Atlas › Modular forms › Weight 4 › Level 69 › 69.4.a

Newspace 69.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(69))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
69.4.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} + 6q^{3} + 2q^{4} - 26q^{5} - 12q^{6} - 10q^{7} + O(q^{8})$	− +	$-1$
69.4.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 6q^{3} + 2q^{4} + 8q^{5} + 6q^{6} - 28q^{7} + O(q^{8})$	+ −	$-1$
69.4.a.c	4	degree 4	$4q - 2q^{2} - 12q^{3} + 26q^{4} - 2q^{5} + 6q^{6} + 32q^{7} + O(q^{8})$	+ +	$+1$
69.4.a.d	4	degree 4	$4q + 4q^{2} + 12q^{3} + 26q^{4} + 4q^{5} + 12q^{6} - 14q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
69.4.a.a	$x^{2} + 82x + 1661$
69.4.a.b	$x^{2} - 22x - 847$
69.4.a.c	$x^{4} + 8x^{3} - 4335x^{2} - 30400x + 1191450$
69.4.a.d	$x^{4} - 16x^{3} - 1491x^{2} + 42164x - 268738$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(69))$	12
Old forms	10
Cusp forms $S_{4}(\Gamma_0(69))$	22
Eisenstein series $E_{4}(\Gamma_0(69))$	4
Modular forms $M_{4}(\Gamma_0(69))$	26

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(69)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 3, 23, 69]: [1, 2, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; 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(69, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 69:4 1000

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