Atlas › Modular forms › Weight 4 › Level 134 › 134.4.a

Newspace 134.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(134))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
134.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 20q^{6} - 34q^{7} + O(q^{8})$	− −	$+1$
134.4.a.b	2	$\mathbb{Q}(\sqrt{37})$	$2q + 4q^{2} - 9q^{3} + 8q^{4} - 6q^{5} - 18q^{6} - 21q^{7} + O(q^{8})$	− +	$-1$
134.4.a.c	3	degree 3	$3q - 6q^{2} - 3q^{3} + 12q^{4} - 6q^{5} + 6q^{6} - 5q^{7} + O(q^{8})$	+ −	$-1$
134.4.a.d	5	degree 5	$5q + 10q^{2} + 2q^{3} + 20q^{4} + 3q^{5} + 4q^{6} + 69q^{7} + O(q^{8})$	− −	$+1$
134.4.a.e	6	degree 6	$6q - 12q^{2} + 24q^{4} + 9q^{5} - 5q^{7} + O(q^{8})$	+ +	$+1$

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 $T$
134.4.a.a	$x - 16$
134.4.a.b	$x^{2} + 15x - 27$
134.4.a.c	$x^{3} + 9x^{2} - x - 8$
134.4.a.d	$x^{5} - 5x^{4} - 468x^{3} + 3186x^{2} + 8270x - 47329$
134.4.a.e	$x^{6} - 9x^{5} - 836x^{4} + 7230x^{3} + 166272x^{2} - 1534153x - 1649494$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(134))$	17
Old forms	32
Cusp forms $S_{4}(\Gamma_0(134))$	49
Eisenstein series $E_{4}(\Gamma_0(134))$	4
Modular forms $M_{4}(\Gamma_0(134))$	53

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(134)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 67, 134]: [1, 2, 18, 53] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(134, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 134:4 1000

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