Atlas › Modular forms › Weight 2 › Level 643 › 643.2.a

Newspace 643.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(643))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
643.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	−	$+1$	643a1 rank 2
643.2.a.b	24	degree 24	$24q - 13q^{2} - 7q^{3} + 23q^{4} - 23q^{5} - 4q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
643.2.a.c	28	degree 28	$28q + 14q^{2} + 5q^{3} + 32q^{4} + 23q^{5} + 2q^{6} + 11q^{7} + O(q^{8})$	−	$+1$

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 $T$
643.2.a.a	$x + 3$
643.2.a.b	$x^{24} + 20x^{23} + 126x^{22} - 3086x^{20} - 9131x^{19} + 18858x^{18} + 119288x^{17} + 20994x^{16} - 635063x^{15} - 563739x^{14} + 1727241x^{13} + 1966296x^{12} - 2827535x^{11} - 2991236x^{10} + 3062908x^{9} + 2122572x^{8} - 2083077x^{7} - 522854x^{6} + 703588x^{5} - 57096x^{4} - 58334x^{3} + 10425x^{2} + 807x - 193$
643.2.a.c	$x^{28} - 19x^{27} + 78x^{26} + 698x^{25} - 6628x^{24} + 3103x^{23} + 142573x^{22} - 399134x^{21} - 1166624x^{20} + 6246761x^{19} + 1952546x^{18} - 46961816x^{17} + 30921337x^{16} + 204445209x^{15} - 246496517x^{14} - 553958930x^{13} + 881577628x^{12} + 972418365x^{11} - 1822848179x^{10} - 1149512388x^{9} + 2286338794x^{8} + 961699176x^{7} - 1697940005x^{6} - 583090578x^{5} + 671548142x^{4} + 230063233x^{3} - 101710074x^{2} - 40612680x - 1982664$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(643))$	53
Old forms	0
Cusp forms $S_{2}(\Gamma_0(643))$	53
Eisenstein series $E_{2}(\Gamma_0(643))$	1
Modular forms $M_{2}(\Gamma_0(643))$	54

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 643]: [0, 54] (certified at ell = 2147476519)
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(643, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 643:2 1000

← 642.2.a · 644.2.a →