Atlas › Modular forms › Weight 2 › Level 77 › 77.2.a

Newspace 77.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(77))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
77.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{4} - q^{5} - q^{7} + O(q^{8})$	+ +	$-1$	77a1 rank 1
77.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 3q^{5} + q^{7} + O(q^{8})$	− +	$+1$	77b1 rank 0
77.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	77c1 rank 0
77.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{4} - 4q^{5} - 10q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$

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 $T$
77.2.a.a	$x + 9$
77.2.a.b	$x - 3$
77.2.a.c	$x - 7$
77.2.a.d	$x^{2} - 6x - 11$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(77))$	5
Old forms	2
Cusp forms $S_{2}(\Gamma_0(77))$	7
Eisenstein series $E_{2}(\Gamma_0(77))$	3
Modular forms $M_{2}(\Gamma_0(77))$	10

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(77)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 7, 11, 77]: [0, 1, 2, 10] (certified at ell = 2147482951)
T = [(2, 1), (3, 3)]; h of degree 5 = 1 x 5 (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(77, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 77:2 1000

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