Atlas › Modular forms › Weight 2 › Level 809 › 809.2.a

Newspace 809.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(809))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
809.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
809.2.a.b	24	degree 24	$24q - 4q^{2} - 11q^{3} + 14q^{4} - 10q^{5} - 12q^{6} - 19q^{7} + O(q^{8})$	+	$-1$
809.2.a.c	41	degree 41	$41q + 12q^{3} + 54q^{4} + 5q^{5} + 4q^{6} + 25q^{7} + O(q^{8})$	−	$+1$

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 $T$
809.2.a.a	$x^{2} - 5$
809.2.a.b	$x^{24} + 15x^{23} + 50x^{22} - 282x^{21} - 1961x^{20} + 351x^{19} + 24497x^{18} + 27763x^{17} - 150480x^{16} - 277948x^{15} + 499685x^{14} + 1247126x^{13} - 873250x^{12} - 3062664x^{11} + 623293x^{10} + 4292247x^{9} + 195840x^{8} - 3412545x^{7} - 531796x^{6} + 1491755x^{5} + 251734x^{4} - 329971x^{3} - 30240x^{2} + 29057x - 2417$
809.2.a.c	$x^{41} - 12x^{40} - 85x^{39} + 1522x^{38} + 1463x^{37} - 83488x^{36} + 97528x^{35} + 2616974x^{34} - 6174083x^{33} - 52139890x^{32} + 169843039x^{31} + 693751999x^{30} - 2871307182x^{29} - 6275828962x^{28} + 32938560235x^{27} + 38052291270x^{26} - 268186838836x^{25} - 142049469551x^{24} + 1585513941046x^{23} + 197111779751x^{22} - 6873268600216x^{21} + 1023215881055x^{20} + 21846306091797x^{19} - 7365776064193x^{18} - 50436387288817x^{17} + 23594719563267x^{16} + 82947339538547x^{15} - 46169484277041x^{14} - 94166535460630x^{13} + 57545578755006x^{12} + 70538309207495x^{11} - 44460267213982x^{10} - 33001374047549x^{9} + 19842525541917x^{8} + 9251314359131x^{7} - 4621576563705x^{6} - 1508903776063x^{5} + 473146481727x^{4} + 125872485363x^{3} - 14515439147x^{2} - 2643641296x + 210257611$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(809))$	67
Old forms	0
Cusp forms $S_{2}(\Gamma_0(809))$	67
Eisenstein series $E_{2}(\Gamma_0(809))$	1
Modular forms $M_{2}(\Gamma_0(809))$	68

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 809]: [0, 68] (certified at ell = 2147474929)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 7 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(809, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 809:2 1000

← 808.2.a · 810.2.a →