Atlas › Modular forms › Weight 4 › Level 136 › 136.4.a

Newspace 136.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(136))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
136.4.a.a	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{3} - 12q^{5} - 36q^{7} + O(q^{8})$	−
136.4.a.b	3	degree 3	$3q - 4q^{3} - 8q^{5} - 2q^{7} + O(q^{8})$	+
136.4.a.c	3	degree 3	$3q + 8q^{3} + 2q^{5} + 12q^{7} + O(q^{8})$	−
136.4.a.d	4	degree 4	$4q - 2q^{3} + 8q^{5} - 22q^{7} + O(q^{8})$	+

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 $T$
136.4.a.a	$x^{2} + 8x + 4$
136.4.a.b	$x^{3} + 12x^{2} - 86x + 124$
136.4.a.c	$x^{3} - 10x^{2} - 236x + 1832$
136.4.a.d	$x^{4} - 6x^{3} - 750x^{2} + 7048x + 32520$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(136))$	12
Old forms	38
Cusp forms $S_{4}(\Gamma_0(136))$	50
Eisenstein series $E_{4}(\Gamma_0(136))$	8
Modular forms $M_{4}(\Gamma_0(136))$	58

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(136)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.121 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, 8, 17, 34, 68, 136]: [1, 2, 3, 5, 6, 16, 30, 58] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(136, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 136:4 1000

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