Atlas › Modular forms › Weight 2 › Level 278 › 278.2.a

Newspace 278.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(278))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
278.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 3q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	278b1 rank 0
278.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$	278a1 rank 1
278.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} - 6q^{7} + O(q^{8})$	+ +	$-1$
278.2.a.d	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$
278.2.a.e	5	degree 5	$5q + 5q^{2} + q^{3} + 5q^{4} - 2q^{5} + q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
278.2.a.a	$x - 7$
278.2.a.b	$x + 5$
278.2.a.c	$x^{2} + 6x + 1$
278.2.a.d	$x^{3} - 3x^{2} - 126x + 57$
278.2.a.e	$x^{5} + 5x^{4} - 91x^{3} - 154x^{2} + 2292x - 2837$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(278))$	12
Old forms	22
Cusp forms $S_{2}(\Gamma_0(278))$	34
Eisenstein series $E_{2}(\Gamma_0(278))$	3
Modular forms $M_{2}(\Gamma_0(278))$	37

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(278)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(139))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.272 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, 2, 139, 278]: [0, 1, 12, 37] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(278, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 278:2 1000

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