Atlas › Modular forms › Weight 2 › Level 765 › 765.2.a

Newspace 765.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(765))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
765.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 2q^{7} + O(q^{8})$	− +	765c1 rank 1
765.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} + 4q^{7} + O(q^{8})$	− +	765b1 rank 0
765.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + 4q^{7} + O(q^{8})$	+ −	765a1 rank 0
765.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} - 2q^{5} + O(q^{8})$	+ −
765.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	− −
765.2.a.f	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{4} + 2q^{5} + O(q^{8})$	− +
765.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ −
765.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+ +
765.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	− −
765.2.a.j	3	degree 3	$3q + 2q^{4} - 3q^{5} + 4q^{7} + O(q^{8})$	+ +
765.2.a.k	3	degree 3	$3q + 6q^{4} - 3q^{5} + O(q^{8})$	+ −
765.2.a.l	3	degree 3	$3q + 6q^{4} + 3q^{5} + O(q^{8})$	− +
765.2.a.m	4	degree 4	$4q - q^{2} + 9q^{4} + 4q^{5} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7} + 5T_{13} + 7T_{19} + 9T_{23}$ 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 $T$
765.2.a.a	$x + 51$
765.2.a.b	$x - 21$
765.2.a.c	$x - 23$
765.2.a.d	$x^{2} + 147x + 4851$
765.2.a.e	$x^{2} + 61x - 601$
765.2.a.f	$x^{2} + 77x + 1479$
765.2.a.g	$x^{2} - 36x - 39$
765.2.a.h	$x^{2} + 131x + 2759$
765.2.a.i	$x^{2} - 26x - 169$
765.2.a.j	$x^{3} - 86x^{2} + 1518x - 5179$
765.2.a.k	$x^{3} - 114x^{2} + 4170x - 48579$
765.2.a.l	$x^{3} - 114x^{2} + 4194x - 50157$
765.2.a.m	$x^{4} - 67x^{3} - 12728x^{2} + 506269x + 32980821$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(765))$	28
Old forms	73
Cusp forms $S_{2}(\Gamma_0(765))$	101
Eisenstein series $E_{2}(\Gamma_0(765))$	15
Modular forms $M_{2}(\Gamma_0(765))$	116

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(765)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(153))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(255))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.572 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, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765]: [0, 1, 1, 2, 4, 2, 8, 8, 10, 20, 40, 112] (certified at ell = 2147483137)
T = [(2, 1), (7, 3), (13, 5), (19, 7), (23, 9)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 8 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(765, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 765:2 1000

← 764.2.a · 766.2.a →