Space of modular forms of level 256, weight 12, and trivial character

• Label: 256.12.a
• Level: $256$
• Weight: $12$
• Character orbit: 256.a
• Rep. character: $\chi_{256}(1,\cdot)$
• Character field: $\Q$
• Dimension: $86$
• Newform subspaces: $17$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$256 = 2^{8}$
Weight:	$k$	$=$	$12$
Character orbit:	$[\chi]$	$=$	256.a (trivial)
Character field:			$\Q$
Newform subspaces:			$17$
Sturm bound:			$384$
Trace bound:			$7$
Distinguishing $T_p$:			$3$, $5$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{12}(\Gamma_0(256))$.

	Total	New	Old
Modular forms	364	90	274
Cusp forms	340	86	254
Eisenstein series	24	4	20

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
$+$		$184$	$46$	$138$		$172$	$44$	$128$		$12$	$2$	$10$
$-$		$180$	$44$	$136$		$168$	$42$	$126$		$12$	$2$	$10$

Trace form

86 * q + 4842022 * q^9 - 4 * q^17 + 761718754 * q^25 + 708584 * q^33 + 4 * q^41 + 20848125510 * q^49 - 4525726280 * q^57 - 3574618080 * q^65 + 92385298388 * q^73 + 244075616654 * q^81 - 104349567180 * q^89 - 291432459620 * q^97

