Atlas › Modular forms › Weight 2 › Level 273 › 273.2.a

Newspace 273.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(273))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
273.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	273a1 rank 1
273.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	273b1 rank 0
273.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
273.2.a.d	3	degree 3	$3q - 2q^{2} - 3q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
273.2.a.e	4	degree 4	$4q + q^{2} + 4q^{3} + 7q^{4} - 3q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
273.2.a.a	$x + 5$
273.2.a.b	$x - 5$
273.2.a.c	$x^{2} - 2x - 1$
273.2.a.d	$x^{3} + 11x^{2} - 15x - 293$
273.2.a.e	$x^{4} + 8x^{3} - 94x^{2} - 460x + 2697$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(273))$	11
Old forms	22
Cusp forms $S_{2}(\Gamma_0(273))$	33
Eisenstein series $E_{2}(\Gamma_0(273))$	7
Modular forms $M_{2}(\Gamma_0(273))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(273)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 7, 13, 21, 39, 91, 273]: [0, 1, 1, 1, 4, 6, 10, 40] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 11 = 1 x 11 (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(273, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 273:2 1000

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