Atlas › Modular forms › Weight 2 › Level 217 › 217.2.a

Newspace 217.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(217))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
217.2.a.a	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 6q^{5} + 6q^{6} + 3q^{7} + O(q^{8})$	− −	$-1$
217.2.a.b	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{7} + O(q^{8})$	+ +	$-1$
217.2.a.c	4	degree 4	$4q + 3q^{3} + 2q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
217.2.a.d	5	degree 5	$5q + 3q^{2} + 3q^{3} + 9q^{4} - 2q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
217.2.a.a	$x^{3} + 6x^{2} + 3x - 1$
217.2.a.b	$x^{3} + 6x^{2} + 9x + 3$
217.2.a.c	$x^{4} - 3x^{3} - 11x^{2} + 20x + 43$
217.2.a.d	$x^{5} - 6x^{4} + 57x^{2} - 115x + 67$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(217))$	15
Old forms	4
Cusp forms $S_{2}(\Gamma_0(217))$	19
Eisenstein series $E_{2}(\Gamma_0(217))$	3
Modular forms $M_{2}(\Gamma_0(217))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(217)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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, 7, 31, 217]: [0, 1, 3, 22] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(217, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 217:2 1000

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