Atlas › Modular forms › Weight 2 › Level 759 › 759.2.a

Newspace 759.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(759))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
759.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	759a1 rank 1
759.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + O(q^{8})$	− − +	$-1$	759b1 rank 1
759.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 2q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
759.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
759.2.a.e	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} - 4q^{5} + q^{6} + O(q^{8})$	+ + +	$-1$
759.2.a.f	3	degree 3	$3q - q^{2} + 3q^{3} + q^{4} - 2q^{5} - q^{6} - 6q^{7} + O(q^{8})$	− + −	$-1$
759.2.a.g	5	degree 5	$5q - 2q^{2} + 5q^{3} + 12q^{4} + 4q^{5} - 2q^{6} + O(q^{8})$	− − −	$+1$
759.2.a.h	6	degree 6	$6q + 2q^{2} - 6q^{3} + 4q^{4} + 6q^{5} - 2q^{6} + O(q^{8})$	+ + −	$+1$
759.2.a.i	8	degree 8	$8q + 2q^{2} - 8q^{3} + 14q^{4} + 6q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	+ − +	$+1$
759.2.a.j	8	degree 8	$8q + 2q^{2} + 8q^{3} + 10q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	− + +	$+1$

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 $T$
759.2.a.a	$x + 1$
759.2.a.b	$x + 7$
759.2.a.c	$x^{2} + 7x + 11$
759.2.a.d	$x^{2} - 14x + 47$
759.2.a.e	$x^{3} + 13x^{2} + 3x - 145$
759.2.a.f	$x^{3} + 7x^{2} + 3x - 19$
759.2.a.g	$x^{5} - 10x^{4} - 195x^{3} + 2121x^{2} + 6014x - 77251$
759.2.a.h	$x^{6} - 20x^{5} - 12x^{4} + 1624x^{3} - 3424x^{2} - 10154x + 23029$
759.2.a.i	$x^{8} - 20x^{7} - 121x^{6} + 4168x^{5} - 9032x^{4} - 211978x^{3} + 1118781x^{2} + 33110x - 5755613$
759.2.a.j	$x^{8} - 2x^{7} - 263x^{6} + 572x^{5} + 18922x^{4} - 31178x^{3} - 373905x^{2} + 505228x + 1238001$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(759))$	39
Old forms	54
Cusp forms $S_{2}(\Gamma_0(759))$	93
Eisenstein series $E_{2}(\Gamma_0(759))$	7
Modular forms $M_{2}(\Gamma_0(759))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(759)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(253))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.388 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, 11, 23, 33, 69, 253, 759]: [0, 1, 2, 3, 6, 10, 26, 100] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(759, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 759:2 1000

← 758.2.a · 760.2.a →