Atlas › Modular forms › Weight 10 › Level 35 › 35.10.a

Newspace 35.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(35))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
35.10.a.a	1	$\mathbb{Q}$	$q + 28q^{2} - 116q^{3} + 272q^{4} + 625q^{5} - 3248q^{6} + 2401q^{7} + O(q^{8})$	− −	$-1$
35.10.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 24q^{2} - 174q^{3} - 720q^{4} + 1250q^{5} + 2952q^{6} + 4802q^{7} + O(q^{8})$	− −	$-1$
35.10.a.c	4	degree 4	$4q - 19q^{2} - 18q^{3} + 1729q^{4} - 2500q^{5} - 144q^{6} - 9604q^{7} + O(q^{8})$	+ +	$-1$
35.10.a.d	5	degree 5	$5q + 2q^{2} + 140q^{3} + 832q^{4} - 3125q^{5} - 144q^{6} + 12005q^{7} + O(q^{8})$	+ −	$+1$
35.10.a.e	6	degree 6	$6q + 15q^{2} - 124q^{3} + 3009q^{4} + 3750q^{5} + 4888q^{6} - 14406q^{7} + O(q^{8})$	− +	$+1$

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 $T$
35.10.a.a	$x + 88$
35.10.a.b	$x^{2} + 198x - 14399$
35.10.a.c	$x^{4} + 37x^{3} - 43117x^{2} + 1361315x + 119627428$
35.10.a.d	$x^{5} - 142x^{4} - 74576x^{3} + 6322840x^{2} + 1239115411x + 3845372466$
35.10.a.e	$x^{6} + 109x^{5} - 116582x^{4} - 13602632x^{3} + 3145846451x^{2} + 412032473931x + 8912066589538$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(35))$	18
Old forms	16
Cusp forms $S_{10}(\Gamma_0(35))$	34
Eisenstein series $E_{10}(\Gamma_0(35))$	4
Modular forms $M_{10}(\Gamma_0(35))$	38

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(35)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 7, 38] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (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(35, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 35:10 1000

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