Atlas › Modular forms › Weight 6 › Level 26 › 26.6.a

Newspace 26.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(26))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
26.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 16q^{4} - 14q^{5} - 170q^{7} + O(q^{8})$	+ +	$-1$
26.6.a.b	2	$\mathbb{Q}(\sqrt{2785})$	$2q - 8q^{2} + 9q^{3} + 32q^{4} - 37q^{5} - 36q^{6} + 327q^{7} + O(q^{8})$	+ −	$+1$
26.6.a.c	2	$\mathbb{Q}(\sqrt{849})$	$2q + 8q^{2} + 9q^{3} + 32q^{4} + 73q^{5} + 36q^{6} + 155q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
26.6.a.a	$x + 42$
26.6.a.b	$x^{2} + 102x - 184$
26.6.a.c	$x^{2} - 228x - 588$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(26))$	5
Old forms	10
Cusp forms $S_{6}(\Gamma_0(26))$	15
Eisenstein series $E_{6}(\Gamma_0(26))$	4
Modular forms $M_{6}(\Gamma_0(26))$	19

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(26)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 13, 26]: [1, 2, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (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(26, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 26:6 1000

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