Atlas › Modular forms › Weight 2 › Level 646 › 646.2.a

Newspace 646.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(646))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
646.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+ + −	$+1$	646a1 rank 0
646.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$	646e1 rank 0
646.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$	646c1 rank 0
646.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	− + −	$-1$	646d1 rank 1
646.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	− + +	$+1$	646b1 rank 0
646.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{5} + 4q^{7} + O(q^{8})$	+ + +	$-1$
646.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{7} + O(q^{8})$	+ + −	$+1$
646.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	− − +	$-1$
646.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + O(q^{8})$	− + +	$+1$
646.2.a.j	3	degree 3	$3q - 3q^{2} - 2q^{3} + 3q^{4} - 4q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$
646.2.a.k	3	degree 3	$3q + 3q^{2} + 6q^{3} + 3q^{4} + 2q^{5} + 6q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
646.2.a.l	4	degree 4	$4q - 4q^{2} + 4q^{4} + 6q^{5} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
646.2.a.a	$x - 30$
646.2.a.b	$x - 8$
646.2.a.c	$x - 44$
646.2.a.d	$x + 30$
646.2.a.e	$x + 20$
646.2.a.f	$x^{2} + 20x - 142$
646.2.a.g	$x^{2} + 20x - 1252$
646.2.a.h	$x^{2} + 94x + 2062$
646.2.a.i	$x^{2} - 32x + 158$
646.2.a.j	$x^{3} + 32x^{2} + 160x - 356$
646.2.a.k	$x^{3} - 36x^{2} + 320x - 512$
646.2.a.l	$x^{4} - 32x^{3} - 1202x^{2} + 12584x + 314096$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(646))$	23
Old forms	64
Cusp forms $S_{2}(\Gamma_0(646))$	87
Eisenstein series $E_{2}(\Gamma_0(646))$	7
Modular forms $M_{2}(\Gamma_0(646))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(646)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(323))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.289 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, 2, 17, 19, 34, 38, 323, 646]: [0, 1, 2, 2, 6, 7, 32, 94] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 23 = 1 x 23 (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(646, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 646:2 1000

← 645.2.a · 647.2.a →