Atlas › Modular forms › Weight 2 › Level 471 › 471.2.a

Newspace 471.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(471))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
471.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$	471a1 rank 1
471.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 2q^{5} - q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
471.2.a.c	3	degree 3	$3q - 3q^{3} + 2q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
471.2.a.d	9	degree 9	$9q + 2q^{2} - 9q^{3} + 8q^{4} + 8q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
471.2.a.e	12	degree 12	$12q - q^{2} + 12q^{3} + 17q^{4} + 4q^{5} - q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$

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 $T$
471.2.a.a	$x + 3$
471.2.a.b	$x^{2} + 3x + 1$
471.2.a.c	$x^{3} + 2x^{2} - 4x - 7$
471.2.a.d	$x^{9} - 10x^{8} + 3x^{7} + 231x^{6} - 669x^{5} + 89x^{4} + 1373x^{3} - 915x^{2} - 296x + 59$
471.2.a.e	$x^{12} - 3x^{11} - 66x^{10} + 207x^{9} + 1441x^{8} - 5006x^{7} - 10184x^{6} + 46490x^{5} - 10091x^{4} - 100439x^{3} + 110924x^{2} - 32453x - 565$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(471))$	27
Old forms	24
Cusp forms $S_{2}(\Gamma_0(471))$	51
Eisenstein series $E_{2}(\Gamma_0(471))$	3
Modular forms $M_{2}(\Gamma_0(471))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(471)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(157))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 157, 471]: [0, 1, 13, 54] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(471, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 471:2 1000

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