Atlas › Modular forms › Weight 10 › Level 40 › 40.10.a

Newspace 40.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(40))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
40.10.a.a	2	$\mathbb{Q}(\sqrt{22})$	$2q - 116q^{3} - 1250q^{5} + 11284q^{7} + O(q^{8})$	+
40.10.a.b	2	$\mathbb{Q}(\sqrt{6049})$	$2q - 92q^{3} + 1250q^{5} - 6908q^{7} + O(q^{8})$	−
40.10.a.c	2	$\mathbb{Q}(\sqrt{46})$	$2q + 108q^{3} - 1250q^{5} - 908q^{7} + O(q^{8})$	+
40.10.a.d	3	degree 3	$3q + 84q^{3} + 1875q^{5} - 5520q^{7} + O(q^{8})$	−

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 $T$
40.10.a.a	$x^{2} - 11168x - 9510144$
40.10.a.b	$x^{2} + 7000x - 19108016$
40.10.a.c	$x^{2} + 800x - 5033216$
40.10.a.d	$x^{3} + 5436x^{2} - 60929616x - 102242806976$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(40))$	9
Old forms	41
Cusp forms $S_{10}(\Gamma_0(40))$	50
Eisenstein series $E_{10}(\Gamma_0(40))$	8
Modular forms $M_{10}(\Gamma_0(40))$	58

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.356 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, 6, 5, 11, 15, 30, 58] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 5 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, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 40:10 1000

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