Atlas › Modular forms › Weight 2 › Level 393 › 393.2.a

Newspace 393.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(393))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
393.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
393.2.a.b	4	degree 4	$4q - 3q^{2} + 4q^{3} + q^{4} - 8q^{5} - 3q^{6} - 8q^{7} + O(q^{8})$	− −	$-1$
393.2.a.c	4	degree 4	$4q - q^{2} - 4q^{3} + q^{4} + q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
393.2.a.d	5	degree 5	$5q + 2q^{2} - 5q^{3} + 8q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
393.2.a.e	6	degree 6	$6q + q^{2} + 6q^{3} + 3q^{4} + 8q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
393.2.a.a	$x^{2} + 2x - 1$
393.2.a.b	$x^{4} + 11x^{3} + 37x^{2} + 29x - 29$
393.2.a.c	$x^{4} + x^{3} - 5x^{2} - 5x - 1$
393.2.a.d	$x^{5} - 38x^{3} + 34x^{2} + 372x - 647$
393.2.a.e	$x^{6} - 9x^{5} + 22x^{4} - 6x^{3} - 30x^{2} + 17x + 7$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(393))$	21
Old forms	22
Cusp forms $S_{2}(\Gamma_0(393))$	43
Eisenstein series $E_{2}(\Gamma_0(393))$	3
Modular forms $M_{2}(\Gamma_0(393))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(393)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(131))^{\oplus 2}$

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, 3, 131, 393]: [0, 1, 12, 46] (certified at ell = 2147482481)
T = [(2, 1), (5, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(393, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 393:2 1000

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