Atlas › Modular forms › Weight 2 › Level 775 › 775.2.a

Newspace 775.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(775))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
775.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + O(q^{8})$	+	775a1 rank 1
775.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} - 2q^{6} - 4q^{7} + O(q^{8})$	−	775b1 rank 0
775.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	−	775c1 rank 0
775.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 6q^{6} + 4q^{7} + O(q^{8})$	−
775.2.a.e	4	degree 4	$4q - q^{2} - q^{3} + 5q^{4} - 4q^{6} - 2q^{7} + O(q^{8})$	+
775.2.a.f	4	degree 4	$4q - 4q^{6} + O(q^{8})$	−
775.2.a.g	4	degree 4	$4q + q^{2} + q^{3} + 9q^{4} + 8q^{6} + O(q^{8})$	−
775.2.a.h	5	degree 5	$5q - 4q^{2} - 3q^{3} + 6q^{4} - q^{6} - 2q^{7} + O(q^{8})$	+
775.2.a.i	5	degree 5	$5q - 4q^{2} - q^{3} + 6q^{4} - q^{6} - 6q^{7} + O(q^{8})$	−
775.2.a.j	5	degree 5	$5q + 4q^{2} + q^{3} + 6q^{4} - q^{6} + 6q^{7} + O(q^{8})$	−
775.2.a.k	5	degree 5	$5q + 4q^{2} + 3q^{3} + 6q^{4} - q^{6} + 2q^{7} + O(q^{8})$	+
775.2.a.l	10	degree 10	$10q + 12q^{4} + 8q^{6} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{3} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
775.2.a.a	$x + 85$
775.2.a.b	$x - 61$
775.2.a.c	$x - 23$
775.2.a.d	$x^{2} - 21x - 2201$
775.2.a.e	$x^{4} + 146x^{3} + 2252x^{2} - 131282x - 2102221$
775.2.a.f	$x^{4} + 104x^{3} + 1790x^{2} - 29576x - 270119$
775.2.a.g	$x^{4} + 72x^{3} - 11422x^{2} - 520400x + 22092517$
775.2.a.h	$x^{5} + 198x^{4} + 12889x^{3} + 289155x^{2} + 673815x - 30002341$
775.2.a.i	$x^{5} + 234x^{4} + 15771x^{3} + 273685x^{2} + 1430221x + 2134863$
775.2.a.j	$x^{5} - 178x^{4} + 12119x^{3} - 396019x^{2} + 6225495x - 37782727$
775.2.a.k	$x^{5} - 294x^{4} + 26397x^{3} - 470017x^{2} - 32491939x + 1027666593$
775.2.a.l	$x^{10} - 36x^{9} - 25636x^{8} + 183698x^{7} + 165552583x^{6} - 757890638x^{5} - 362041869433x^{4} + 341524624070x^{3} + 258779193990712x^{2} + 1623926611842794x - 22541755003100483$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(775))$	47
Old forms	28
Cusp forms $S_{2}(\Gamma_0(775))$	75
Eisenstein series $E_{2}(\Gamma_0(775))$	11
Modular forms $M_{2}(\Gamma_0(775))$	86

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(775)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(155))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.451 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, 5, 25, 31, 155, 775]: [0, 1, 3, 3, 18, 82] (certified at ell = 2147482921)
T = [(2, 1), (3, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 13 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(775, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 775:2 1000

← 774.2.a · 776.2.a →