Atlas › Modular forms › Weight 2 › Level 860 › 860.2.a

Newspace 860.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(860))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
860.2.a.a	3	degree 3	$3q - 3q^{3} + 3q^{5} - 3q^{7} + O(q^{8})$	− +
860.2.a.b	3	degree 3	$3q + 3q^{3} - 3q^{5} + 3q^{7} + O(q^{8})$	+ +
860.2.a.c	4	degree 4	$4q - q^{3} - 4q^{5} - 7q^{7} + O(q^{8})$	+ −
860.2.a.d	4	degree 4	$4q + q^{3} + 4q^{5} + 7q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
860.2.a.a	$x^{3} + 6x^{2} + 9x + 1$
860.2.a.b	$x^{3} - 6x^{2} - 9x + 17$
860.2.a.c	$x^{4} + 8x^{3} + 9x^{2} - 35x - 54$
860.2.a.d	$x^{4} - 8x^{3} + 11x^{2} + 33x - 54$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(860))$	14
Old forms	113
Cusp forms $S_{2}(\Gamma_0(860))$	127
Eisenstein series $E_{2}(\Gamma_0(860))$	11
Modular forms $M_{2}(\Gamma_0(860))$	138

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(860)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(172))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(215))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(430))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 4, 5, 10, 20, 43, 86, 172, 215, 430, 860]: [0, 1, 2, 1, 3, 6, 4, 13, 25, 24, 70, 138] (certified at ell = 2147483353)
T = [(3, 1), (7, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 2 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(860, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 860:2 1000

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