Atlas › Modular forms › Weight 2 › Level 878 › 878.2.a

Newspace 878.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(878))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
878.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} + 3q^{6} + O(q^{8})$	+ +	$-1$
878.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
878.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{4} - 3q^{5} - 6q^{7} + O(q^{8})$	− +	$+1$
878.2.a.d	2	$\mathbb{Q}(\sqrt{21})$	$2q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
878.2.a.e	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
878.2.a.f	3	degree 3	$3q + 3q^{2} - 5q^{3} + 3q^{4} - 5q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
878.2.a.g	4	degree 4	$4q - 4q^{2} + q^{3} + 4q^{4} - q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
878.2.a.h	4	degree 4	$4q + 4q^{2} - q^{3} + 4q^{4} - 6q^{5} - q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
878.2.a.i	7	degree 7	$7q + 7q^{2} + 3q^{3} + 7q^{4} + 4q^{5} + 3q^{6} + 14q^{7} + O(q^{8})$	− +	$+1$
878.2.a.j	8	degree 8	$8q - 8q^{2} + q^{3} + 8q^{4} - 2q^{5} - q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
878.2.a.a	$x^{2} + 12x + 31$
878.2.a.b	$x^{2} - 5x - 5$
878.2.a.c	$x^{2} + 9x - 11$
878.2.a.d	$x^{2} - 4x - 17$
878.2.a.e	$x^{3} - 5x^{2} - 36x - 37$
878.2.a.f	$x^{3} + 5x^{2} - 36x - 167$
878.2.a.g	$x^{4} - x^{3} - 90x^{2} + 443x - 460$
878.2.a.h	$x^{4} + 19x^{3} + 78x^{2} - 229x - 1361$
878.2.a.i	$x^{7} - 15x^{6} - 131x^{5} + 2992x^{4} - 10711x^{3} - 15230x^{2} + 36742x + 5431$
878.2.a.j	$x^{8} + 5x^{7} - 319x^{6} - 1782x^{5} + 30324x^{4} + 205756x^{3} - 711231x^{2} - 7256761x - 12464981$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(878))$	37
Old forms	72
Cusp forms $S_{2}(\Gamma_0(878))$	109
Eisenstein series $E_{2}(\Gamma_0(878))$	3
Modular forms $M_{2}(\Gamma_0(878))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(878)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(439))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.387 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, 2, 439, 878]: [0, 1, 37, 112] (certified at ell = 2147481151)
T = [(3, 1), (5, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 6 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(878, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 878:2 1000

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