Atlas › Modular forms › Weight 2 › Level 862 › 862.2.a

Newspace 862.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(862))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
862.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} - q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	862b1 rank 1
862.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	862a1 rank 1
862.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$	862f1 rank 1
862.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	862e1 rank 1
862.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− +	$+1$	862c1 rank 0
862.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	862d1 rank 0
862.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$
862.2.a.h	5	degree 5	$5q + 5q^{2} + 5q^{3} + 5q^{4} + 3q^{5} + 5q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$
862.2.a.i	6	degree 6	$6q - 6q^{2} - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
862.2.a.j	6	degree 6	$6q + 6q^{2} - 8q^{3} + 6q^{4} - 8q^{5} - 8q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
862.2.a.k	10	degree 10	$10q - 10q^{2} + 4q^{3} + 10q^{4} + 4q^{5} - 4q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
862.2.a.a	$x + 13$
862.2.a.b	$x - 35$
862.2.a.c	$x + 19$
862.2.a.d	$x + 83$
862.2.a.e	$x - 62$
862.2.a.f	$x - 41$
862.2.a.g	$x^{2} - 26x - 551$
862.2.a.h	$x^{5} - 51x^{4} - 535x^{3} + 49875x^{2} - 132866x - 9498413$
862.2.a.i	$x^{6} + 116x^{5} + 1490x^{4} - 234688x^{3} - 9416921x^{2} - 115487755x - 367227989$
862.2.a.j	$x^{6} + 120x^{5} + 3764x^{4} - 29132x^{3} - 2225363x^{2} - 1846907x + 181300913$
862.2.a.k	$x^{10} - 74x^{9} - 7611x^{8} + 663114x^{7} + 12540977x^{6} - 1745223691x^{5} + 4387070676x^{4} + 1660877196061x^{3} - 17035723731669x^{2} - 418932479344174x + 4055128352184206$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(862))$	35
Old forms	72
Cusp forms $S_{2}(\Gamma_0(862))$	107
Eisenstein series $E_{2}(\Gamma_0(862))$	3
Modular forms $M_{2}(\Gamma_0(862))$	110

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(862)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(431))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.47 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, 431, 862]: [0, 1, 37, 110] (certified at ell = 2147481491)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 9 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(862, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 862:2 1000

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