$		$15$	$2$	$13$		$3$	$0$	$3$
$-$	$+$	$-$		$18$	$4$	$14$		$15$	$4$	$11$		$3$	$0$	$3$
$-$	$-$	$+$		$21$	$4$	$17$		$18$	$4$	$14$		$3$	$0$	$3$
Plus space		$+$		$42$	$9$	$33$		$36$	$9$	$27$		$6$	$0$	$6$
Minus space		$-$		$36$	$6$	$30$		$30$	$6$	$24$		$6$	$0$	$6$

Trace form

15 * q - 6 * q^3 + 2 * q^5 + 36 * q^7 + 135 * q^9 - 33 * q^11 - 46 * q^13 + 66 * q^15 - 26 * q^17 + 204 * q^19 + 136 * q^21 + 138 * q^23 + 397 * q^25 + 138 * q^27 - 198 * q^29 - 318 * q^31 + 228 * q^35 + 330 * q^37 + 120 * q^39 + 118 * q^41 + 336 * q^43 - 350 * q^45 + 848 * q^47 + 375 * q^49 - 44 * q^51 - 286 * q^53 + 220 * q^55 + 680 * q^57 - 1738 * q^59 - 1302 * q^61 - 416 * q^63 + 116 * q^65 - 570 * q^67 - 1544 * q^69 + 390 * q^71 - 154 * q^73 + 1292 * q^75 - 3180 * q^79 + 295 * q^81 - 1448 * q^83 - 1348 * q^85 - 1384 * q^87 + 26 * q^89 + 368 * q^91 - 2136 * q^93 - 2064 * q^95 + 1490 * q^97 - 891 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(176))$ 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	11
176.4.a.a	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	$\Q$	None	✓				88.4.a.b	$1$	$1$	$0$	$-7$	$9$	$-2$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-7q^{3}+9q^{5}-2q^{7}+22q^{9}+11q^{11}+\cdots$
176.4.a.b	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	$\Q$	None	✓				22.4.a.b	$1$	$0$	$0$	$-4$	$14$	$8$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-4q^{3}+14q^{5}+8q^{7}-11q^{9}+11q^{11}+\cdots$
176.4.a.c	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	$\Q$	None	✓				22.4.a.c	$1$	$1$	$0$	$-1$	$-3$	$10$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-3q^{5}+10q^{7}-26q^{9}-11q^{11}+\cdots$
176.4.a.d	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	$\Q$	None	✓				88.4.a.a	$1$	$1$	$0$	$1$	$-7$	$6$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-7q^{5}+6q^{7}-26q^{9}+11q^{11}+\cdots$
176.4.a.e	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	$\Q$	None	✓				44.4.a.a	$1$	$0$	$0$	$5$	$-7$	$26$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+5q^{3}-7q^{5}+26q^{7}-2q^{9}+11q^{11}+\cdots$
176.4.a.f	$176$	$4$	176.a	1.a	$1$	$1$	$1$	$10.384$	$\Q$	None	✓				22.4.a.a	$1$	$1$	$0$	$7$	$-19$	$-14$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+7q^{3}-19q^{5}-14q^{7}+22q^{9}-11q^{11}+\cdots$
176.4.a.g	$176$	$4$	176.a	1.a	$1$	$2$	$2$	$10.384$	$\Q(\sqrt{97})$	None	✓		✓		44.4.a.b	$1$	$1$	$0$	$-9$	$11$	$-10$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-4-\beta )q^{3}+(6-\beta )q^{5}+(-8+6\beta )q^{7}+\cdots$
176.4.a.h	$176$	$4$	176.a	1.a	$1$	$2$	$2$	$10.384$	$\Q(\sqrt{5})$	None	✓				88.4.a.c	$1$	$0$	$0$	$2$	$-6$	$56$		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(-3+4\beta )q^{5}+(28+\beta )q^{7}+\cdots$
176.4.a.i	$176$	$4$	176.a	1.a	$1$	$2$	$2$	$10.384$	$\Q(\sqrt{3})$	None	✓		✓		11.4.a.a	$1$	$0$	$0$	$2$	$2$	$-20$		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(1+2\beta )q^{5}+(-10+\beta )q^{7}+\cdots$
176.4.a.j	$176$	$4$	176.a	1.a	$1$	$3$	$3$	$10.384$	3.3.11109.1	None	✓		✓		88.4.a.d	$1$	$0$	$0$	$-2$	$8$	$-24$		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(176))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(176)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$^{\oplus 5}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(22))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(44))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(88))$$^{\oplus 2}$