Atlas › Modular forms › Weight 2 › Level 321 › 321.2.a

Newspace 321.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(321))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
321.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} + 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
321.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 6q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
321.2.a.c	6	degree 6	$6q + 3q^{2} + 6q^{3} + 7q^{4} + 6q^{5} + 3q^{6} + O(q^{8})$	− +	$+1$
321.2.a.d	7	degree 7	$7q - 7q^{3} + 14q^{4} - 8q^{5} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
321.2.a.a	$x^{2} - x - 1$
321.2.a.b	$x^{2} + 7x + 11$
321.2.a.c	$x^{6} - 9x^{5} + 22x^{4} - 5x^{3} - 40x^{2} + 41x - 9$
321.2.a.d	$x^{7} + 8x^{6} - 15x^{5} - 191x^{4} - 133x^{3} + 613x^{2} + 664x + 167$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(321))$	17
Old forms	18
Cusp forms $S_{2}(\Gamma_0(321))$	35
Eisenstein series $E_{2}(\Gamma_0(321))$	3
Modular forms $M_{2}(\Gamma_0(321))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(321)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 107, 321]: [0, 1, 10, 38] (certified at ell = 2147482621)
T = [(2, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(321, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 321:2 1000

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