Atlas › Modular forms › Weight 2 › Level 411 › 411.2.a

Newspace 411.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(411))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
411.2.a.a	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
411.2.a.b	3	degree 3	$3q + q^{2} - 3q^{3} - q^{4} - 2q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
411.2.a.c	3	degree 3	$3q + 6q^{2} + 3q^{3} + 6q^{4} + 2q^{5} + 6q^{6} + O(q^{8})$	− +	$+1$
411.2.a.d	5	degree 5	$5q - q^{2} + 5q^{3} + 5q^{4} + 6q^{5} - q^{6} - q^{7} + O(q^{8})$	− +	$+1$
411.2.a.e	9	degree 9	$9q - 9q^{3} + 14q^{4} + 6q^{5} + 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
411.2.a.a	$x^{3} + 9x^{2} + 24x + 19$
411.2.a.b	$x^{3} + x^{2} - 2x - 1$
411.2.a.c	$x^{3} - 8x^{2} + 17x - 5$
411.2.a.d	$x^{5} - 5x^{4} + 13x^{2} - 10x + 2$
411.2.a.e	$x^{9} - 6x^{8} - 36x^{7} + 230x^{6} + 246x^{5} - 2262x^{4} + 1502x^{3} + 4157x^{2} - 5590x + 1790$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(411))$	23
Old forms	22
Cusp forms $S_{2}(\Gamma_0(411))$	45
Eisenstein series $E_{2}(\Gamma_0(411))$	3
Modular forms $M_{2}(\Gamma_0(411))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(411)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(137))^{\oplus 2}$

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, 3, 137, 411]: [0, 1, 12, 48] (certified at ell = 2147483249)
T = [(2, 1), (5, 1)]; h of degree 23 = 1 x 23 (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(411, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 411:2 1000

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