Atlas › Modular forms › Weight 2 › Level 682 › 682.2.a

Newspace 682.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(682))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
682.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	682a1 rank 1
682.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	− − +	$-1$	682b1 rank 1
682.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{7} + O(q^{8})$	+ − +	$+1$
682.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$
682.2.a.e	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{4} - 2q^{7} + O(q^{8})$	− − −	$+1$
682.2.a.f	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} + q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$
682.2.a.g	4	degree 4	$4q - 4q^{2} - 3q^{3} + 4q^{4} - 3q^{5} + 3q^{6} + O(q^{8})$	+ + +	$-1$
682.2.a.h	4	degree 4	$4q - 4q^{2} - 3q^{3} + 4q^{4} - q^{5} + 3q^{6} - 6q^{7} + O(q^{8})$	+ − −	$-1$
682.2.a.i	6	degree 6	$6q + 6q^{2} + q^{3} + 6q^{4} + 5q^{5} + q^{6} + q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ 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 $T$
682.2.a.a	$x + 62$
682.2.a.b	$x + 22$
682.2.a.c	$x^{2} - 64x - 434$
682.2.a.d	$x^{2} - 42x + 198$
682.2.a.e	$x^{2} - 100x + 2446$
682.2.a.f	$x^{3} - 57x^{2} + 442x + 4001$
682.2.a.g	$x^{4} - x^{3} - 3050x^{2} - 731x + 1109456$
682.2.a.h	$x^{4} + 121x^{3} - 442x^{2} - 227539x + 3512596$
682.2.a.i	$x^{6} - 51x^{5} - 7700x^{4} + 185787x^{3} + 10273516x^{2} - 199187170x + 873301916$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(682))$	25
Old forms	68
Cusp forms $S_{2}(\Gamma_0(682))$	93
Eisenstein series $E_{2}(\Gamma_0(682))$	7
Modular forms $M_{2}(\Gamma_0(682))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(682)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(341))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.448 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, 11, 22, 31, 62, 341, 682]: [0, 1, 2, 5, 3, 10, 34, 100] (certified at ell = 2147482951)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 7 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(682, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 682:2 1000

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