Atlas › Modular forms › Weight 2 › Level 649 › 649.2.a

Newspace 649.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(649))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
649.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	− −	$-1$	649a1 rank 1
649.2.a.b	2	$\mathbb{Q}(\sqrt{37})$	$2q - 2q^{2} - 4q^{3} - 2q^{4} - q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− −	$-1$
649.2.a.c	4	degree 4	$4q + q^{2} + q^{3} + 3q^{4} - 5q^{5} - 8q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
649.2.a.d	11	degree 11	$11q + 4q^{2} + 2q^{3} + 8q^{4} - q^{5} + 9q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
649.2.a.e	12	degree 12	$12q - 4q^{2} - 4q^{3} + 8q^{4} - 5q^{5} - 3q^{6} - 10q^{7} + O(q^{8})$	+ +	$-1$
649.2.a.f	17	degree 17	$17q + 3q^{2} + 4q^{3} + 23q^{4} + 7q^{5} - 5q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{17}$ 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 $T$
649.2.a.a	$x + 39$
649.2.a.b	$x^{2} + 6x - 324$
649.2.a.c	$x^{4} + 71x^{3} + 97x^{2} - 52561x - 450239$
649.2.a.d	$x^{11} - 82x^{10} - 12022x^{9} + 1184342x^{8} + 22800726x^{7} - 4042940087x^{6} + 32353834854x^{5} + 3885137217180x^{4} - 56028719873876x^{3} - 994061940277488x^{2} + 12983016841403232x + 67862976861413264$
649.2.a.e	$x^{12} + 187x^{11} + 5849x^{10} - 768978x^{9} - 54364538x^{8} - 427210528x^{7} + 42372373311x^{6} + 980392745421x^{5} + 2900673407330x^{4} - 56929210685166x^{3} - 198032612555975x^{2} - 13224954150616x + 764489292368$
649.2.a.f	$x^{17} - 12x^{16} - 26746x^{15} + 798505x^{14} + 260182258x^{13} - 10906681000x^{12} - 1173380969327x^{11} + 61598455616964x^{10} + 2517473168057563x^{9} - 167656485621870880x^{8} - 2217138230295877859x^{7} + 226986158331107788833x^{6} + 40809040809689999828x^{5} - 143073953131882365724796x^{4} + 746848332653814837936820x^{3} + 34771512514108240174900720x^{2} - 132654962130733971841453408x - 2999432250775016512729227120$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(649))$	47
Old forms	12
Cusp forms $S_{2}(\Gamma_0(649))$	59
Eisenstein series $E_{2}(\Gamma_0(649))$	3
Modular forms $M_{2}(\Gamma_0(649))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(649)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 11, 59, 649]: [0, 2, 6, 62] (certified at ell = 2147482481)
T = [(2, 1), (3, 3), (7, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(649, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 649:2 1000

← 648.2.a · 650.2.a →