Atlas › Modular forms › Weight 2 › Level 469 › 469.2.a

Newspace 469.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(469))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
469.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - q^{4} + q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+ +	$-1$	469b1 rank 1
469.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - 3q^{5} + q^{6} - q^{7} + O(q^{8})$	+ +	$-1$	469a1 rank 1
469.2.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} - q^{3} + 5q^{4} - 3q^{5} - 9q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
469.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - q^{3} - 2q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
469.2.a.e	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	− −	$-1$
469.2.a.f	3	degree 3	$3q - q^{2} - q^{3} + q^{4} - 9q^{5} + 5q^{6} + 3q^{7} + O(q^{8})$	− −	$-1$
469.2.a.g	5	degree 5	$5q + 2q^{2} - 2q^{3} + 4q^{4} - 4q^{5} - 14q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
469.2.a.h	7	degree 7	$7q + q^{2} + 6q^{3} + 11q^{4} + 6q^{5} + 10q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$
469.2.a.i	9	degree 9	$9q - q^{2} + 8q^{3} + 9q^{4} + 8q^{5} - 9q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
469.2.a.a	$x + 10$
469.2.a.b	$x - 4$
469.2.a.c	$x^{2} + 2x - 16$
469.2.a.d	$x^{2} + x - 38$
469.2.a.e	$x^{3} + 12x^{2} + 36x + 24$
469.2.a.f	$x^{3} + 4x^{2} - 60x + 50$
469.2.a.g	$x^{5} + 4x^{4} - 20x^{3} - 72x^{2} + 48x + 128$
469.2.a.h	$x^{7} - 19x^{6} - 15x^{5} + 2259x^{4} - 14306x^{3} + 8308x^{2} + 152944x - 330224$
469.2.a.i	$x^{9} - 23x^{8} + 71x^{7} + 1709x^{6} - 11218x^{5} - 24902x^{4} + 275754x^{3} - 94044x^{2} - 1624608x + 1884208$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(469))$	33
Old forms	10
Cusp forms $S_{2}(\Gamma_0(469))$	43
Eisenstein series $E_{2}(\Gamma_0(469))$	3
Modular forms $M_{2}(\Gamma_0(469))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(469)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 7, 67, 469]: [0, 1, 6, 46] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 33 = 1 x 33 (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(469, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 469:2 1000

← 468.2.a · 470.2.a →