Atlas › Modular forms › Weight 8 › Level 26 › 26.8.a

Newspace 26.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(26))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
26.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 39q^{3} + 64q^{4} + 385q^{5} + 312q^{6} - 293q^{7} + O(q^{8})$	+ −	$-1$
26.8.a.b	1	$\mathbb{Q}$	$q + 8q^{2} - 87q^{3} + 64q^{4} + 321q^{5} - 696q^{6} - 181q^{7} + O(q^{8})$	− −	$+1$
26.8.a.c	1	$\mathbb{Q}$	$q + 8q^{2} - 27q^{3} + 64q^{4} - 245q^{5} - 216q^{6} - 587q^{7} + O(q^{8})$	− +	$-1$
26.8.a.d	2	$\mathbb{Q}(\sqrt{105})$	$2q - 16q^{2} - 12q^{3} + 128q^{4} - 146q^{5} + 96q^{6} - 1780q^{7} + O(q^{8})$	+ +	$+1$
26.8.a.e	2	$\mathbb{Q}(\sqrt{2305})$	$2q + 16q^{2} + 87q^{3} + 128q^{4} + 215q^{5} + 696q^{6} + 705q^{7} + O(q^{8})$	− −	$+1$

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 $T$
26.8.a.a	$x - 346$
26.8.a.b	$x - 234$
26.8.a.c	$x + 272$
26.8.a.d	$x^{2} + 158x - 187904$
26.8.a.e	$x^{2} - 302x + 2056$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(26))$	7
Old forms	16
Cusp forms $S_{8}(\Gamma_0(26))$	23
Eisenstein series $E_{8}(\Gamma_0(26))$	4
Modular forms $M_{8}(\Gamma_0(26))$	27

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(26)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 3, 9, 27] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 3 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 26:8 1000

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