Atlas › Modular forms › Weight 2 › Level 440 › 440.2.a

Newspace 440.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(440))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
440.2.a.a	1	$\mathbb{Q}$	$q - q^{5} - 2q^{7} + O(q^{8})$	+ +	440a1 rank 1
440.2.a.b	1	$\mathbb{Q}$	$q - q^{5} - 2q^{7} + O(q^{8})$	+ −	440b1 rank 1
440.2.a.c	1	$\mathbb{Q}$	$q + q^{5} + 4q^{7} + O(q^{8})$	− +	440c1 rank 0
440.2.a.d	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} + q^{7} + O(q^{8})$	− +	440d1 rank 0
440.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} + 2q^{5} - q^{7} + O(q^{8})$	− −
440.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - 2q^{5} + 3q^{7} + O(q^{8})$	+ −
440.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - 2q^{5} + 5q^{7} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
440.2.a.a	$x + 166$
440.2.a.b	$x + 54$
440.2.a.c	$x - 80$
440.2.a.d	$x + 70$
440.2.a.e	$x^{2} - 68x - 544$
440.2.a.f	$x^{2} - 216x + 11052$
440.2.a.g	$x^{2} - 22x - 712$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(440))$	10
Old forms	55
Cusp forms $S_{2}(\Gamma_0(440))$	65
Eisenstein series $E_{2}(\Gamma_0(440))$	15
Modular forms $M_{2}(\Gamma_0(440))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(440)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(110))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(220))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 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, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440]: [0, 1, 2, 1, 3, 3, 2, 6, 5, 10, 9, 8, 16, 22, 42, 80] (certified at ell = 2147482921)
T = [(3, 1), (7, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 10 = 1 x 10 (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(440, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 440:2 1000

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