Atlas › Modular forms › Weight 2 › Level 641 › 641.2.a

Newspace 641.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(641))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
641.2.a.a	20	degree 20	$20q - 3q^{2} - 9q^{3} + 9q^{4} - 5q^{5} - 11q^{6} - 24q^{7} + O(q^{8})$	+	$-1$
641.2.a.b	33	degree 33	$33q + 7q^{3} + 44q^{4} + q^{5} + 11q^{6} + 28q^{7} + O(q^{8})$	−	$+1$

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 $T$
641.2.a.a	$x^{20} + 12x^{19} + 25x^{18} - 205x^{17} - 832x^{16} + 998x^{15} + 7346x^{14} - 1141x^{13} - 31037x^{12} - 1441x^{11} + 71812x^{10} - 3899x^{9} - 88679x^{8} + 23245x^{7} + 46845x^{6} - 21549x^{5} - 3473x^{4} + 1884x^{3} + 109x^{2} - 29x + 1$
641.2.a.b	$x^{33} - 7x^{32} - 109x^{31} + 844x^{30} + 4793x^{29} - 43066x^{28} - 108176x^{27} + 1227201x^{26} + 1280303x^{25} - 21791939x^{24} - 5816382x^{23} + 255531689x^{22} - 39404004x^{21} - 2043980421x^{20} + 740387445x^{19} + 11326101189x^{18} - 4833523791x^{17} - 43503821109x^{16} + 16893852890x^{15} + 114097105530x^{14} - 31680541284x^{13} - 197470146947x^{12} + 24568695297x^{11} + 212731611627x^{10} + 9068601522x^{9} - 130249412361x^{8} - 27034348035x^{7} + 38028669128x^{6} + 14209959484x^{5} - 2757904350x^{4} - 1947828035x^{3} - 264562436x^{2} - 4314432x + 731243$

Orbit

Characteristic polynomial of

T

641.2.a.a

x^{20} + 12x^{19} + 25x^{18} - 205x^{17} - 832x^{16} + 998x^{15} + 7346x^{14} - 1141x^{13} - 31037x^{12} - 1441x^{11} + 71812x^{10} - 3899x^{9} - 88679x^{8} + 23245x^{7} + 46845x^{6} - 21549x^{5} - 3473x^{4} + 1884x^{3} + 109x^{2} - 29x + 1

641.2.a.b

x^{33} - 7x^{32} - 109x^{31} + 844x^{30} + 4793x^{29} - 43066x^{28} - 108176x^{27} + 1227201x^{26} + 1280303x^{25} - 21791939x^{24} - 5816382x^{23} + 255531689x^{22} - 39404004x^{21} - 2043980421x^{20} + 740387445x^{19} + 11326101189x^{18} - 4833523791x^{17} - 43503821109x^{16} + 16893852890x^{15} + 114097105530x^{14} - 31680541284x^{13} - 197470146947x^{12} + 24568695297x^{11} + 212731611627x^{10} + 9068601522x^{9} - 130249412361x^{8} - 27034348035x^{7} + 38028669128x^{6} + 14209959484x^{5} - 2757904350x^{4} - 1947828035x^{3} - 264562436x^{2} - 4314432x + 731243

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(641))$	53
Old forms	0
Cusp forms $S_{2}(\Gamma_0(641))$	53
Eisenstein series $E_{2}(\Gamma_0(641))$	1
Modular forms $M_{2}(\Gamma_0(641))$	54

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 641]: [0, 54] (certified at ell = 2147479681)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(641, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 641:2 1000

← 640.2.a · 642.2.a →