Atlas › Modular forms › Weight 2 › Level 638 › 638.2.a

Newspace 638.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(638))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
638.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} + 3q^{6} + O(q^{8})$	+ − −	$-1$
638.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + q^{3} + 2q^{4} + 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
638.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 4q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$
638.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
638.2.a.e	4	degree 4	$4q - 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ + −	$+1$
638.2.a.f	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	− + +	$+1$
638.2.a.g	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
638.2.a.h	5	degree 5	$5q - 5q^{2} + 4q^{3} + 5q^{4} + 4q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
638.2.a.a	$x^{2} + 3x + 1$
638.2.a.b	$x^{2} - 3x + 1$
638.2.a.c	$x^{2} + 7x + 9$
638.2.a.d	$x^{2} + 5x + 5$
638.2.a.e	$x^{4} + 2x^{3} - 13x^{2} - 22x - 7$
638.2.a.f	$x^{4} - 2x^{3} - 25x^{2} + 86x - 63$
638.2.a.g	$x^{4} - 6x^{3} + 3x^{2} + 14x + 5$
638.2.a.h	$x^{5} - 8x^{4} - 5x^{3} + 132x^{2} - 171x + 54$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(638))$	25
Old forms	62
Cusp forms $S_{2}(\Gamma_0(638))$	87
Eisenstein series $E_{2}(\Gamma_0(638))$	7
Modular forms $M_{2}(\Gamma_0(638))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(638)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(319))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 2, 11, 22, 29, 58, 319, 638]: [0, 1, 2, 5, 3, 9, 32, 94] (certified at ell = 2147482681)
T = [(3, 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(638, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 638:2 1000

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