Atlas › Modular forms › Weight 2 › Level 699 › 699.2.a

Newspace 699.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(699))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
699.2.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− +	$+1$	699a1 rank 0
699.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 6q^{4} - 5q^{5} + 5q^{7} + O(q^{8})$	+ −	$+1$
699.2.a.c	5	degree 5	$5q - 2q^{2} + 5q^{3} - 9q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
699.2.a.d	5	degree 5	$5q - 5q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
699.2.a.e	13	degree 13	$13q + q^{2} - 13q^{3} + 19q^{4} + 10q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
699.2.a.f	13	degree 13	$13q + 2q^{2} + 13q^{3} + 16q^{4} + 12q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
699.2.a.a	$x - 5$
699.2.a.b	$x^{2} + 15x + 25$
699.2.a.c	$x^{5} + 29x^{4} + 294x^{3} + 1243x^{2} + 1897x + 163$
699.2.a.d	$x^{5} + 9x^{4} - 8x^{3} - 139x^{2} - 205x - 83$
699.2.a.e	$x^{13} - 31x^{12} + 11x^{11} + 7885x^{10} - 50827x^{9} - 653149x^{8} + 5788786x^{7} + 22872192x^{6} - 221191998x^{5} - 516188202x^{4} + 2803313839x^{3} + 8351523380x^{2} + 7234680932x + 2018087653$
699.2.a.f	$x^{13} - 38x^{12} + 320x^{11} + 4517x^{10} - 85171x^{9} + 193574x^{8} + 3874911x^{7} - 25460981x^{6} + 17698133x^{5} + 183262305x^{4} - 208437886x^{3} - 431740882x^{2} + 183007787x + 4073001$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(699))$	39
Old forms	38
Cusp forms $S_{2}(\Gamma_0(699))$	77
Eisenstein series $E_{2}(\Gamma_0(699))$	3
Modular forms $M_{2}(\Gamma_0(699))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(699)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(233))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 3, 233, 699]: [0, 1, 20, 80] (certified at ell = 2147482481)
T = [(2, 1), (5, 3)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(699, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 699:2 1000

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