Atlas › Modular forms › Weight 2 › Level 265 › 265.2.a

Newspace 265.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(265))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
265.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} + 2q^{7} + O(q^{8})$	+ +	$-1$	265a1 rank 1
265.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 6q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
265.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{3} - q^{4} + 2q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
265.2.a.d	2	$\mathbb{Q}(\sqrt{21})$	$2q - q^{2} - q^{3} + 7q^{4} - 2q^{5} - 10q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
265.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + q^{3} + 3q^{4} + 2q^{5} + 6q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
265.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{3} + 2q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	+ −	$+1$
265.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - 3q^{3} + 3q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
265.2.a.h	4	degree 4	$4q + 4q^{2} - 2q^{3} + 4q^{4} + 4q^{5} - 6q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
265.2.a.a	$x + 1$
265.2.a.b	$x^{2} + 8x - 16$
265.2.a.c	$x^{2} + 4x - 1$
265.2.a.d	$x^{2} + 4x - 17$
265.2.a.e	$x^{2} - 2x - 51$
265.2.a.f	$x^{2} - 12x + 33$
265.2.a.g	$x^{2} + 6x - 11$
265.2.a.h	$x^{4} + 2x^{3} - 49x^{2} - 32x + 544$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(265))$	17
Old forms	8
Cusp forms $S_{2}(\Gamma_0(265))$	25
Eisenstein series $E_{2}(\Gamma_0(265))$	3
Modular forms $M_{2}(\Gamma_0(265))$	28

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(265)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 53, 265]: [0, 1, 5, 28] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 17 = 1 x 17 (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(265, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 265:2 1000

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