Atlas › Modular forms › Weight 2 › Level 431 › 431.2.a

Newspace 431.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(431))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
431.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+	$-1$	431a1 rank 1
431.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} - q^{4} - 3q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	−	$+1$	431b1 rank 0
431.2.a.c	3	degree 3	$3q + q^{3} + 4q^{4} + q^{5} - 3q^{6} + O(q^{8})$	−	$+1$
431.2.a.d	3	degree 3	$3q + q^{2} - q^{3} + 3q^{4} - 3q^{5} - 9q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
431.2.a.e	4	degree 4	$4q - q^{2} - 3q^{3} - q^{4} - 5q^{5} + 5q^{6} + 2q^{7} + O(q^{8})$	+	$-1$
431.2.a.f	24	degree 24	$24q + q^{2} + q^{3} + 33q^{4} + 13q^{5} + 17q^{6} + 8q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
431.2.a.a	$x - 2$
431.2.a.b	$x - 8$
431.2.a.c	$x^{3} - 3x^{2} - 68x + 289$
431.2.a.d	$x^{3} + 2x^{2} - 16x - 24$
431.2.a.e	$x^{4} + 10x^{3} - 9x^{2} - 190x + 109$
431.2.a.f	$x^{24} - 4x^{23} - 547x^{22} + 2113x^{21} + 126101x^{20} - 451086x^{19} - 16156477x^{18} + 50965066x^{17} + 1276409252x^{16} - 3327977393x^{15} - 65142631706x^{14} + 127769456159x^{13} + 2179977658082x^{12} - 2753837252287x^{11} - 47219887506411x^{10} + 27322019836141x^{9} + 631086685409461x^{8} + 7488142619012x^{7} - 4690129691489764x^{6} - 1862090641190448x^{5} + 15762907809605344x^{4} + 5412692447679040x^{3} - 19513376951187008x^{2} + 2819863398264832x + 730731043717120$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(431))$	36
Old forms	0
Cusp forms $S_{2}(\Gamma_0(431))$	36
Eisenstein series $E_{2}(\Gamma_0(431))$	1
Modular forms $M_{2}(\Gamma_0(431))$	37

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 431]: [0, 37] (certified at ell = 2147481491)
T = [(2, 1), (3, 3)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 5 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(431, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 431:2 1000

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