Atlas › Modular forms › Weight 6 › Level 65 › 65.6.a

Newspace 65.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(65))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
65.6.a.a	1	$\mathbb{Q}$	$q + 5q^{2} + 6q^{3} - 7q^{4} - 25q^{5} + 30q^{6} - 244q^{7} + O(q^{8})$	+ +	$-1$
65.6.a.b	3	degree 3	$3q - 2q^{2} - 16q^{3} - 10q^{4} + 75q^{5} + 22q^{6} - 208q^{7} + O(q^{8})$	− −	$-1$
65.6.a.c	4	degree 4	$4q - 9q^{2} - 4q^{3} + 93q^{4} - 100q^{5} - 74q^{6} - 136q^{7} + O(q^{8})$	+ +	$-1$
65.6.a.d	6	degree 6	$6q + 38q^{3} + 134q^{4} - 150q^{5} + 318q^{6} + 220q^{7} + O(q^{8})$	+ −	$+1$
65.6.a.e	6	degree 6	$6q + 2q^{2} + 20q^{3} + 134q^{4} + 150q^{5} - 52q^{6} + 172q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
65.6.a.a	$x - 11$
65.6.a.b	$x^{3} + 18x^{2} - 339x - 2016$
65.6.a.c	$x^{4} + 13x^{3} - 760x^{2} + 1728x + 50544$
65.6.a.d	$x^{6} - 38x^{5} - 747x^{4} + 24900x^{3} + 251532x^{2} - 4153328x - 39408252$
65.6.a.e	$x^{6} - 22x^{5} - 1115x^{4} + 18140x^{3} + 346736x^{2} - 3753264x - 23139596$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(65))$	20
Old forms	12
Cusp forms $S_{6}(\Gamma_0(65))$	32
Eisenstein series $E_{6}(\Gamma_0(65))$	4
Modular forms $M_{6}(\Gamma_0(65))$	36

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(65)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 7, 36] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 4 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 65:6 1000

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