Atlas › Modular forms › Weight 2 › Level 728 › 728.2.a

Newspace 728.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(728))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
728.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - q^{5} + q^{7} + O(q^{8})$	− +	728d1 rank 1
728.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{7} + O(q^{8})$	− +	728a1 rank 0
728.2.a.c	1	$\mathbb{Q}$	$q - q^{5} - q^{7} + O(q^{8})$	+ −	728c1 rank 1
728.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + 3q^{5} + q^{7} + O(q^{8})$	− +	728b1 rank 0
728.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	− −
728.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{7} + O(q^{8})$	+ +
728.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ −
728.2.a.h	4	degree 4	$4q + q^{3} + 4q^{7} + O(q^{8})$	− −
728.2.a.i	4	degree 4	$4q + q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	+ +

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 $T$
728.2.a.a	$x + 5$
728.2.a.b	$x + 1$
728.2.a.c	$x + 3$
728.2.a.d	$x - 11$
728.2.a.e	$x^{2} + 10x - 7$
728.2.a.f	$x^{2} + 2x - 11$
728.2.a.g	$x^{2} + 2x - 67$
728.2.a.h	$x^{4} - x^{3} - 151x^{2} + 633x - 82$
728.2.a.i	$x^{4} - 7x^{3} - 171x^{2} + 807x + 3578$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(728))$	18
Old forms	87
Cusp forms $S_{2}(\Gamma_0(728))$	105
Eisenstein series $E_{2}(\Gamma_0(728))$	15
Modular forms $M_{2}(\Gamma_0(728))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(728)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(104))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(182))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(364))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 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, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, 728]: [0, 1, 2, 1, 3, 1, 4, 5, 7, 10, 12, 10, 18, 32, 62, 120] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (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(728, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 728:2 1000

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