Atlas › Modular forms › Weight 8 › Level 14 › 14.8.a

Newspace 14.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(14))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
14.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 82q^{3} + 64q^{4} + 448q^{5} + 656q^{6} - 343q^{7} + O(q^{8})$	+ +	$+1$
14.8.a.b	1	$\mathbb{Q}$	$q + 8q^{2} - 66q^{3} + 64q^{4} - 400q^{5} - 528q^{6} - 343q^{7} + O(q^{8})$	− +	$-1$
14.8.a.c	2	$\mathbb{Q}(\sqrt{1969})$	$2q + 16q^{2} + 70q^{3} + 128q^{4} + 126q^{5} + 560q^{6} + 686q^{7} + O(q^{8})$	− −	$+1$

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 $T$
14.8.a.a	$x - 366$
14.8.a.b	$x + 466$
14.8.a.c	$x^{2} - 196x - 116412$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(14))$	4
Old forms	8
Cusp forms $S_{8}(\Gamma_0(14))$	12
Eisenstein series $E_{8}(\Gamma_0(14))$	4
Modular forms $M_{8}(\Gamma_0(14))$	16

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(14)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.258 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, 7, 14]: [1, 3, 5, 16] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(14, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 14:8 1000

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