Atlas › Modular forms › Weight 2 › Level 511 › 511.2.a

Newspace 511.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(511))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
511.2.a.a	3	degree 3	$3q + 6q^{3} + 4q^{4} + 3q^{5} + 3q^{7} + O(q^{8})$	− +	$+1$
511.2.a.b	3	degree 3	$3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + O(q^{8})$	+ −	$+1$
511.2.a.c	6	degree 6	$6q - 3q^{2} - 5q^{3} + 3q^{4} - 6q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	− −	$-1$
511.2.a.d	6	degree 6	$6q - 3q^{2} + q^{3} + 3q^{4} - 2q^{5} - 4q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
511.2.a.e	9	degree 9	$9q + 2q^{2} - q^{3} + 6q^{4} + 7q^{5} + 4q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$
511.2.a.f	10	degree 10	$10q + 4q^{2} - q^{3} + 14q^{4} + q^{5} + 2q^{6} - 10q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
511.2.a.a	$x^{3} - 6x^{2} + 7x + 3$
511.2.a.b	$x^{3} - x^{2} - 4x - 1$
511.2.a.c	$x^{6} + 8x^{5} + 14x^{4} - 20x^{3} - 55x^{2} - 28x - 1$
511.2.a.d	$x^{6} + 2x^{5} - 8x^{4} - 10x^{3} + 13x^{2} + 2x - 1$
511.2.a.e	$x^{9} - x^{8} - 38x^{7} + 37x^{6} + 393x^{5} - 143x^{4} - 1485x^{3} - 232x^{2} + 1408x + 277$
511.2.a.f	$x^{10} - 3x^{9} - 49x^{8} + 118x^{7} + 798x^{6} - 1259x^{5} - 5370x^{4} + 4227x^{3} + 13282x^{2} - 3900x - 7551$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(511))$	37
Old forms	10
Cusp forms $S_{2}(\Gamma_0(511))$	47
Eisenstein series $E_{2}(\Gamma_0(511))$	3
Modular forms $M_{2}(\Gamma_0(511))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(511)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.2 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 7, 73, 511]: [0, 1, 6, 50] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(511, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 511:2 1000

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