Atlas › Modular forms › Weight 2 › Level 381 › 381.2.a

Newspace 381.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(381))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
381.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} - 2q^{7} + O(q^{8})$	− −	$-1$	381a1 rank 1
381.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$	381b1 rank 0
381.2.a.c	5	degree 5	$5q - q^{2} - 5q^{3} + q^{4} - 5q^{5} + q^{6} + O(q^{8})$	+ +	$-1$
381.2.a.d	5	degree 5	$5q + 2q^{2} - 5q^{3} + 6q^{4} + q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$
381.2.a.e	9	degree 9	$9q - 2q^{2} + 9q^{3} + 14q^{4} - 4q^{5} - 2q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$

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 $T$
381.2.a.a	$x + 1$
381.2.a.b	$x - 5$
381.2.a.c	$x^{5} + 6x^{4} - 30x^{2} - 14x + 19$
381.2.a.d	$x^{5} - 3x^{4} - 11x^{3} + 45x^{2} - 35x - 1$
381.2.a.e	$x^{9} + 6x^{8} - 27x^{7} - 136x^{6} + 287x^{5} + 877x^{4} - 1228x^{3} - 1253x^{2} + 620x + 77$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(381))$	21
Old forms	20
Cusp forms $S_{2}(\Gamma_0(381))$	41
Eisenstein series $E_{2}(\Gamma_0(381))$	3
Modular forms $M_{2}(\Gamma_0(381))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(381)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(127))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 127, 381]: [0, 1, 11, 44] (certified at ell = 2147483647)
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(381, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 381:2 1000

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