Atlas › Modular forms › Weight 2 › Level 237 › 237.2.a

Newspace 237.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(237))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
237.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
237.2.a.b	4	degree 4	$4q - 3q^{2} - 4q^{3} + 3q^{4} - 4q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
237.2.a.c	7	degree 7	$7q + 2q^{2} + 7q^{3} + 12q^{4} - 2q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
237.2.a.a	$x^{2} - 2x - 1$
237.2.a.b	$x^{4} + 7x^{3} + 8x^{2} - 30x - 53$
237.2.a.c	$x^{7} - 36x^{5} + 50x^{4} + 104x^{3} + 41x^{2} + x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(237))$	13
Old forms	12
Cusp forms $S_{2}(\Gamma_0(237))$	25
Eisenstein series $E_{2}(\Gamma_0(237))$	3
Modular forms $M_{2}(\Gamma_0(237))$	28

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(237)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.18 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, 79, 237]: [0, 1, 7, 28] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 13 = 1 x 13 (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(237, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 237:2 1000

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