Atlas › Modular forms › Weight 8 › Level 38 › 38.8.a

Newspace 38.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(38))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
38.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} + 77q^{3} + 64q^{4} + 440q^{5} - 616q^{6} + 951q^{7} + O(q^{8})$	+ +	$+1$
38.8.a.b	2	$\mathbb{Q}(\sqrt{2737})$	$2q - 16q^{2} - 61q^{3} + 128q^{4} + 175q^{5} + 488q^{6} - 2592q^{7} + O(q^{8})$	+ +	$+1$
38.8.a.c	2	$\mathbb{Q}(\sqrt{17953})$	$2q - 16q^{2} - 11q^{3} + 128q^{4} - 69q^{5} + 88q^{6} - 348q^{7} + O(q^{8})$	+ −	$-1$
38.8.a.d	2	$\mathbb{Q}(\sqrt{633})$	$2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + O(q^{8})$	− +	$-1$
38.8.a.e	4	degree 4	$4q + 32q^{2} + 12q^{3} + 256q^{4} - 279q^{5} + 96q^{6} + 2485q^{7} + O(q^{8})$	− −	$+1$

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 $T$
38.8.a.a	$x - 517$
38.8.a.b	$x^{2} - 114x - 171919$
38.8.a.c	$x^{2} + 80x - 16353$
38.8.a.d	$x^{2} - 86x - 122219$
38.8.a.e	$x^{4} + 267x^{3} - 212659x^{2} - 31505691x + 5724320994$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(38))$	11
Old forms	22
Cusp forms $S_{8}(\Gamma_0(38))$	33
Eisenstein series $E_{8}(\Gamma_0(38))$	4
Modular forms $M_{8}(\Gamma_0(38))$	37

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(38)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 19, 38]: [1, 3, 12, 37] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (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(38, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 38:8 1000

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