Atlas › Modular forms › Weight 2 › Level 694 › 694.2.a

Newspace 694.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(694))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
694.2.a.a	5	degree 5	$5q + 5q^{2} - 5q^{3} + 5q^{4} - 7q^{5} - 5q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
694.2.a.b	7	degree 7	$7q - 7q^{2} - 3q^{3} + 7q^{4} + 10q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+ −	$+1$
694.2.a.c	7	degree 7	$7q - 7q^{2} + 7q^{4} - 9q^{5} + 3q^{7} + O(q^{8})$	+ +	$-1$
694.2.a.d	9	degree 9	$9q + 9q^{2} + 6q^{3} + 9q^{4} + 6q^{5} + 6q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
694.2.a.a	$x^{5} + 12x^{4} + 51x^{3} + 88x^{2} + 48x + 1$
694.2.a.b	$x^{7} - 7x^{6} - 8x^{5} + 119x^{4} - 91x^{3} - 379x^{2} + 441x - 117$
694.2.a.c	$x^{7} + 9x^{6} + 8x^{5} - 65x^{4} - 123x^{3} + 5x^{2} + 81x + 5$
694.2.a.d	$x^{9} - 12x^{8} + 32x^{7} + 120x^{6} - 752x^{5} + 1125x^{4} + 215x^{3} - 1817x^{2} + 1274x - 181$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(694))$	28
Old forms	58
Cusp forms $S_{2}(\Gamma_0(694))$	86
Eisenstein series $E_{2}(\Gamma_0(694))$	3
Modular forms $M_{2}(\Gamma_0(694))$	89

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(694)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(347))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 347, 694]: [0, 1, 30, 89] (certified at ell = 2147481179)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 3 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(694, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 694:2 1000

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