Atlas › Modular forms › Weight 2 › Level 783 › 783.2.a

Newspace 783.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(783))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
783.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} + 2q^{5} - 3q^{7} + O(q^{8})$	+
783.2.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 5q^{4} - 4q^{5} - 3q^{7} + O(q^{8})$	+
783.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	−
783.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} + 4q^{5} - 3q^{7} + O(q^{8})$	−
783.2.a.e	3	degree 3	$3q - 2q^{2} + q^{7} + O(q^{8})$	+
783.2.a.f	3	degree 3	$3q + 2q^{2} + q^{7} + O(q^{8})$	−
783.2.a.g	4	degree 4	$4q - 3q^{2} + 7q^{4} - 4q^{5} - q^{7} + O(q^{8})$	−
783.2.a.h	4	degree 4	$4q + 3q^{2} + 7q^{4} + 4q^{5} - q^{7} + O(q^{8})$	+
783.2.a.i	8	degree 8	$8q - q^{2} + 11q^{4} + 2q^{5} + 5q^{7} + O(q^{8})$	−
783.2.a.j	8	degree 8	$8q + q^{2} + 11q^{4} - 2q^{5} + 5q^{7} + O(q^{8})$	+

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 $T$
783.2.a.a	$x^{2} - x - 1$
783.2.a.b	$x^{2} + 5x + 2$
783.2.a.c	$x^{2} + x - 1$
783.2.a.d	$x^{2} - 5x + 2$
783.2.a.e	$x^{3} + 2x^{2} - 8x - 8$
783.2.a.f	$x^{3} - 2x^{2} - 8x + 8$
783.2.a.g	$x^{4} + 7x^{3} + 6x^{2} - 32x - 40$
783.2.a.h	$x^{4} - 7x^{3} + 6x^{2} + 32x - 40$
783.2.a.i	$x^{8} - x^{7} - 55x^{6} + 13x^{5} + 809x^{4} - 321x^{3} - 3815x^{2} + 2827x + 2205$
783.2.a.j	$x^{8} + x^{7} - 55x^{6} - 13x^{5} + 809x^{4} + 321x^{3} - 3815x^{2} - 2827x + 2205$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(783))$	38
Old forms	47
Cusp forms $S_{2}(\Gamma_0(783))$	85
Eisenstein series $E_{2}(\Gamma_0(783))$	11
Modular forms $M_{2}(\Gamma_0(783))$	96

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(783)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(261))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.251 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, 27, 29, 87, 261, 783]: [0, 1, 2, 4, 3, 12, 32, 92] (certified at ell = 2147483269)
T = [(2, 1), (5, 1)]; h of degree 38 = 1 x 38 (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(783, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 783:2 1000

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