Atlas › Modular forms › Weight 4 › Level 65 › 65.4.a

Newspace 65.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(65))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
65.4.a.a	1	$\mathbb{Q}$	$q + 5q^{2} + 2q^{3} + 17q^{4} - 5q^{5} + 10q^{6} - 12q^{7} + O(q^{8})$	+ +	$+1$
65.4.a.b	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} - 4q^{3} - 2q^{4} + 10q^{5} - 8q^{6} - 20q^{7} + O(q^{8})$	− +	$-1$
65.4.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} - 7q^{4} - 10q^{5} + 34q^{6} + 58q^{7} + O(q^{8})$	+ +	$+1$
65.4.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{2} - 10q^{3} - 2q^{4} - 10q^{5} - 38q^{6} - 36q^{7} + O(q^{8})$	+ −	$-1$
65.4.a.e	5	degree 5	$5q + 8q^{3} + 34q^{4} + 25q^{5} - 42q^{6} + 38q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
65.4.a.a	$x - 11$
65.4.a.b	$x^{2} + 14x + 25$
65.4.a.c	$x^{2} - x - 718$
65.4.a.d	$x^{2} + 26x - 23$
65.4.a.e	$x^{5} - 24x^{4} - 523x^{3} + 14542x^{2} + 8712x - 1165956$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(65))$	12
Old forms	8
Cusp forms $S_{4}(\Gamma_0(65))$	20
Eisenstein series $E_{4}(\Gamma_0(65))$	4
Modular forms $M_{4}(\Gamma_0(65))$	24

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(65)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.103 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, 5, 13, 65]: [1, 3, 5, 24] (certified at ell = 2147483629)
T = [(2, 1), (3, 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(65, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 65:4 1000

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