Atlas › Modular forms › Weight 2 › Level 249 › 249.2.a

Newspace 249.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(249))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
249.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + O(q^{8})$	+ +	$-1$	249a1 rank 1
249.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$	249b1 rank 1
249.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 6q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
249.2.a.d	4	degree 4	$4q + 2q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 2q^{6} + O(q^{8})$	− +	$+1$
249.2.a.e	5	degree 5	$5q - 3q^{2} - 5q^{3} + 7q^{4} - 2q^{5} + 3q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
249.2.a.a	$x - 2$
249.2.a.b	$x + 2$
249.2.a.c	$x^{2} + 20x + 92$
249.2.a.d	$x^{4} - 20x^{3} + 128x^{2} - 304x + 224$
249.2.a.e	$x^{5} + 9x^{4} - 112x^{3} - 560x^{2} + 4368x - 4064$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(249))$	13
Old forms	14
Cusp forms $S_{2}(\Gamma_0(249))$	27
Eisenstein series $E_{2}(\Gamma_0(249))$	3
Modular forms $M_{2}(\Gamma_0(249))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(249)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 83, 249]: [0, 1, 8, 30] (certified at ell = 2147483323)
T = [(2, 1), (5, 3)]; h of degree 13 = 1 x 13 (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(249, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 249:2 1000

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