Atlas › Modular forms › Weight 2 › Level 209 › 209.2.a

Newspace 209.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(209))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
209.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	209a1 rank 1
209.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
209.2.a.c	5	degree 5	$5q + 2q^{2} + q^{3} + 6q^{4} - 5q^{5} - 2q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
209.2.a.d	7	degree 7	$7q - q^{2} + 2q^{3} + 15q^{4} + 2q^{5} - 2q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
209.2.a.a	$x - 3$
209.2.a.b	$x^{2} + 6x + 1$
209.2.a.c	$x^{5} - 5x^{4} - 75x^{3} + 559x^{2} - 991x + 125$
209.2.a.d	$x^{7} - 5x^{6} - 140x^{5} + 656x^{4} + 5492x^{3} - 20358x^{2} - 58557x + 70863$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(209))$	15
Old forms	4
Cusp forms $S_{2}(\Gamma_0(209))$	19
Eisenstein series $E_{2}(\Gamma_0(209))$	3
Modular forms $M_{2}(\Gamma_0(209))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(209)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

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, 11, 19, 209]: [0, 2, 2, 22] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; 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(209, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 209:2 1000

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