Atlas › Modular forms › Weight 6 › Level 35 › 35.6.a

Newspace 35.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(35))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
35.6.a.a	1	$\mathbb{Q}$	$q - 8q^{2} + q^{3} + 32q^{4} + 25q^{5} - 8q^{6} + 49q^{7} + O(q^{8})$	− −	$-1$
35.6.a.b	2	$\mathbb{Q}(\sqrt{65})$	$2q + q^{2} + 3q^{3} - 31q^{4} - 50q^{5} - 96q^{6} - 98q^{7} + O(q^{8})$	+ +	$-1$
35.6.a.c	3	degree 3	$3q - 6q^{2} + 26q^{3} + 112q^{4} - 75q^{5} - 96q^{6} + 147q^{7} + O(q^{8})$	+ −	$+1$
35.6.a.d	4	degree 4	$4q + 7q^{2} + 14q^{3} + 49q^{4} + 100q^{5} + 136q^{6} - 196q^{7} + O(q^{8})$	− +	$+1$

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 $T$
35.6.a.a	$x + 7$
35.6.a.b	$x^{2} - 4x - 61$
35.6.a.c	$x^{3} - 20x^{2} - 417x + 8074$
35.6.a.d	$x^{4} - 21x^{3} - 877x^{2} + 19823x - 99526$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(35))$	10
Old forms	8
Cusp forms $S_{6}(\Gamma_0(35))$	18
Eisenstein series $E_{6}(\Gamma_0(35))$	4
Modular forms $M_{6}(\Gamma_0(35))$	22

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(35)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 3, 5, 22] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 35:6 1000

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