Atlas › Modular forms › Weight 2 › Level 495 › 495.2.a

Newspace 495.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(495))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
495.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} + O(q^{8})$	+ −	495a1 rank 1
495.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	− +
495.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− −
495.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	− −
495.2.a.e	3	degree 3	$3q + q^{2} + 5q^{4} - 3q^{5} + O(q^{8})$	+ +
495.2.a.f	4	degree 4	$4q - 2q^{2} + 8q^{4} - 4q^{5} + 4q^{7} + O(q^{8})$	+ −
495.2.a.g	4	degree 4	$4q + 2q^{2} + 8q^{4} + 4q^{5} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7} + 5T_{19} + 7T_{23} + 9T_{29}$ 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 $T$
495.2.a.a	$x + 103$
495.2.a.b	$x^{2} + 50x - 4177$
495.2.a.c	$x^{2} - 32x + 253$
495.2.a.d	$x^{2} - 42x - 9$
495.2.a.e	$x^{3} - 131x^{2} + 5263x - 66697$
495.2.a.f	$x^{4} - 122x^{3} - 6866x^{2} + 730106x - 8665589$
495.2.a.g	$x^{4} + 58x^{3} - 19386x^{2} - 2139746x - 58942269$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(495))$	18
Old forms	47
Cusp forms $S_{2}(\Gamma_0(495))$	65
Eisenstein series $E_{2}(\Gamma_0(495))$	15
Modular forms $M_{2}(\Gamma_0(495))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(495)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(99))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(165))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.424 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, 11, 15, 33, 45, 55, 99, 165, 495]: [0, 1, 1, 2, 2, 4, 6, 8, 8, 14, 28, 76] (certified at ell = 2147482921)
T = [(2, 1), (7, 3), (19, 5), (23, 7), (29, 9)]; h of degree 18 = 1 x 18 (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(495, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 495:2 1000

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