Atlas › Modular forms › Weight 12 › Level 26 › 26.12.a

Newspace 26.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(26))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
26.12.a.a	2	$\mathbb{Q}(\sqrt{10921})$	$2q - 64q^{2} - 243q^{3} + 2048q^{4} - 4985q^{5} + 7776q^{6} + 10155q^{7} + O(q^{8})$	+ −	$-1$
26.12.a.b	2	$\mathbb{Q}(\sqrt{16545})$	$2q + 64q^{2} - 243q^{3} + 2048q^{4} - 2035q^{5} - 7776q^{6} - 27601q^{7} + O(q^{8})$	− +	$-1$
26.12.a.c	3	degree 3	$3q - 96q^{2} + 3072q^{4} - 1464q^{5} + 68514q^{7} + O(q^{8})$	+ +	$+1$
26.12.a.d	4	degree 4	$4q + 128q^{2} + 4096q^{4} + 694q^{5} - 5844q^{7} + O(q^{8})$	− −	$+1$

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 $T$
26.12.a.a	$x^{2} + 5228x - 4350108$
26.12.a.b	$x^{2} + 2278x - 41736224$
26.12.a.c	$x^{3} + 1464x^{2} - 119280756x - 333302387264$
26.12.a.d	$x^{4} - 694x^{3} - 177859540x^{2} - 289416160712x + 691689639135360$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(26))$	11
Old forms	26
Cusp forms $S_{12}(\Gamma_0(26))$	37
Eisenstein series $E_{12}(\Gamma_0(26))$	4
Modular forms $M_{12}(\Gamma_0(26))$	41

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(26)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.546 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]: [2, 4, 15, 41] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 5 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, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 26:12 1000

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