Atlas › Modular forms › Weight 4 › Level 236 › 236.4.a

Newspace 236.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(236))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
236.4.a.a	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} - 3q^{7} + O(q^{8})$	+
236.4.a.b	4	degree 4	$4q + 10q^{5} + 35q^{7} + O(q^{8})$	−
236.4.a.c	5	degree 5	$5q - 2q^{3} - 6q^{5} - 38q^{7} + O(q^{8})$	+
236.4.a.d	5	degree 5	$5q + 6q^{3} - 14q^{5} - 6q^{7} + O(q^{8})$	−

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 $T$
236.4.a.a	$x - 4$
236.4.a.b	$x^{4} - 10x^{3} - 544x^{2} + 6720x - 9216$
236.4.a.c	$x^{5} + 8x^{4} - 213x^{3} - 3089x^{2} - 14016x - 21364$
236.4.a.d	$x^{5} + 8x^{4} - 531x^{3} - 4391x^{2} + 66476x + 546096$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(236))$	15
Old forms	72
Cusp forms $S_{4}(\Gamma_0(236))$	87
Eisenstein series $E_{4}(\Gamma_0(236))$	6
Modular forms $M_{4}(\Gamma_0(236))$	93

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(236)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(118))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 4, 59, 118, 236]: [1, 2, 3, 16, 47, 93] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(236, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 236:4 1000

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