Atlas › Modular forms › Weight 2 › Level 413 › 413.2.a

Newspace 413.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(413))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
413.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} + 6q^{4} + 2q^{5} + 5q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
413.2.a.b	3	degree 3	$3q - 3q^{2} + 3q^{3} - 3q^{4} - 3q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
413.2.a.c	5	degree 5	$5q - 2q^{2} - 5q^{3} + q^{5} + 3q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
413.2.a.d	5	degree 5	$5q - 7q^{3} - 5q^{5} - 3q^{6} + 5q^{7} + O(q^{8})$	− −	$-1$
413.2.a.e	5	degree 5	$5q + 4q^{2} + 7q^{3} + 12q^{4} - 3q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$
413.2.a.f	9	degree 9	$9q + 7q^{3} + 8q^{4} + 3q^{5} + q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$

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 $T$
413.2.a.a	$x^{2} + x - 11$
413.2.a.b	$x^{3} - 4x + 1$
413.2.a.c	$x^{5} + 7x^{4} + 8x^{3} - 19x^{2} - 29x + 1$
413.2.a.d	$x^{5} + 7x^{4} + 12x^{3} + x^{2} - 5x + 1$
413.2.a.e	$x^{5} - 11x^{4} + 32x^{3} + 9x^{2} - 117x + 55$
413.2.a.f	$x^{9} - 7x^{8} - 7x^{7} + 122x^{6} - 60x^{5} - 683x^{4} + 439x^{3} + 1392x^{2} - 521x - 457$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(413))$	29
Old forms	10
Cusp forms $S_{2}(\Gamma_0(413))$	39
Eisenstein series $E_{2}(\Gamma_0(413))$	3
Modular forms $M_{2}(\Gamma_0(413))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(413)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 59, 413]: [0, 1, 6, 42] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (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(413, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 413:2 1000

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