Atlas › Modular forms › Weight 2 › Level 365 › 365.2.a

Newspace 365.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(365))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
365.2.a.a	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{3} + 2q^{4} + 2q^{5} + 6q^{7} + O(q^{8})$	− +	$+1$
365.2.a.b	3	degree 3	$3q - q^{2} - 4q^{3} - q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− −	$-1$
365.2.a.c	5	degree 5	$5q - q^{2} - 6q^{3} + q^{4} - 5q^{5} + 3q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
365.2.a.d	7	degree 7	$7q - q^{2} + 2q^{3} + 11q^{4} + 7q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$
365.2.a.e	8	degree 8	$8q + 2q^{2} + 8q^{3} + 10q^{4} - 8q^{5} + 5q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
365.2.a.a	$x^{2} - 4x + 1$
365.2.a.b	$x^{3} + 5x^{2} + 6x + 1$
365.2.a.c	$x^{5} + 7x^{4} + 5x^{3} - 20x^{2} - 6x + 11$
365.2.a.d	$x^{7} - x^{6} - 25x^{5} + 34x^{4} + 115x^{3} - 132x^{2} + 28x - 1$
365.2.a.e	$x^{8} - 10x^{7} + 14x^{6} + 121x^{5} - 309x^{4} - 357x^{3} + 1110x^{2} + 421x - 883$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(365))$	25
Old forms	10
Cusp forms $S_{2}(\Gamma_0(365))$	35
Eisenstein series $E_{2}(\Gamma_0(365))$	3
Modular forms $M_{2}(\Gamma_0(365))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(365)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 73, 365]: [0, 1, 6, 38] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (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(365, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 365:2 1000

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