Atlas › Modular forms › Weight 2 › Level 269 › 269.2.a

Newspace 269.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(269))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
269.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - 4q^{7} + O(q^{8})$	+	$-1$	269a1 rank 1
269.2.a.b	5	degree 5	$5q - q^{2} - 5q^{3} + q^{4} - 4q^{5} - 4q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
269.2.a.c	16	degree 16	$16q + q^{2} + 5q^{3} + 25q^{4} - q^{5} - 2q^{6} + 11q^{7} + O(q^{8})$	−	$+1$

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 $T$
269.2.a.a	$x$
269.2.a.b	$x^{5} + 6x^{4} + 7x^{3} - 6x^{2} - 10x - 3$
269.2.a.c	$x^{16} - 6x^{15} - 43x^{14} + 314x^{13} + 462x^{12} - 5829x^{11} + 2229x^{10} + 45324x^{9} - 56637x^{8} - 137976x^{7} + 239275x^{6} + 160761x^{5} - 347291x^{4} - 80588x^{3} + 167274x^{2} + 27003x + 626$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(269))$	22
Old forms	0
Cusp forms $S_{2}(\Gamma_0(269))$	22
Eisenstein series $E_{2}(\Gamma_0(269))$	1
Modular forms $M_{2}(\Gamma_0(269))$	23

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 269]: [0, 23] (certified at ell = 2147482661)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 3 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(269, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 269:2 1000

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