Atlas › Modular forms › Weight 2 › Level 405 › 405.2.a

Newspace 405.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(405))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
405.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - q^{5} + O(q^{8})$	+	405f1 rank 1
405.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 3q^{7} + O(q^{8})$	−	405d1 rank 1
405.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} + 2q^{7} + O(q^{8})$	+	405b1 rank 1
405.2.a.d	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} + 2q^{7} + O(q^{8})$	−	405a1 rank 0
405.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} - 3q^{7} + O(q^{8})$	+	405c1 rank 1
405.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + q^{5} + O(q^{8})$	−	405e1 rank 0
405.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 4q^{4} + 2q^{5} - 6q^{7} + O(q^{8})$	−
405.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 4q^{4} - 2q^{5} - 6q^{7} + O(q^{8})$	+
405.2.a.i	3	degree 3	$3q - q^{2} + 5q^{4} - 3q^{5} + 5q^{7} + O(q^{8})$	+
405.2.a.j	3	degree 3	$3q + q^{2} + 5q^{4} + 3q^{5} + 5q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
405.2.a.a	$x + 13$
405.2.a.b	$x + 90$
405.2.a.c	$x + 171$
405.2.a.d	$x - 135$
405.2.a.e	$x + 34$
405.2.a.f	$x + 15$
405.2.a.g	$x^{2} + 84x + 441$
405.2.a.h	$x^{2} - 52x + 1$
405.2.a.i	$x^{3} - 100x^{2} - 6224x - 42816$
405.2.a.j	$x^{3} - 3456x - 62208$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(405))$	16
Old forms	27
Cusp forms $S_{2}(\Gamma_0(405))$	43
Eisenstein series $E_{2}(\Gamma_0(405))$	23
Modular forms $M_{2}(\Gamma_0(405))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(405)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(135))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 27, 45, 81, 135, 405]: [0, 1, 1, 2, 4, 4, 8, 10, 20, 56] (certified at ell = 2147482801)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 5 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(405, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 405:2 1000

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