Atlas › Modular forms › Weight 2 › Level 328 › 328.2.a

Newspace 328.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(328))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
328.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} - 2q^{7} + O(q^{8})$	+	328a1 rank 1
328.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	−	328b1 rank 0
328.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{7} + O(q^{8})$	+
328.2.a.d	3	degree 3	$3q - 4q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	−
328.2.a.e	3	degree 3	$3q - 2q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
328.2.a.a	$x + 2$
328.2.a.b	$x - 4$
328.2.a.c	$x^{2} - 2x - 2$
328.2.a.d	$x^{3} + 6x^{2} + 8x - 2$
328.2.a.e	$x^{3} - 20x + 34$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(328))$	10
Old forms	29
Cusp forms $S_{2}(\Gamma_0(328))$	39
Eisenstein series $E_{2}(\Gamma_0(328))$	7
Modular forms $M_{2}(\Gamma_0(328))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(328)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(164))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 4, 8, 41, 82, 164, 328]: [0, 1, 2, 3, 4, 12, 24, 46] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(328, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 328:2 1000

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