Atlas › Modular forms › Weight 2 › Level 738 › 738.2.a

Newspace 738.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(738))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
738.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+ +	738b1 rank 0
738.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + 2q^{7} + O(q^{8})$	+ −	738d1 rank 1
738.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 4q^{7} + O(q^{8})$	+ +	738a1 rank 1
738.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+ +	738c1 rank 0
738.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 2q^{7} + O(q^{8})$	− +	738f1 rank 1
738.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} + 2q^{7} + O(q^{8})$	− −	738j1 rank 0
738.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 4q^{7} + O(q^{8})$	− −	738e1 rank 1
738.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	− −	738i1 rank 0
738.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− −	738h1 rank 0
738.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− −	738g1 rank 0
738.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{7} + O(q^{8})$	+ −
738.2.a.l	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} + 4q^{7} + O(q^{8})$	+ −
738.2.a.m	3	degree 3	$3q + 3q^{2} + 3q^{4} + 3q^{5} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
738.2.a.a	$x + 24$
738.2.a.b	$x + 54$
738.2.a.c	$x + 26$
738.2.a.d	$x - 48$
738.2.a.e	$x + 12$
738.2.a.f	$x - 26$
738.2.a.g	$x + 68$
738.2.a.h	$x - 28$
738.2.a.i	$x - 16$
738.2.a.j	$x$
738.2.a.k	$x^{2} + 12x - 476$
738.2.a.l	$x^{3} - 44x^{2} + 16x + 64$
738.2.a.m	$x^{3} - 50x^{2} + 448x - 512$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(738))$	18
Old forms	101
Cusp forms $S_{2}(\Gamma_0(738))$	119
Eisenstein series $E_{2}(\Gamma_0(738))$	15
Modular forms $M_{2}(\Gamma_0(738))$	134

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(738)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(246))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(369))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.623 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, 3, 6, 9, 18, 41, 82, 123, 246, 369, 738]: [0, 1, 1, 3, 2, 5, 4, 12, 16, 46, 44, 130] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 18 = 1 x 18 (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(738, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 738:2 1000

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