Atlas › Modular forms › Weight 2 › Level 648 › 648.2.a

Newspace 648.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(648))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
648.2.a.a	1	$\mathbb{Q}$	$q - q^{5} - 3q^{7} + O(q^{8})$	648b1 rank 1
648.2.a.b	1	$\mathbb{Q}$	$q - q^{5} + O(q^{8})$	648a1 rank 1
648.2.a.c	1	$\mathbb{Q}$	$q + q^{5} - 3q^{7} + O(q^{8})$	648d1 rank 1
648.2.a.d	1	$\mathbb{Q}$	$q + q^{5} + O(q^{8})$	648c1 rank 0
648.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{5} + O(q^{8})$
648.2.a.f	2	$\mathbb{Q}(\sqrt{33})$	$2q - q^{5} + 3q^{7} + O(q^{8})$
648.2.a.g	2	$\mathbb{Q}(\sqrt{33})$	$2q + q^{5} + 3q^{7} + O(q^{8})$
648.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{5} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
648.2.a.a	$x + 10$
648.2.a.b	$x + 1$
648.2.a.c	$x + 8$
648.2.a.d	$x - 1$
648.2.a.e	$x^{2} + 4x - 71$
648.2.a.f	$x^{2} - 8x - 116$
648.2.a.g	$x^{2} - 10x - 8$
648.2.a.h	$x^{2} - 4x - 143$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(648))$	12
Old forms	73
Cusp forms $S_{2}(\Gamma_0(648))$	85
Eisenstein series $E_{2}(\Gamma_0(648))$	47
Modular forms $M_{2}(\Gamma_0(648))$	132

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.279 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, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648]: [0, 1, 1, 2, 3, 3, 2, 5, 5, 8, 4, 9, 11, 16, 10, 21, 29, 40, 57, 112] (certified at ell = 2147483179)
T = [(5, 1), (7, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(648, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 648:2 1000

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