Atlas › Modular forms › Weight 12 › Level 25 › 25.12.a

Newspace 25.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(25))$

Newforms

Label	Dim	Coefficient field	Trace form
25.12.a.a	1	$\mathbb{Q}$	$q - 34q^{2} + 792q^{3} - 892q^{4} - 26928q^{6} + 17556q^{7} + O(q^{8})$
25.12.a.b	1	$\mathbb{Q}$	$q + 24q^{2} - 252q^{3} - 1472q^{4} - 6048q^{6} + 16744q^{7} + O(q^{8})$
25.12.a.c	2	$\mathbb{Q}(\sqrt{151})$	$2q + 20q^{2} + 220q^{3} + 6976q^{4} + 60184q^{6} - 57900q^{7} + O(q^{8})$
25.12.a.d	4	degree 4	$4q - 55q^{2} + 20q^{3} + 5557q^{4} - 31187q^{6} - 96400q^{7} + O(q^{8})$
25.12.a.e	4	degree 4	$4q + 72q^{4} - 1752q^{6} + O(q^{8})$
25.12.a.f	4	degree 4	$4q + 55q^{2} - 20q^{3} + 5557q^{4} - 31187q^{6} + 96400q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
25.12.a.a	$x - 758$
25.12.a.b	$x + 228$
25.12.a.c	$x^{2} - 240x - 203644$
25.12.a.d	$x^{4} + 35x^{3} - 428953x^{2} + 55701685x + 10321346496$
25.12.a.e	$x^{4} - 343948x^{2} + 12206704576$
25.12.a.f	$x^{4} - 35x^{3} - 428953x^{2} - 55701685x + 10321346496$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(25))$	16
Old forms	9
Cusp forms $S_{12}(\Gamma_0(25))$	25
Eisenstein series $E_{12}(\Gamma_0(25))$	6
Modular forms $M_{12}(\Gamma_0(25))$	31

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(25)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 3}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.524 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, 5, 25]: [2, 7, 29] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(25, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 25:12 1000

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