Atlas › Modular forms › Weight 2 › Level 447 › 447.2.a

Newspace 447.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(447))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
447.2.a.a	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
447.2.a.b	3	degree 3	$3q - q^{2} - 3q^{3} - q^{4} + q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
447.2.a.c	9	degree 9	$9q + 4q^{2} + 9q^{3} + 10q^{4} + 8q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− +	$+1$
447.2.a.d	10	degree 10	$10q + 3q^{2} - 10q^{3} + 13q^{4} + 4q^{5} - 3q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
447.2.a.a	$x^{3} + 9x^{2} + 24x + 17$
447.2.a.b	$x^{3} + x^{2} - 2x - 1$
447.2.a.c	$x^{9} - 12x^{8} + 32x^{7} + 115x^{6} - 773x^{5} + 1443x^{4} - 729x^{3} - 692x^{2} + 737x - 121$
447.2.a.d	$x^{10} - 7x^{9} - 42x^{8} + 353x^{7} + 362x^{6} - 5134x^{5} - 170x^{4} + 28405x^{3} - 3617x^{2} - 48254x + 3115$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(447))$	25
Old forms	24
Cusp forms $S_{2}(\Gamma_0(447))$	49
Eisenstein series $E_{2}(\Gamma_0(447))$	3
Modular forms $M_{2}(\Gamma_0(447))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(447)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(149))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 149, 447]: [0, 1, 13, 52] (certified at ell = 2147481629)
T = [(2, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(447, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 447:2 1000

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