Atlas › Modular forms › Weight 2 › Level 248 › 248.2.a

Newspace 248.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(248))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
248.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} - 3q^{7} + O(q^{8})$	+	248a1 rank 1
248.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} + O(q^{8})$	+	248b1 rank 0
248.2.a.c	1	$\mathbb{Q}$	$q - 3q^{5} - 3q^{7} + O(q^{8})$	−	248c1 rank 1
248.2.a.d	2	$\mathbb{Q}(\sqrt{33})$	$2q + 4q^{3} + 3q^{5} + q^{7} + O(q^{8})$	+
248.2.a.e	3	degree 3	$3q + 2q^{3} - 3q^{5} + 5q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
248.2.a.a	$x + 1$
248.2.a.b	$x$
248.2.a.c	$x + 3$
248.2.a.d	$x^{2} - 7x + 4$
248.2.a.e	$x^{3} + x^{2} - 24x + 38$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(248))$	8
Old forms	21
Cusp forms $S_{2}(\Gamma_0(248))$	29
Eisenstein series $E_{2}(\Gamma_0(248))$	7
Modular forms $M_{2}(\Gamma_0(248))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(248)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 4, 8, 31, 62, 124, 248]: [0, 1, 2, 3, 3, 10, 19, 36] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 1 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(248, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 248:2 1000

← 247.2.a · 249.2.a →