Atlas › Modular forms › Weight 2 › Level 578 › 578.2.a

Newspace 578.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(578))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
578.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	−	578a1 rank 0
578.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + O(q^{8})$	+
578.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + O(q^{8})$	−
578.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + O(q^{8})$	−
578.2.a.e	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{6} - 9q^{7} + O(q^{8})$	+
578.2.a.f	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + 9q^{7} + O(q^{8})$	+
578.2.a.g	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	−
578.2.a.h	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} + 6q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	−
578.2.a.i	4	degree 4	$4q - 4q^{2} + 4q^{4} + O(q^{8})$	+

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 $T$
578.2.a.a	$x - 2$
578.2.a.b	$x^{2} - 18$
578.2.a.c	$x^{2} - 98$
578.2.a.d	$x^{2} - 32$
578.2.a.e	$x^{3} + 3x^{2} - 60x - 233$
578.2.a.f	$x^{3} - 3x^{2} - 60x + 233$
578.2.a.g	$x^{3} + 21x^{2} + 126x + 179$
578.2.a.h	$x^{3} - 21x^{2} + 126x - 179$
578.2.a.i	$x^{4} - 196x^{2} + 4802$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(578))$	23
Old forms	36
Cusp forms $S_{2}(\Gamma_0(578))$	59
Eisenstein series $E_{2}(\Gamma_0(578))$	35
Modular forms $M_{2}(\Gamma_0(578))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(578)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(289))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 17, 34, 289, 578]: [0, 1, 2, 6, 26, 78] (certified at ell = 2147483249)
T = [(3, 1), (5, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 4 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(578, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 578:2 1000

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