Atlas › Modular forms › Weight 2 › Level 866 › 866.2.a

Newspace 866.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(866))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
866.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + O(q^{8})$	− −	$-1$	866a1 rank 1
866.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 4q^{5} - 2q^{7} + O(q^{8})$	− −	$-1$
866.2.a.c	5	degree 5	$5q - 5q^{2} + 5q^{4} - 2q^{5} - 5q^{7} + O(q^{8})$	+ +	$-1$
866.2.a.d	13	degree 13	$13q - 13q^{2} + 2q^{3} + 13q^{4} - 2q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$
866.2.a.e	16	degree 16	$16q + 16q^{2} + 4q^{3} + 16q^{4} + 6q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
866.2.a.a	$x + 7$
866.2.a.b	$x^{2} + 50x + 463$
866.2.a.c	$x^{5} + 45x^{4} - 538x^{3} - 35210x^{2} - 350619x - 689077$
866.2.a.d	$x^{13} - 47x^{12} - 4829x^{11} + 285083x^{10} + 3450662x^{9} - 362834512x^{8} - 36090644x^{7} + 192931678792x^{6} - 452459144960x^{5} - 49063689550264x^{4} + 94383148640912x^{3} + 5645972976137472x^{2} - 4123803377868832x - 212480998294164832$
866.2.a.e	$x^{16} - 79x^{15} - 4150x^{14} + 517974x^{13} - 6139105x^{12} - 714493189x^{11} + 24238403898x^{10} + 15555672356x^{9} - 11011081891072x^{8} + 118014235621660x^{7} + 1370174002643976x^{6} - 28343344544714512x^{5} + 25347976242819696x^{4} + 1897166162055925088x^{3} - 8586329708242123776x^{2} - 25637543316553961888x + 164317941233419690240$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(866))$	37
Old forms	70
Cusp forms $S_{2}(\Gamma_0(866))$	107
Eisenstein series $E_{2}(\Gamma_0(866))$	3
Modular forms $M_{2}(\Gamma_0(866))$	110

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(866)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(433))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.422 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, 433, 866]: [0, 1, 36, 110] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(866, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 866:2 1000

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