Atlas › Modular forms › Weight 12 › Level 13 › 13.12.a

Newspace 13.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(13))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
13.12.a.a	5	degree 5	$5q - 41q^{2} - 496q^{3} + 2993q^{4} - 2542q^{5} + 35441q^{6} - 36296q^{7} + O(q^{8})$	−	$-1$
13.12.a.b	6	degree 6	$6q + 55q^{2} + 476q^{3} + 15309q^{4} + 3312q^{5} - 57797q^{6} - 4176q^{7} + O(q^{8})$	+	$+1$

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 $T$
13.12.a.a	$x^{5} + 537x^{4} - 427063x^{3} - 222063897x^{2} - 31352452530x - 1392053521176$
13.12.a.b	$x^{6} - 531x^{5} - 784571x^{4} + 368209683x^{3} + 170603219290x^{2} - 63992195659536x - 6815180829926288$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(13))$	11
Old forms	2
Cusp forms $S_{12}(\Gamma_0(13))$	13
Eisenstein series $E_{12}(\Gamma_0(13))$	2
Modular forms $M_{12}(\Gamma_0(13))$	15

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.49 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, 13]: [2, 15] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 4 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(13, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 13:12 1000

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