Atlas › Modular forms › Weight 8 › Level 35 › 35.8.a

Newspace 35.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(35))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
35.8.a.a	2	$\mathbb{Q}(\sqrt{11})$	$2q + 16q^{2} - 30q^{3} - 40q^{4} + 250q^{5} - 768q^{6} - 686q^{7} + O(q^{8})$	− +	$-1$
35.8.a.b	3	degree 3	$3q - 23q^{2} - 50q^{3} + 233q^{4} - 375q^{5} + 1144q^{6} + 1029q^{7} + O(q^{8})$	+ −	$-1$
35.8.a.c	4	degree 4	$4q - 2q^{2} - 37q^{3} + 280q^{4} - 500q^{5} + 1144q^{6} - 1372q^{7} + O(q^{8})$	+ +	$+1$
35.8.a.d	5	degree 5	$5q + 11q^{2} + 65q^{3} + 553q^{4} + 625q^{5} + 96q^{6} + 1715q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
35.8.a.a	$x^{2} + 14x - 1051$
35.8.a.b	$x^{3} + 73x^{2} - 3293x + 26131$
35.8.a.c	$x^{4} + 39x^{3} - 6349x^{2} - 265597x - 2620702$
35.8.a.d	$x^{5} - 76x^{4} - 3266x^{3} + 227230x^{2} - 625391x - 43857426$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(35))$	14
Old forms	12
Cusp forms $S_{8}(\Gamma_0(35))$	26
Eisenstein series $E_{8}(\Gamma_0(35))$	4
Modular forms $M_{8}(\Gamma_0(35))$	30

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(35)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 5, 7, 35]: [1, 5, 5, 30] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (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(35, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 35:8 1000

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