Atlas › Modular forms › Weight 2 › Level 747 › 747.2.a

Newspace 747.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(747))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
747.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{5} + O(q^{8})$	+	747a1 rank 1
747.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	−	747e1 rank 1
747.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + O(q^{8})$	−	747c1 rank 1
747.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 2q^{5} - 3q^{7} + O(q^{8})$	−	747d1 rank 1
747.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 2q^{5} + O(q^{8})$	−	747b1 rank 0
747.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 6q^{5} - 4q^{7} + O(q^{8})$	+
747.2.a.g	4	degree 4	$4q - 2q^{2} + 4q^{4} - 6q^{5} + O(q^{8})$	−
747.2.a.h	5	degree 5	$5q + 3q^{2} + 7q^{4} + 2q^{5} + 8q^{7} + O(q^{8})$	+
747.2.a.i	6	degree 6	$6q - 4q^{2} + 10q^{4} - 8q^{5} + O(q^{8})$	+
747.2.a.j	6	degree 6	$6q - q^{2} + 7q^{4} - 2q^{5} + 3q^{7} + O(q^{8})$	+
747.2.a.k	6	degree 6	$6q + 4q^{2} + 10q^{4} + 8q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
747.2.a.a	$x + 1$
747.2.a.b	$x + 5$
747.2.a.c	$x - 7$
747.2.a.d	$x + 95$
747.2.a.e	$x - 85$
747.2.a.f	$x^{2} - 50x - 943$
747.2.a.g	$x^{4} + 82x^{3} + 1180x^{2} - 26882x - 519125$
747.2.a.h	$x^{5} + x^{4} - 9326x^{3} + 169186x^{2} + 4751909x - 7592699$
747.2.a.i	$x^{6} + 164x^{5} + 4079x^{4} - 560432x^{3} - 38847685x^{2} - 884683508x - 6766548419$
747.2.a.j	$x^{6} - 151x^{5} + 1946x^{4} + 575233x^{3} - 21706090x^{2} - 215569688x + 12883734091$
747.2.a.k	$x^{6} - 24x^{5} - 8437x^{4} + 159304x^{3} + 16897123x^{2} - 93594464x - 8165940343$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(747))$	34
Old forms	47
Cusp forms $S_{2}(\Gamma_0(747))$	81
Eisenstein series $E_{2}(\Gamma_0(747))$	7
Modular forms $M_{2}(\Gamma_0(747))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(747)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(249))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.428 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, 3, 9, 83, 249, 747]: [0, 1, 2, 8, 30, 86] (certified at ell = 2147483323)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 9 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(747, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 747:2 1000

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