Atlas › Modular forms › Weight 2 › Level 415 › 415.2.a

Newspace 415.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(415))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
415.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} - q^{4} + q^{5} + 3q^{6} + q^{7} + O(q^{8})$	− +	$+1$	415a1 rank 0
415.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{3} - q^{4} + 2q^{5} - 2q^{6} + O(q^{8})$	− −	$-1$
415.2.a.c	6	degree 6	$6q + 2q^{2} + 3q^{3} + 2q^{4} - 6q^{5} + 4q^{6} + O(q^{8})$	+ −	$+1$
415.2.a.d	7	degree 7	$7q - 3q^{2} - 5q^{3} + 7q^{4} - 7q^{5} - 6q^{7} + O(q^{8})$	+ +	$-1$
415.2.a.e	11	degree 11	$11q + 18q^{4} + 11q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$

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 $T$
415.2.a.a	$x - 10$
415.2.a.b	$x^{2} + 4x - 1$
415.2.a.c	$x^{6} - 11x^{5} - 80x^{4} + 840x^{3} + 1448x^{2} - 13371x - 8779$
415.2.a.d	$x^{7} + 18x^{6} + 48x^{5} - 469x^{4} - 921x^{3} + 5365x^{2} - 4606x + 964$
415.2.a.e	$x^{11} - 248x^{9} + 335x^{8} + 22121x^{7} - 48198x^{6} - 856784x^{5} + 2210696x^{4} + 13158705x^{3} - 31622405x^{2} - 43975783x - 12032336$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(415))$	27
Old forms	14
Cusp forms $S_{2}(\Gamma_0(415))$	41
Eisenstein series $E_{2}(\Gamma_0(415))$	3
Modular forms $M_{2}(\Gamma_0(415))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(415)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 5, 83, 415]: [0, 1, 8, 44] (certified at ell = 2147483077)
T = [(2, 1), (3, 3)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 4 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(415, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 415:2 1000

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