Atlas › Modular forms › Weight 2 › Level 247 › 247.2.a

Newspace 247.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(247))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
247.2.a.a	$x^{2} + x - 11$
247.2.a.b	$x^{3} + 6x^{2} + 3x - 1$
247.2.a.c	$x^{4} + 4x^{3} + x^{2} - 7x - 4$
247.2.a.d	$x^{5} - 3x^{4} - 13x^{3} + 22x^{2} + 11x - 16$
247.2.a.e	$x^{5} - 7x^{4} + 8x^{3} + 34x^{2} - 84x + 49$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(247))$	19
Old forms	2
Cusp forms $S_{2}(\Gamma_0(247))$	21
Eisenstein series $E_{2}(\Gamma_0(247))$	3
Modular forms $M_{2}(\Gamma_0(247))$	24

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(247)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 13, 19, 247]: [0, 1, 2, 24] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (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(247, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 247:2 1000

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