Atlas › Modular forms › Weight 2 › Level 377 › 377.2.a

Newspace 377.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(377))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
377.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} + O(q^{8})$	− −	$-1$	377a1 rank 1
377.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} + 6q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
377.2.a.c	5	degree 5	$5q - 3q^{2} - 4q^{3} + 5q^{4} + 2q^{5} - 2q^{6} - 15q^{7} + O(q^{8})$	− −	$-1$
377.2.a.d	5	degree 5	$5q - q^{2} - 4q^{3} + q^{4} - 2q^{5} + 2q^{6} - 11q^{7} + O(q^{8})$	+ +	$-1$
377.2.a.e	7	degree 7	$7q + 3q^{2} + 2q^{3} + 11q^{4} - 2q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
377.2.a.f	9	degree 9	$9q + q^{2} + 9q^{4} + 2q^{5} - 2q^{6} + 17q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
377.2.a.a	$x - 1$
377.2.a.b	$x^{2} - 2x - 11$
377.2.a.c	$x^{5} + 7x^{4} + 6x^{3} - 41x^{2} - 75x - 31$
377.2.a.d	$x^{5} + 5x^{4} - 2x^{3} - 7x^{2} - x + 1$
377.2.a.e	$x^{7} - 5x^{6} - 9x^{5} + 78x^{4} - 51x^{3} - 282x^{2} + 465x - 161$
377.2.a.f	$x^{9} - x^{8} - 30x^{7} + 25x^{6} + 222x^{5} - 64x^{4} - 338x^{3} + 193x^{2} + 17x - 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(377))$	29
Old forms	4
Cusp forms $S_{2}(\Gamma_0(377))$	33
Eisenstein series $E_{2}(\Gamma_0(377))$	3
Modular forms $M_{2}(\Gamma_0(377))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(377)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 13, 29, 377]: [0, 1, 3, 36] (certified at ell = 2147483353)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 3 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(377, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 377:2 1000

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