Atlas › Modular forms › Weight 2 › Level 671 › 671.2.a

Newspace 671.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(671))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
671.2.a.a	5	degree 5	$5q - 2q^{2} - 2q^{5} - 5q^{6} - q^{7} + O(q^{8})$	− −	$-1$
671.2.a.b	6	degree 6	$6q - q^{3} + 2q^{4} - q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
671.2.a.c	19	degree 19	$19q + 5q^{2} + 23q^{4} + q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$
671.2.a.d	21	degree 21	$21q + 3q^{3} + 32q^{4} + 7q^{5} + 5q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
671.2.a.a	$x^{5} + 2x^{4} - 4x^{3} - 3x^{2} + 2x + 1$
671.2.a.b	$x^{6} + x^{5} - 12x^{4} + x^{3} + 19x^{2} + 7x - 1$
671.2.a.c	$x^{19} - 5x^{18} - 62x^{17} + 332x^{16} + 1304x^{15} - 7837x^{14} - 10914x^{13} + 81680x^{12} + 35317x^{11} - 397276x^{10} - 68313x^{9} + 911444x^{8} + 306508x^{7} - 997474x^{6} - 652748x^{5} + 327694x^{4} + 432936x^{3} + 127619x^{2} + 1964x - 3011$
671.2.a.d	$x^{21} - 3x^{20} - 89x^{19} + 261x^{18} + 3104x^{17} - 8373x^{16} - 57470x^{15} + 131239x^{14} + 641963x^{13} - 1101240x^{12} - 4543308x^{11} + 4886972x^{10} + 20148283x^{9} - 9616526x^{8} - 52467170x^{7} + 480572x^{6} + 71279372x^{5} + 21215513x^{4} - 41473562x^{3} - 19610254x^{2} + 5027208x + 2168723$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(671))$	51
Old forms	10
Cusp forms $S_{2}(\Gamma_0(671))$	61
Eisenstein series $E_{2}(\Gamma_0(671))$	3
Modular forms $M_{2}(\Gamma_0(671))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(671)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.224 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, 61, 671]: [0, 2, 5, 64] (certified at ell = 2147482921)
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(671, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 671:2 1000

← 670.2.a · 672.2.a →