Atlas › Modular forms › Weight 2 › Level 685 › 685.2.a

Newspace 685.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(685))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
685.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + 3q^{7} + O(q^{8})$	+ +	$-1$	685a1 rank 1
685.2.a.b	8	degree 8	$8q - 3q^{2} - 5q^{3} + 3q^{4} + 8q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
685.2.a.c	11	degree 11	$11q - 6q^{2} - 7q^{3} + 16q^{4} - 11q^{5} + 2q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
685.2.a.d	11	degree 11	$11q + 6q^{2} + 7q^{3} + 12q^{4} - 11q^{5} + 6q^{7} + O(q^{8})$	+ −	$+1$
685.2.a.e	14	degree 14	$14q + 5q^{2} + 5q^{3} + 17q^{4} + 14q^{5} + O(q^{8})$	− +	$+1$

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 $T$
685.2.a.a	$x - 1$
685.2.a.b	$x^{8} + 8x^{7} + 15x^{6} - 21x^{5} - 65x^{4} + 14x^{3} + 57x^{2} - 10x - 1$
685.2.a.c	$x^{11} + 13x^{10} + 40x^{9} - 105x^{8} - 627x^{7} - 177x^{6} + 1889x^{5} + 846x^{4} - 1770x^{3} - 578x^{2} + 413x - 43$
685.2.a.d	$x^{11} - 13x^{10} + 44x^{9} + 73x^{8} - 609x^{7} + 429x^{6} + 2083x^{5} - 2972x^{4} - 1358x^{3} + 3248x^{2} + 109x - 1043$
685.2.a.e	$x^{14} - 10x^{13} - 2x^{12} + 303x^{11} - 713x^{10} - 2265x^{9} + 8604x^{8} + 4153x^{7} - 35880x^{6} + 8459x^{5} + 58507x^{4} - 26018x^{3} - 29434x^{2} + 9262x + 829$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(685))$	45
Old forms	22
Cusp forms $S_{2}(\Gamma_0(685))$	67
Eisenstein series $E_{2}(\Gamma_0(685))$	3
Modular forms $M_{2}(\Gamma_0(685))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(685)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(137))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 5, 137, 685]: [0, 1, 12, 70] (certified at ell = 2147483249)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(685, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 685:2 1000

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