Atlas › Modular forms › Weight 6 › Level 90 › 90.6.a

Newspace 90.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(90))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
90.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 16q^{4} - 25q^{5} + 32q^{7} + O(q^{8})$	+ +
90.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + 16q^{4} + 25q^{5} - 118q^{7} + O(q^{8})$	+ −
90.6.a.c	1	$\mathbb{Q}$	$q - 4q^{2} + 16q^{4} + 25q^{5} + 98q^{7} + O(q^{8})$	+ −
90.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} + 16q^{4} - 25q^{5} - 172q^{7} + O(q^{8})$	− +
90.6.a.e	1	$\mathbb{Q}$	$q + 4q^{2} + 16q^{4} - 25q^{5} + 98q^{7} + O(q^{8})$	− +
90.6.a.f	1	$\mathbb{Q}$	$q + 4q^{2} + 16q^{4} + 25q^{5} - 22q^{7} + O(q^{8})$	− −
90.6.a.g	1	$\mathbb{Q}$	$q + 4q^{2} + 16q^{4} + 25q^{5} + 164q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + T_{11}$ 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 $T$
90.6.a.a	$x - 20$
90.6.a.b	$x + 310$
90.6.a.c	$x + 256$
90.6.a.d	$x + 304$
90.6.a.e	$x - 452$
90.6.a.f	$x - 746$
90.6.a.g	$x + 556$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(90))$	7
Old forms	75
Cusp forms $S_{6}(\Gamma_0(90))$	82
Eisenstein series $E_{6}(\Gamma_0(90))$	16
Modular forms $M_{6}(\Gamma_0(90))$	98

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90]: [1, 2, 3, 3, 7, 6, 9, 12, 17, 34, 32, 94] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 7 = 1 x 7 (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(90, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 90:6 1000

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