Atlas › Modular forms › Weight 2 › Level 972 › 972.2.a

Newspace 972.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(972))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
972.2.a.a	1	$\mathbb{Q}$	$q - 4q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	972d1 rank 1
972.2.a.b	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	972c1 rank 1
972.2.a.c	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	972a1 rank 0
972.2.a.d	1	$\mathbb{Q}$	$q + 5q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	972b1 rank 0
972.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{7} + O(q^{8})$
972.2.a.f	3	degree 3	$3q - 3q^{7} + O(q^{8})$
972.2.a.g	3	degree 3	$3q - 3q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
972.2.a.a	$x - 23$
972.2.a.b	$x + 52$
972.2.a.c	$x - 11$
972.2.a.d	$x - 50$
972.2.a.e	$x^{2} + 2x - 647$
972.2.a.f	$x^{3} + 75x^{2} + 1272x + 6211$
972.2.a.g	$x^{3} - 15x^{2} - 1122x - 9377$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(972))$	12
Old forms	124
Cusp forms $S_{2}(\Gamma_0(972))$	136
Eisenstein series $E_{2}(\Gamma_0(972))$	53
Modular forms $M_{2}(\Gamma_0(972))$	189

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.449 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, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972]: [0, 1, 1, 2, 3, 2, 5, 5, 4, 9, 11, 10, 21, 29, 28, 57, 83, 165] (certified at ell = 2147483179)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 12 = 1 x 12 (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(972, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 972:2 1000

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