Atlas › Modular forms › Weight 2 › Level 535 › 535.2.a

Newspace 535.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(535))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
535.2.a.a	3	degree 3	$3q - 2q^{2} + 3q^{5} - q^{7} + O(q^{8})$	− −	$-1$
535.2.a.b	8	degree 8	$8q - 4q^{2} - 2q^{3} + 8q^{4} - 8q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
535.2.a.c	9	degree 9	$9q + 5q^{2} + 2q^{3} + 7q^{4} - 9q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
535.2.a.d	15	degree 15	$15q + 4q^{2} + 22q^{4} + 15q^{5} + 4q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
535.2.a.a	$x^{3} + 2x^{2} - x - 1$
535.2.a.b	$x^{8} + 6x^{7} - 4x^{6} - 48x^{5} + 13x^{4} + 82x^{3} - 19x^{2} - 10x + 2$
535.2.a.c	$x^{9} - 7x^{8} - 8x^{7} + 111x^{6} - 21x^{5} - 477x^{4} + 117x^{3} + 386x^{2} - 134x + 3$
535.2.a.d	$x^{15} - 4x^{14} - 60x^{13} + 270x^{12} + 1123x^{11} - 5930x^{10} - 7029x^{9} + 51873x^{8} + 7448x^{7} - 184032x^{6} + 10344x^{5} + 277387x^{4} + 14810x^{3} - 143263x^{2} - 16210x + 17602$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(535))$	35
Old forms	18
Cusp forms $S_{2}(\Gamma_0(535))$	53
Eisenstein series $E_{2}(\Gamma_0(535))$	3
Modular forms $M_{2}(\Gamma_0(535))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(535)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 5, 107, 535]: [0, 1, 10, 56] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 4 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(535, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 535:2 1000

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