Atlas › Modular forms › Weight 2 › Level 736 › 736.2.a

Newspace 736.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(736))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
736.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 4q^{5} + 4q^{7} + O(q^{8})$	−
736.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{5} - 6q^{7} + O(q^{8})$	−
736.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{5} + 6q^{7} + O(q^{8})$	+
736.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 4q^{5} - 4q^{7} + O(q^{8})$	+
736.2.a.e	3	degree 3	$3q - 4q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	+
736.2.a.f	3	degree 3	$3q + 4q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	−
736.2.a.g	4	degree 4	$4q - 2q^{3} + 6q^{5} + 4q^{7} + O(q^{8})$	−
736.2.a.h	4	degree 4	$4q + 2q^{3} + 6q^{5} - 4q^{7} + O(q^{8})$	+

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 $T$
736.2.a.a	$x^{2} + 14x + 17$
736.2.a.b	$x^{2} - 6x - 3$
736.2.a.c	$x^{2} - 6x - 39$
736.2.a.d	$x^{2} + 10x + 17$
736.2.a.e	$x^{3} + 10x^{2} - 23x - 304$
736.2.a.f	$x^{3} + 2x^{2} - 83x - 8$
736.2.a.g	$x^{4} - 16x^{3} - 15x^{2} + 1170x - 4196$
736.2.a.h	$x^{4} - 20x^{3} + 33x^{2} + 474x - 164$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(736))$	22
Old forms	67
Cusp forms $S_{2}(\Gamma_0(736))$	89
Eisenstein series $E_{2}(\Gamma_0(736))$	15
Modular forms $M_{2}(\Gamma_0(736))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(736)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(92))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(184))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(368))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.272 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, 4, 8, 16, 23, 32, 46, 92, 184, 368, 736]: [0, 1, 2, 3, 4, 3, 6, 8, 15, 28, 52, 100] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 4 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(736, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 736:2 1000

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