Atlas › Modular forms › Weight 2 › Level 865 › 865.2.a

Newspace 865.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(865))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
865.2.a.a	9	degree 9	$9q - 3q^{2} - 4q^{3} + q^{4} + 9q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
865.2.a.b	13	degree 13	$13q + 3q^{2} + 4q^{3} + 9q^{4} - 13q^{5} + 5q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
865.2.a.c	16	degree 16	$16q - 6q^{2} - 8q^{3} + 18q^{4} - 16q^{5} - 5q^{6} - 12q^{7} + O(q^{8})$	+ +	$-1$
865.2.a.d	19	degree 19	$19q + 3q^{2} + 4q^{3} + 27q^{4} + 19q^{5} + 7q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

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 $T$
865.2.a.a	$x^{9} + 7x^{8} + 7x^{7} - 36x^{6} - 47x^{5} + 73x^{4} + 53x^{3} - 72x^{2} + 16x - 1$
865.2.a.b	$x^{13} - 7x^{12} - 25x^{11} + 220x^{10} + 148x^{9} - 2152x^{8} - 248x^{7} + 8370x^{6} + 1169x^{5} - 12570x^{4} - 4681x^{3} + 4658x^{2} + 2524x + 277$
865.2.a.c	$x^{16} + 14x^{15} + 46x^{14} - 171x^{13} - 1152x^{12} - 62x^{11} + 9051x^{10} + 8418x^{9} - 32267x^{8} - 40780x^{7} + 55758x^{6} + 75295x^{5} - 47006x^{4} - 58468x^{3} + 17197x^{2} + 15515x - 1457$
865.2.a.d	$x^{19} - 7x^{18} - 55x^{17} + 474x^{16} + 809x^{15} - 11507x^{14} + 1196x^{13} + 128667x^{12} - 100794x^{11} - 750144x^{10} + 773207x^{9} + 2376124x^{8} - 2468501x^{7} - 4000775x^{6} + 3518312x^{5} + 3278448x^{4} - 2089557x^{3} - 1104063x^{2} + 388846x + 57811$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(865))$	57
Old forms	28
Cusp forms $S_{2}(\Gamma_0(865))$	85
Eisenstein series $E_{2}(\Gamma_0(865))$	3
Modular forms $M_{2}(\Gamma_0(865))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(865)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(173))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 5, 173, 865]: [0, 1, 15, 88] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(865, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 865:2 1000

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