Atlas › Modular forms › Weight 2 › Level 465 › 465.2.a

Newspace 465.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(465))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
465.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$	465b1 rank 1
465.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	465a1 rank 1
465.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$
465.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 2q^{5} - 6q^{7} + O(q^{8})$	+ + +	$-1$
465.2.a.e	3	degree 3	$3q + q^{2} - 3q^{3} + 5q^{4} - 3q^{5} - q^{6} + 8q^{7} + O(q^{8})$	+ + −	$+1$
465.2.a.f	3	degree 3	$3q + q^{2} + 3q^{3} + q^{4} + 3q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
465.2.a.g	3	degree 3	$3q + 3q^{2} + 3q^{3} + 5q^{4} - 3q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
465.2.a.h	4	degree 4	$4q + 2q^{2} - 4q^{3} + 8q^{4} + 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
465.2.a.a	$x + 13$
465.2.a.b	$x + 5$
465.2.a.c	$x^{2} + 14x + 41$
465.2.a.d	$x^{2} + 18x + 69$
465.2.a.e	$x^{3} - 25x^{2} + 187x - 411$
465.2.a.f	$x^{3} - 7x^{2} + 3x + 19$
465.2.a.g	$x^{3} - 9x^{2} - 109x - 115$
465.2.a.h	$x^{4} - 14x^{3} - 84x^{2} + 1374x - 3613$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(465))$	19
Old forms	42
Cusp forms $S_{2}(\Gamma_0(465))$	61
Eisenstein series $E_{2}(\Gamma_0(465))$	7
Modular forms $M_{2}(\Gamma_0(465))$	68

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(465)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(155))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 15, 31, 93, 155, 465]: [0, 1, 1, 4, 3, 12, 18, 68] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(465, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 465:2 1000

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