Decomposition of $S_{12}^{\mathrm{new}}(\Gamma_0(256))$ 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
256.12.a.a	$256$	$12$	256.a	1.a	$1$	$1$	$1$	$196.696$	$\Q$	$\Q(\sqrt{-2})$	✓				64.12.b.a	$2$	$1$	$0$	$-394$	$0$	$0$		$-$	$-$	$1$	$N(\mathrm{U}(1))$	$q-394q^{3}-21911q^{9}+141906q^{11}+\cdots$
256.12.a.b	$256$	$12$	256.a	1.a	$1$	$1$	$1$	$196.696$	$\Q$	$\Q(\sqrt{-1})$	✓				128.12.b.b	$2$	$0$	$0$	$0$	$-5284$	$0$		$+$	$+$	$1$	$N(\mathrm{U}(1))$	$q-5284q^{5}-3^{11}q^{9}+492092q^{13}+\cdots$
256.12.a.c	$256$	$12$	256.a	1.a	$1$	$1$	$1$	$196.696$	$\Q$	$\Q(\sqrt{-1})$	✓				128.12.b.b	$2$	$1$	$0$	$0$	$5284$	$0$		$-$	$-$	$1$	$N(\mathrm{U}(1))$	$q+5284q^{5}-3^{11}q^{9}-492092q^{13}+\cdots$
256.12.a.d	$256$	$12$	256.a	1.a	$1$	$1$	$1$	$196.696$	$\Q$	$\Q(\sqrt{-2})$	✓				64.12.b.a	$2$	$0$	$0$	$394$	$0$	$0$		$+$	$+$	$1$	$N(\mathrm{U}(1))$	$q+394q^{3}-21911q^{9}-141906q^{11}+\cdots$
256.12.a.e	$256$	$12$	256.a	1.a	$1$	$2$	$2$	$196.696$	$\Q(\sqrt{22155})$	None	✓				64.12.b.b	$2$	$0$	$0$	$-900$	$0$	$0$		$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q-450q^{3}+\beta q^{5}+2\beta q^{7}+25353q^{9}+\cdots$
256.12.a.f	$256$	$12$	256.a	1.a	$1$	$2$	$2$	$196.696$	$\Q(\sqrt{2})$	$\Q(\sqrt{-2})$	✓				128.12.b.a	$4$	$1$	$0$	$0$	$0$	$0$		$-$	$-$	$2$	$N(\mathrm{U}(1))$	$q+263\beta q^{3}+376205q^{9}-374351\beta q^{11}+\cdots$
256.12.a.g	$256$	$12$	256.a	1.a	$1$	$2$	$2$	$196.696$	$\Q(\sqrt{22155})$	None	✓				64.12.b.b	$2$	$1$	$0$	$900$	$0$	$0$		$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q+450q^{3}-\beta q^{5}+2\beta q^{7}+25353q^{9}+\cdots$
256.12.a.h	$256$	$12$	256.a	1.a	$1$	$4$	$4$	$196.696$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				128.12.b.c	$2$	$1$	$0$	$0$	$-21040$	$0$		$-$	$-$	$2^{17}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-5260+5\beta _{3})q^{5}+(13\beta _{1}+\cdots)q^{7}+\cdots$
256.12.a.i	$256$	$12$	256.a	1.a	$1$	$4$	$4$	$196.696$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				128.12.b.c	$2$	$0$	$0$	$0$	$21040$	$0$		$+$	$+$	$2^{17}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(5260-5\beta _{3})q^{5}+(-13\beta _{1}+\cdots)q^{7}+\cdots$
256.12.a.j	$256$	$12$	256.a	1.a	$1$	$6$	$6$	$196.696$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓				128.12.b.f	$2$	$0$	$0$	$0$	$-3256$	$0$		$+$	$+$	$2^{41}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-543-\beta _{4})q^{5}+(5\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$
256.12.a.k	$256$	$12$	256.a	1.a	$1$	$6$	$6$	$196.696$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓				128.12.b.f	$2$	$1$	$0$	$0$	$3256$	$0$		$-$	$-$	$2^{41}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(543+\beta _{4})q^{5}+(-5\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$
256.12.a.l	$256$	$12$	256.a	1.a	$1$	$8$	$8$	$196.696$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				64.12.b.c	$2$	$1$	$0$	$-992$	$0$	$0$		$-$	$-$	$2^{53}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q+(-124+\beta _{1})q^{3}+\beta _{3}q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots$
256.12.a.m	$256$	$12$	256.a	1.a	$1$	$8$	$8$	$196.696$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				128.12.b.d	$4$	$1$	$0$	$0$	$0$	$0$		$-$	$-$	$2^{54}\cdot 3^{6}$	$\mathrm{SU}(2)$	$q+(\beta _{3}-\beta _{4})q^{3}-\beta _{5}q^{5}-\beta _{1}q^{7}+(-23859+\cdots)q^{9}+\cdots$
256.12.a.n	$256$	$12$	256.a	1.a	$1$	$8$	$8$	$196.696$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				64.12.b.c	$2$	$0$	$0$	$992$	$0$	$0$		$+$	$+$	$2^{53}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q+(124-\beta _{1})q^{3}+\beta _{3}q^{5}+(\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots$
256.12.a.o	$256$	$12$	256.a	1.a	$1$	$10$	$10$	$196.696$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓		✓		8.12.b.a	$2$	$1$	$0$	$0$	$0$	$-33616$		$-$	$-$	$2^{85}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(2\beta _{1}+\beta _{4})q^{5}+(-3362+\cdots)q^{7}+\cdots$
256.12.a.p	$256$	$12$	256.a	1.a	$1$	$10$	$10$	$196.696$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓				8.12.b.a	$2$	$0$	$0$	$0$	$0$	$33616$		$+$	$+$	$2^{85}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(-2\beta _{1}-\beta _{4})q^{5}+(3362+\cdots)q^{7}+\cdots$
256.12.a.q	$256$	$12$	256.a	1.a	$1$	$12$	$12$	$196.696$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓		✓		128.12.b.e	$4$	$0$	$0$	$0$	$0$	$0$		$+$	$+$	$2^{126}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+\beta _{6}q^{5}+\beta _{7}q^{7}+(96002+\cdots)q^{9}+\cdots$

Decomposition of $S_{12}^{\mathrm{old}}(\Gamma_0(256))$ into lower level spaces

$S_{12}^{\mathrm{old}}(\Gamma_0(256)) \simeq$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))$$^{\oplus 9}$$\oplus$$S_{12}^{\mathrm{new}}(\Gamma_0(4))$$^{\oplus 7}$$\oplus$$S_{12}^{\mathrm{new}}(\Gamma_0(8))$$^{\oplus 6}$$\oplus$$S_{12}^{\mathrm{new}}(\Gamma_0(16))$$^{\oplus 5}$$\oplus$$S_{12}^{\mathrm{new}}(\Gamma_0(32))$$^{\oplus 4}$$\oplus$$S_{12}^{\mathrm{new}}(\Gamma_0(64))$$^{\oplus 3}$$\oplus$$S_{12}^{\mathrm{new}}(\Gamma_0(128))$$^{\oplus 2}$