Atlas › Modular forms › Weight 4 › Level 177 › 177.4.a

Newspace 177.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(177))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
177.4.a.a	7	degree 7	$7q - 8q^{2} + 21q^{3} + 22q^{4} - 28q^{5} - 24q^{6} - 59q^{7} + O(q^{8})$	− +	$-1$
177.4.a.b	7	degree 7	$7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} + O(q^{8})$	+ −	$-1$
177.4.a.c	8	degree 8	$8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + O(q^{8})$	+ +	$+1$
177.4.a.d	8	degree 8	$8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + O(q^{8})$	− −	$+1$

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 $T$
177.4.a.a	$x^{7} + 36x^{6} + 154x^{5} - 5947x^{4} - 37467x^{3} + 371126x^{2} + 1504000x - 8997599$
177.4.a.b	$x^{7} + 2x^{6} - 314x^{5} - 1321x^{4} + 18591x^{3} + 86182x^{2} + 55434x - 87215$
177.4.a.c	$x^{8} + 10x^{7} - 662x^{6} - 5405x^{5} + 136869x^{4} + 966014x^{3} - 9551010x^{2} - 57276451x + 95723082$
177.4.a.d	$x^{8} - 48x^{7} + 534x^{6} + 5617x^{5} - 115037x^{4} - 18626x^{3} + 6231436x^{2} - 8667983x - 84466198$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(177))$	30
Old forms	28
Cusp forms $S_{4}(\Gamma_0(177))$	58
Eisenstein series $E_{4}(\Gamma_0(177))$	4
Modular forms $M_{4}(\Gamma_0(177))$	62

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(177)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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, 59, 177]: [1, 2, 16, 62] (certified at ell = 2147482481)
T = [(2, 1), (5, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(177, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 177:4 1000

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