Atlas › Modular forms › Weight 2 › Level 706 › 706.2.a

Newspace 706.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(706))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
706.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 3q^{5} - 2q^{6} + O(q^{8})$	+ +	$-1$	706a1 rank 1
706.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + O(q^{8})$	− −	$-1$	706d1 rank 1
706.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 4q^{5} + 2q^{7} + O(q^{8})$	− −	$-1$	706c1 rank 1
706.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	706b1 rank 1
706.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 4q^{5} + 8q^{7} + O(q^{8})$	− +	$+1$
706.2.a.f	5	degree 5	$5q - 5q^{2} - 2q^{3} + 5q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
706.2.a.g	8	degree 8	$8q - 8q^{2} + 2q^{3} + 8q^{4} + 5q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
706.2.a.h	10	degree 10	$10q + 10q^{2} + 2q^{3} + 10q^{4} + 3q^{5} + 2q^{6} - 12q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
706.2.a.a	$x + 7$
706.2.a.b	$x + 2$
706.2.a.c	$x + 12$
706.2.a.d	$x + 3$
706.2.a.e	$x^{2} - 12x + 28$
706.2.a.f	$x^{5} + 2x^{4} - 94x^{3} - 24x^{2} + 1540x + 1646$
706.2.a.g	$x^{8} - 17x^{7} - 102x^{6} + 2596x^{5} - 4397x^{4} - 90535x^{3} + 475748x^{2} - 768958x + 388126$
706.2.a.h	$x^{10} - 11x^{9} - 314x^{8} + 4006x^{7} + 23207x^{6} - 408519x^{5} + 96706x^{4} + 12583048x^{3} - 36244104x^{2} + 5890280x + 2890922$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(706))$	29
Old forms	58
Cusp forms $S_{2}(\Gamma_0(706))$	87
Eisenstein series $E_{2}(\Gamma_0(706))$	3
Modular forms $M_{2}(\Gamma_0(706))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(706)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(353))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 353, 706]: [0, 1, 30, 90] (certified at ell = 2147480897)
T = [(3, 1), (5, 3)]; h of degree 29 = 1 x 29 (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(706, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 706:2 1000

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