Atlas › Modular forms › Weight 2 › Level 799 › 799.2.a

Newspace 799.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(799))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
799.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	799b1 rank 1
799.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} + 4q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$	799a1 rank 0
799.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} - q^{5} - 5q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
799.2.a.d	8	degree 8	$8q + q^{2} - 7q^{3} + 7q^{4} - 10q^{5} + 2q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
799.2.a.e	12	degree 12	$12q - 4q^{2} - q^{3} + 8q^{4} - 11q^{5} - q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
799.2.a.f	17	degree 17	$17q + 3q^{2} + q^{3} + 19q^{4} + 11q^{5} + 5q^{6} + q^{7} + O(q^{8})$	− +	$+1$
799.2.a.g	20	degree 20	$20q + 4q^{2} + 3q^{3} + 24q^{4} + 13q^{5} + 11q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ 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 $T$
799.2.a.a	$x + 9$
799.2.a.b	$x - 11$
799.2.a.c	$x^{2} + 76x + 1199$
799.2.a.d	$x^{8} + 151x^{7} + 4438x^{6} - 317748x^{5} - 21040182x^{4} - 309469901x^{3} + 3896635431x^{2} + 134292769928x + 890716588271$
799.2.a.e	$x^{12} + 109x^{11} + 806x^{10} - 247804x^{9} - 6179857x^{8} + 151199376x^{7} + 5618616209x^{6} - 11349241052x^{5} - 1507416977731x^{4} - 8353576311653x^{3} + 44415857678457x^{2} + 285273914102020x - 19979807770049$
799.2.a.f	$x^{17} - 140x^{16} - 3803x^{15} + 1326062x^{14} - 26531600x^{13} - 4453271133x^{12} + 186288292084x^{11} + 5816514427435x^{10} - 419448589691954x^{9} - 421675032020617x^{8} + 410004844558822969x^{7} - 5102096086265869552x^{6} - 156513792034819904472x^{5} + 3450591291917968727165x^{4} + 11022219367597009713506x^{3} - 616516351134528977510381x^{2} + 1860397705981978243248869x + 17733328387528411007449913$
799.2.a.g	$x^{20} - 147x^{19} - 8278x^{18} + 2267100x^{17} - 42653475x^{16} - 10845599284x^{15} + 541699343743x^{14} + 15206243901214x^{13} - 1534665254050088x^{12} + 10703030337182072x^{11} + 1502021849942256205x^{10} - 31343730664727996104x^{9} - 507937072213646181294x^{8} + 19368722563226758295436x^{7} - 21804468886010623471035x^{6} - 4204366297860108705480446x^{5} + 36606586740145421618785723x^{4} + 163847564754158241267420763x^{3} - 3314848479908423738976562643x^{2} + 12589682422287260129246873108x - 8395243417324645960907831403$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(799))$	61
Old forms	10
Cusp forms $S_{2}(\Gamma_0(799))$	71
Eisenstein series $E_{2}(\Gamma_0(799))$	3
Modular forms $M_{2}(\Gamma_0(799))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(799)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.41 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, 17, 47, 799]: [0, 2, 5, 74] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 15 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(799, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 799:2 1000

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