Atlas › Modular forms › Weight 8 › Level 62 › 62.8.a

Newspace 62.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(62))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
62.8.a.a	3	degree 3	$3q - 24q^{2} - 26q^{3} + 192q^{4} + 152q^{5} + 208q^{6} - 1504q^{7} + O(q^{8})$	+ −	$-1$
62.8.a.b	4	degree 4	$4q + 32q^{2} - 68q^{3} + 256q^{4} - 334q^{5} - 544q^{6} - 948q^{7} + O(q^{8})$	− +	$-1$
62.8.a.c	5	degree 5	$5q - 40q^{2} + 28q^{3} + 320q^{4} + 152q^{5} - 224q^{6} - 132q^{7} + O(q^{8})$	+ +	$+1$
62.8.a.d	6	degree 6	$6q + 48q^{2} + 40q^{3} + 384q^{4} + 166q^{5} + 320q^{6} + 1796q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
62.8.a.a	$x^{3} - 126x^{2} - 64789x + 7367766$
62.8.a.b	$x^{4} + 402x^{3} - 35931x^{2} - 3522414x + 236938226$
62.8.a.c	$x^{5} - 180x^{4} - 345943x^{3} + 74916772x^{2} + 29646125654x - 7277639530380$
62.8.a.d	$x^{6} - 206x^{5} - 432723x^{4} + 79925822x^{3} + 38170622654x^{2} - 8313065403912x + 404457085305912$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(62))$	18
Old forms	36
Cusp forms $S_{8}(\Gamma_0(62))$	54
Eisenstein series $E_{8}(\Gamma_0(62))$	4
Modular forms $M_{8}(\Gamma_0(62))$	58

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(62)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 31, 62]: [1, 3, 19, 58] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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(62, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 62:8 1000

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