Atlas › Modular forms › Weight 4 › Level 85 › 85.4.a

Newspace 85.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(85))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
85.4.a.a	1	$\mathbb{Q}$	$q + 3q^{2} - 7q^{3} + q^{4} + 5q^{5} - 21q^{6} - 22q^{7} + O(q^{8})$	− +	$-1$
85.4.a.b	1	$\mathbb{Q}$	$q + 3q^{2} - 5q^{3} + q^{4} - 5q^{5} - 15q^{6} - 22q^{7} + O(q^{8})$	+ −	$-1$
85.4.a.c	1	$\mathbb{Q}$	$q + 3q^{2} + 10q^{3} + q^{4} + 5q^{5} + 30q^{6} - 22q^{7} + O(q^{8})$	− −	$+1$
85.4.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{2} - 2q^{3} - 2q^{4} + 10q^{5} + 10q^{6} + 2q^{7} + O(q^{8})$	− +	$-1$
85.4.a.e	3	degree 3	$3q - 6q^{2} - 4q^{3} + 10q^{4} - 15q^{5} + 36q^{6} + 8q^{7} + O(q^{8})$	+ −	$-1$
85.4.a.f	3	degree 3	$3q + 3q^{2} + 9q^{3} - q^{4} - 15q^{5} + 33q^{6} + 34q^{7} + O(q^{8})$	+ +	$+1$
85.4.a.g	5	degree 5	$5q + 2q^{2} - q^{3} + 34q^{4} + 25q^{5} - 5q^{6} + 10q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
85.4.a.a	$x + 18$
85.4.a.b	$x + 12$
85.4.a.c	$x - 33$
85.4.a.d	$x^{2} + 10x - 23$
85.4.a.e	$x^{3} + 18x^{2} - 503x - 1212$
85.4.a.f	$x^{3} - 30x^{2} - 295x + 9832$
85.4.a.g	$x^{5} + x^{4} - 494x^{3} + 1804x^{2} + 34324x - 106120$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(85))$	16
Old forms	10
Cusp forms $S_{4}(\Gamma_0(85))$	26
Eisenstein series $E_{4}(\Gamma_0(85))$	4
Modular forms $M_{4}(\Gamma_0(85))$	30

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(85)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 17, 85]: [1, 3, 6, 30] (certified at ell = 2147483489)
T = [(2, 1), (3, 3)]; h of degree 16 = 1 x 16 (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(85, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 85:4 1000

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