Atlas › Modular forms › Weight 2 › Level 534 › 534.2.a

Newspace 534.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(534))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
534.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	534a1 rank 1
534.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ + +	$-1$
534.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
534.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
534.2.a.e	4	degree 4	$4q - 4q^{2} + 4q^{3} + 4q^{4} + 3q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
534.2.a.f	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
534.2.a.a	$x + 4$
534.2.a.b	$x^{2} - 2$
534.2.a.c	$x^{2} - x - 29$
534.2.a.d	$x^{2} - 3x - 9$
534.2.a.e	$x^{4} - 5x^{3} - 15x^{2} + 54x + 54$
534.2.a.f	$x^{4} - 5x^{3} + x^{2} + 10x + 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(534))$	15
Old forms	72
Cusp forms $S_{2}(\Gamma_0(534))$	87
Eisenstein series $E_{2}(\Gamma_0(534))$	7
Modular forms $M_{2}(\Gamma_0(534))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(534)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(178))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(267))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.359 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, 2, 3, 6, 89, 178, 267, 534]: [0, 1, 1, 3, 8, 24, 32, 94] (certified at ell = 2147482921)
T = [(5, 1), (7, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(534, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 534:2 1000

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