Atlas › Modular forms › Weight 2 › Level 429 › 429.2.a

Newspace 429.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(429))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
429.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} + O(q^{8})$	+ − −	$-1$	429a1 rank 1
429.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + O(q^{8})$	− + −	$-1$	429b1 rank 1
429.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$
429.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+ + +	$-1$
429.2.a.e	3	degree 3	$3q - q^{2} - 3q^{3} + 5q^{4} - 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
429.2.a.f	3	degree 3	$3q + q^{2} + 3q^{3} + q^{4} + q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$
429.2.a.g	3	degree 3	$3q + 3q^{2} + 3q^{3} + 5q^{4} + 4q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
429.2.a.h	4	degree 4	$4q - 2q^{2} - 4q^{3} + 8q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$

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 $T$
429.2.a.a	$x + 1$
429.2.a.b	$x + 7$
429.2.a.c	$x^{2} + 14x + 17$
429.2.a.d	$x^{2} + 6x - 3$
429.2.a.e	$x^{3} + 7x^{2} - 113x - 387$
429.2.a.f	$x^{3} - x^{2} - 45x + 25$
429.2.a.g	$x^{3} - 15x^{2} + 47x - 37$
429.2.a.h	$x^{4} + 2x^{3} - 108x^{2} - 150x + 23$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(429))$	19
Old forms	34
Cusp forms $S_{2}(\Gamma_0(429))$	53
Eisenstein series $E_{2}(\Gamma_0(429))$	7
Modular forms $M_{2}(\Gamma_0(429))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(429)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(143))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.316 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, 11, 13, 33, 39, 143, 429]: [0, 1, 2, 1, 6, 6, 16, 60] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 19 = 1 x 19 (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(429, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 429:2 1000

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