Atlas › Modular forms › Weight 2 › Level 445 › 445.2.a

Newspace 445.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(445))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
445.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$
445.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} + 2q^{5} + 6q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
445.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} - 4q^{6} + O(q^{8})$	− +	$+1$
445.2.a.d	4	degree 4	$4q + q^{2} - 2q^{3} + 3q^{4} - 4q^{5} - 6q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$
445.2.a.e	4	degree 4	$4q + q^{2} + 4q^{3} + 3q^{4} + 4q^{5} - 6q^{6} + 12q^{7} + O(q^{8})$	− +	$+1$
445.2.a.f	7	degree 7	$7q - 4q^{2} - 8q^{3} + 8q^{4} + 7q^{5} - 2q^{6} - 16q^{7} + O(q^{8})$	− −	$-1$
445.2.a.g	8	degree 8	$8q + q^{2} + 6q^{3} + 7q^{4} - 8q^{5} + 6q^{6} - 6q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
445.2.a.a	$x^{2} + 8x - 29$
445.2.a.b	$x^{2} - 6x - 39$
445.2.a.c	$x^{2} - 2x - 7$
445.2.a.d	$x^{4} + 5x^{3} - 11x^{2} - 65x - 61$
445.2.a.e	$x^{4} - 13x^{3} + 7x^{2} + 413x - 1235$
445.2.a.f	$x^{7} + 28x^{6} + 243x^{5} + 288x^{4} - 5216x^{3} - 18307x^{2} - 1176x + 29691$
445.2.a.g	$x^{8} - 19x^{7} - 11x^{6} + 1875x^{5} - 5942x^{4} - 47629x^{3} + 202749x^{2} + 57461x - 271357$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(445))$	29
Old forms	14
Cusp forms $S_{2}(\Gamma_0(445))$	43
Eisenstein series $E_{2}(\Gamma_0(445))$	3
Modular forms $M_{2}(\Gamma_0(445))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(445)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.289 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, 89, 445]: [0, 1, 8, 46] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 29 = 1 x 29 (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(445, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 445:2 1000

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