Atlas › Modular forms › Weight 2 › Level 71 › 71.2.a

Newspace 71.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(71))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
71.2.a.a	3	degree 3	$3q - q^{2} + q^{3} + 3q^{4} + 5q^{5} - 9q^{6} + 2q^{7} + O(q^{8})$	−	$+1$
71.2.a.b	3	degree 3	$3q - q^{3} + 4q^{4} - 3q^{5} + 9q^{6} + 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
71.2.a.a	$x^{3} - 2x^{2} - 16x + 24$
71.2.a.b	$x^{3} + 3x^{2} - 104x + 75$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(71))$	6
Old forms	0
Cusp forms $S_{2}(\Gamma_0(71))$	6
Eisenstein series $E_{2}(\Gamma_0(71))$	1
Modular forms $M_{2}(\Gamma_0(71))$	7

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.078 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, 71]: [0, 7] (certified at ell = 2147483171)
T = [(2, 1), (3, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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(71, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 71:2 1000

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