Atlas › Modular forms › Weight 6 › Level 69 › 69.6.a

Newspace 69.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(69))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
69.6.a.a	2	$\mathbb{Q}(\sqrt{29})$	$2q - 18q^{3} + 168q^{4} + 94q^{5} - 118q^{7} + O(q^{8})$	+ −	$+1$
69.6.a.b	3	degree 3	$3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} + O(q^{8})$	− −	$-1$
69.6.a.c	4	degree 4	$4q + 4q^{2} - 36q^{3} - 46q^{4} - 122q^{5} - 36q^{6} + 62q^{7} + O(q^{8})$	+ −	$+1$
69.6.a.d	4	degree 4	$4q + 4q^{2} - 36q^{3} + 26q^{4} + 22q^{5} - 36q^{6} - 62q^{7} + O(q^{8})$	+ +	$-1$
69.6.a.e	5	degree 5	$5q + 8q^{2} + 45q^{3} + 118q^{4} + 94q^{5} + 72q^{6} + 272q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
69.6.a.a	$x^{2} - 94x + 1948$
69.6.a.b	$x^{3} + 64x^{2} - 3357x - 213192$
69.6.a.c	$x^{4} + 118x^{3} - 2943x^{2} - 393660x + 7479540$
69.6.a.d	$x^{4} - 26x^{3} - 6125x^{2} + 180354x + 1747692$
69.6.a.e	$x^{5} - 102x^{4} - 8595x^{3} + 1056452x^{2} - 4696824x - 1084414968$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(69))$	18
Old forms	20
Cusp forms $S_{6}(\Gamma_0(69))$	38
Eisenstein series $E_{6}(\Gamma_0(69))$	4
Modular forms $M_{6}(\Gamma_0(69))$	42

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(69)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 3, 11, 42] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 69:6 1000

← 68.6.a · 70.6.a →