Atlas › Modular forms › Weight 2 › Level 327 › 327.2.a

Newspace 327.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(327))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
327.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	327a1 rank 1
327.2.a.b	3	degree 3	$3q - 3q^{2} - 3q^{3} + 5q^{4} - 3q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
327.2.a.c	6	degree 6	$6q + 4q^{2} - 6q^{3} + 8q^{4} + 5q^{5} - 4q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
327.2.a.d	9	degree 9	$9q + 3q^{2} + 9q^{3} + 13q^{4} + q^{5} + 3q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

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 $T$
327.2.a.a	$x + 2$
327.2.a.b	$x^{3} + 6x^{2} + 8x - 2$
327.2.a.c	$x^{6} - 9x^{5} + 5x^{4} + 141x^{3} - 455x^{2} + 529x - 214$
327.2.a.d	$x^{9} - 4x^{8} - 36x^{7} + 144x^{6} + 338x^{5} - 1438x^{4} - 1079x^{3} + 5202x^{2} + 964x - 5866$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(327))$	19
Old forms	16
Cusp forms $S_{2}(\Gamma_0(327))$	35
Eisenstein series $E_{2}(\Gamma_0(327))$	3
Modular forms $M_{2}(\Gamma_0(327))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(327)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 109, 327]: [0, 1, 9, 38] (certified at ell = 2147482801)
T = [(2, 1), (5, 1)]; h of degree 19 = 1 x 19 (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(327, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 327:2 1000

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