Atlas › Modular forms › Weight 2 › Level 486 › 486.2.a

Newspace 486.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(486))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
486.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} + 2q^{7} + O(q^{8})$	+	486b1 rank 1
486.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{7} + O(q^{8})$	+	486a1 rank 1
486.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	+	486c1 rank 0
486.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	−	486f1 rank 1
486.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{7} + O(q^{8})$	−	486d1 rank 0
486.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	−	486e1 rank 0
486.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{4} + 6q^{7} + O(q^{8})$	+
486.2.a.h	3	degree 3	$3q + 3q^{2} + 3q^{4} + 6q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
486.2.a.a	$x + 323$
486.2.a.b	$x + 281$
486.2.a.c	$x - 319$
486.2.a.d	$x - 25$
486.2.a.e	$x + 137$
486.2.a.f	$x + 245$
486.2.a.g	$x^{3} - 723x^{2} + 125850x + 497933$
486.2.a.h	$x^{3} + 591x^{2} + 26976x - 6079267$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(486))$	12
Old forms	52
Cusp forms $S_{2}(\Gamma_0(486))$	64
Eisenstein series $E_{2}(\Gamma_0(486))$	35
Modular forms $M_{2}(\Gamma_0(486))$	99

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(486)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(162))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(243))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.983 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, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486]: [0, 1, 1, 3, 2, 5, 4, 11, 10, 29, 28, 83] (certified at ell = 2147483179)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 12 = 1 x 12 (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(486, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 486:2 1000

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