Atlas › Modular forms › Weight 12 › Level 17 › 17.12.a

Newspace 17.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(17))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
17.12.a.a	6	degree 6	$6q - 9q^{2} - 476q^{3} + 4613q^{4} - 12884q^{5} + 552q^{6} - 23436q^{7} + O(q^{8})$	−	$-1$
17.12.a.b	8	degree 8	$8q + 55q^{2} + 496q^{3} + 10757q^{4} + 8592q^{5} + 17194q^{6} + 95288q^{7} + O(q^{8})$	+	$+1$

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 $T$
17.12.a.a	$x^{6} + 485x^{5} - 571030x^{4} - 268719892x^{3} + 80932345936x^{2} + 36890201127040x + 101392495911936$
17.12.a.b	$x^{8} - 551x^{7} - 837876x^{6} + 391948436x^{5} + 192257457996x^{4} - 77857488689676x^{3} - 10513316544768032x^{2} + 2989821396207767984x + 318453699583382083008$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(17))$	14
Old forms	2
Cusp forms $S_{12}(\Gamma_0(17))$	16
Eisenstein series $E_{12}(\Gamma_0(17))$	2
Modular forms $M_{12}(\Gamma_0(17))$	18

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(17)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.504 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, 17]: [2, 18] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (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(17, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 17:12 1000

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