Atlas › Modular forms › Weight 8 › Level 40 › 40.8.a

Newspace 40.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(40))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
40.8.a.a	1	$\mathbb{Q}$	$q - 36q^{3} + 125q^{5} + 776q^{7} + O(q^{8})$	−
40.8.a.b	2	$\mathbb{Q}(\sqrt{46})$	$2q + 4q^{3} - 250q^{5} + 844q^{7} + O(q^{8})$	+
40.8.a.c	2	$\mathbb{Q}(\sqrt{6})$	$2q + 36q^{3} - 250q^{5} - 404q^{7} + O(q^{8})$	+
40.8.a.d	2	$\mathbb{Q}(\sqrt{601})$	$2q + 76q^{3} + 250q^{5} + 796q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
40.8.a.a	$x - 740$
40.8.a.b	$x^{2} - 848x - 774080$
40.8.a.c	$x^{2} + 368x - 1442240$
40.8.a.d	$x^{2} - 872x + 36240$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(40))$	7
Old forms	31
Cusp forms $S_{8}(\Gamma_0(40))$	38
Eisenstein series $E_{8}(\Gamma_0(40))$	8
Modular forms $M_{8}(\Gamma_0(40))$	46

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(40)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 6}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 4, 5, 8, 10, 20, 40]: [1, 3, 5, 5, 9, 13, 24, 46] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 7 = 1 x 7 (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(40, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 40:8 1000

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