Atlas › Modular forms › Weight 2 › Level 427 › 427.2.a

Newspace 427.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(427))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
427.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	+ +	$-1$	427c1 rank 1
427.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} + 4q^{5} + q^{7} + O(q^{8})$	− +	$+1$	427a1 rank 0
427.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - 4q^{5} + q^{6} + q^{7} + O(q^{8})$	− −	$-1$	427b1 rank 1
427.2.a.d	6	degree 6	$6q - 5q^{2} - 8q^{3} + 9q^{4} - 9q^{5} + q^{6} + 6q^{7} + O(q^{8})$	− −	$-1$
427.2.a.e	6	degree 6	$6q - 5q^{2} + 9q^{4} - 5q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
427.2.a.f	7	degree 7	$7q + 4q^{2} + q^{3} + 8q^{4} + 7q^{5} + 2q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$
427.2.a.g	9	degree 9	$9q + 5q^{2} - q^{3} + 13q^{4} + 9q^{5} - 4q^{6} - 9q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
427.2.a.a	$x - 2$
427.2.a.b	$x - 6$
427.2.a.c	$x - 4$
427.2.a.d	$x^{6} + 29x^{5} + 281x^{4} + 914x^{3} - 480x^{2} - 5841x - 4527$
427.2.a.e	$x^{6} + 5x^{5} - 79x^{4} - 246x^{3} + 1392x^{2} + 2445x - 4075$
427.2.a.f	$x^{7} - 7x^{6} - 105x^{5} + 746x^{4} + 2742x^{3} - 20597x^{2} - 5905x + 87878$
427.2.a.g	$x^{9} - 2x^{8} - 168x^{7} + 201x^{6} + 9052x^{5} - 7005x^{4} - 175428x^{3} + 129957x^{2} + 924662x - 1134608$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(427))$	31
Old forms	8
Cusp forms $S_{2}(\Gamma_0(427))$	39
Eisenstein series $E_{2}(\Gamma_0(427))$	3
Modular forms $M_{2}(\Gamma_0(427))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(427)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 61, 427]: [0, 1, 5, 42] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 31 = 1 x 31 (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(427, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 427:2 1000

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