Atlas › Modular forms › Weight 2 › Level 662 › 662.2.a

Newspace 662.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(662))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
662.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	662a1 rank 1
662.2.a.b	4	degree 4	$4q - 4q^{2} - q^{3} + 4q^{4} - 5q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
662.2.a.c	4	degree 4	$4q + 4q^{2} - 5q^{3} + 4q^{4} - 3q^{5} - 5q^{6} - 8q^{7} + O(q^{8})$	− −	$-1$
662.2.a.d	9	degree 9	$9q - 9q^{2} + 6q^{3} + 9q^{4} + 5q^{5} - 6q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
662.2.a.e	10	degree 10	$10q + 10q^{2} + 6q^{3} + 10q^{4} + 6q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
662.2.a.a	$x - 19$
662.2.a.b	$x^{4} + 8x^{3} - 994x^{2} - 4167x + 74169$
662.2.a.c	$x^{4} + 96x^{3} + 2466x^{2} + 10751x - 116839$
662.2.a.d	$x^{9} - 16x^{8} - 4003x^{7} + 95158x^{6} + 3000690x^{5} - 71664412x^{4} - 587795154x^{3} + 14015379500x^{2} - 10320674300x - 71627755125$
662.2.a.e	$x^{10} - 95x^{9} + 1497x^{8} + 117175x^{7} - 4877020x^{6} + 40658774x^{5} + 847684182x^{4} - 16794797054x^{3} + 78506834288x^{2} - 15304176387x - 281393088443$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(662))$	28
Old forms	54
Cusp forms $S_{2}(\Gamma_0(662))$	82
Eisenstein series $E_{2}(\Gamma_0(662))$	3
Modular forms $M_{2}(\Gamma_0(662))$	85

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(662)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(331))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 331, 662]: [0, 1, 28, 85] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 7 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(662, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 662:2 1000

← 661.2.a · 663.2.a →