Atlas › Modular forms › Weight 12 › Level 19 › 19.12.a

Newspace 19.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(19))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
19.12.a.a	7	degree 7	$7q - 9q^{2} + 10q^{3} + 6661q^{4} - 14307q^{5} - 66435q^{6} - 2209q^{7} + O(q^{8})$	−	$-1$
19.12.a.b	9	degree 9	$9q + 87q^{2} + 496q^{3} + 11781q^{4} + 2114q^{5} + 84181q^{6} - 19080q^{7} + O(q^{8})$	+	$+1$

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 $T$
19.12.a.a	$x^{7} - x^{6} - 737511x^{5} - 12159567x^{4} + 131948713182x^{3} - 4449565121016x^{2} - 5287613705621856x + 459570856744281600$
19.12.a.b	$x^{9} - 583x^{8} - 1042721x^{7} + 466702653x^{6} + 396196150308x^{5} - 103292325596454x^{4} - 59342910232666924x^{3} + 3714367199125112392x^{2} + 1458964133140235097008x - 50351612334872711003904$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(19))$	16
Old forms	2
Cusp forms $S_{12}(\Gamma_0(19))$	18
Eisenstein series $E_{12}(\Gamma_0(19))$	2
Modular forms $M_{12}(\Gamma_0(19))$	20

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(19)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.506 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, 19]: [2, 20] (certified at ell = 2147483647)
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(19, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 19:12 1000

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