Atlas › Modular forms › Weight 2 › Level 549 › 549.2.a

Newspace 549.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(549))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
549.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{7} + O(q^{8})$	+	549b1 rank 1
549.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{7} + O(q^{8})$	+	549a1 rank 1
549.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 3q^{5} + q^{7} + O(q^{8})$	+	549c1 rank 0
549.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} - 2q^{7} + O(q^{8})$	−
549.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	+
549.2.a.f	3	degree 3	$3q - q^{2} + q^{4} - 6q^{5} + O(q^{8})$	−
549.2.a.g	3	degree 3	$3q - q^{2} + q^{4} + q^{5} - 3q^{7} + O(q^{8})$	−
549.2.a.h	6	degree 6	$6q + 10q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	+
549.2.a.i	6	degree 6	$6q + 14q^{4} + 6q^{7} + O(q^{8})$	−

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 $T$
549.2.a.a	$x + 1$
549.2.a.b	$x - 1$
549.2.a.c	$x - 4$
549.2.a.d	$x^{2} - 12$
549.2.a.e	$x^{2} - 4x + 2$
549.2.a.f	$x^{3} + 7x^{2} + 13x + 5$
549.2.a.g	$x^{3} - 4x - 2$
549.2.a.h	$x^{6} + 2x^{5} - 28x^{4} - 44x^{3} - 15x^{2} + 4x + 2$
549.2.a.i	$x^{6} - 38x^{4} + 355x^{2} - 484$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(549))$	25
Old forms	34
Cusp forms $S_{2}(\Gamma_0(549))$	59
Eisenstein series $E_{2}(\Gamma_0(549))$	7
Modular forms $M_{2}(\Gamma_0(549))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(549)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(183))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 9, 61, 183, 549]: [0, 1, 2, 5, 22, 64] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(549, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 549:2 1000

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