Atlas › Modular forms › Weight 2 › Level 667 › 667.2.a

Newspace 667.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(667))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
667.2.a.a	10	degree 10	$10q - 3q^{2} - 9q^{3} + 9q^{4} - 10q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− −	$-1$
667.2.a.b	12	degree 12	$12q - 3q^{2} - 3q^{3} + 11q^{4} - 16q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
667.2.a.c	13	degree 13	$13q + 4q^{2} + 3q^{3} + 12q^{4} + 16q^{5} + q^{7} + O(q^{8})$	+ −	$+1$
667.2.a.d	16	degree 16	$16q + 3q^{2} + 5q^{3} + 21q^{4} + 16q^{5} + 6q^{6} + q^{7} + O(q^{8})$	− +	$+1$

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 $T$
667.2.a.a	$x^{10} + 12x^{9} + 35x^{8} - 91x^{7} - 600x^{6} - 532x^{5} + 1517x^{4} + 2661x^{3} + 262x^{2} - 937x + 23$
667.2.a.b	$x^{12} + 6x^{11} - 17x^{10} - 109x^{9} + 134x^{8} + 668x^{7} - 689x^{6} - 1459x^{5} + 1850x^{4} + 509x^{3} - 1553x^{2} + 752x - 116$
667.2.a.c	$x^{13} - 7x^{12} - 21x^{11} + 216x^{10} + 49x^{9} - 2272x^{8} + 885x^{7} + 10546x^{6} - 4733x^{5} - 21991x^{4} + 7354x^{3} + 18207x^{2} - 2672x - 3616$
667.2.a.d	$x^{16} - 8x^{15} - 36x^{14} + 415x^{13} + 67x^{12} - 7173x^{11} + 7661x^{10} + 54271x^{9} - 85937x^{8} - 192086x^{7} + 352855x^{6} + 284971x^{5} - 550741x^{4} - 97488x^{3} + 162524x^{2} + 4832x - 11584$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(667))$	51
Old forms	8
Cusp forms $S_{2}(\Gamma_0(667))$	59
Eisenstein series $E_{2}(\Gamma_0(667))$	3
Modular forms $M_{2}(\Gamma_0(667))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(667)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.215 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, 23, 29, 667]: [0, 3, 3, 62] (certified at ell = 2147482877)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 5 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(667, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 667:2 1000

← 666.2.a · 668.2.a →