Atlas › Modular forms › Weight 2 › Level 637 › 637.2.a

Newspace 637.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(637))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
637.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + 3q^{5} + O(q^{8})$	−	637d1 rank 1
637.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} + 3q^{5} + O(q^{8})$	+	637b1 rank 0
637.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + O(q^{8})$	+	637a1 rank 1
637.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + O(q^{8})$	−	637c1 rank 1
637.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} - 5q^{6} + O(q^{8})$	−
637.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} + 5q^{6} + O(q^{8})$	+
637.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 6q^{5} + 4q^{6} + O(q^{8})$	−
637.2.a.h	3	degree 3	$3q - 2q^{2} - 4q^{3} + 6q^{4} - 5q^{5} - 2q^{6} + O(q^{8})$	−
637.2.a.i	3	degree 3	$3q - 2q^{2} + 4q^{3} + 6q^{4} + 5q^{5} + 2q^{6} + O(q^{8})$	+
637.2.a.j	3	degree 3	$3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{5} - 4q^{6} + O(q^{8})$	+
637.2.a.k	5	degree 5	$5q + 4q^{2} + 8q^{4} - 2q^{5} + 5q^{6} + O(q^{8})$	+
637.2.a.l	5	degree 5	$5q + 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} + O(q^{8})$	−
637.2.a.m	6	degree 6	$6q - 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + O(q^{8})$	+
637.2.a.n	6	degree 6	$6q + 8q^{3} + 4q^{4} + 6q^{5} + 4q^{6} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{3} + 3T_{5} + 4T_{11} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
637.2.a.a	$x + 45$
637.2.a.b	$x - 63$
637.2.a.c	$x - 4$
637.2.a.d	$x - 18$
637.2.a.e	$x^{2} + 439x + 35929$
637.2.a.f	$x^{2} + 247x + 11471$
637.2.a.g	$x^{2} + 250x + 14473$
637.2.a.h	$x^{3} + 183x^{2} - 389x - 541003$
637.2.a.i	$x^{3} - 65x^{2} - 42789x - 289091$
637.2.a.j	$x^{3} - 407x^{2} + 48559x - 1389097$
637.2.a.k	$x^{5} - 23x^{4} - 60173x^{3} - 535258x^{2} + 514276644x - 15649538538$
637.2.a.l	$x^{5} - 293x^{4} + 11919x^{3} + 2477304x^{2} - 105686532x - 7680829586$
637.2.a.m	$x^{6} + 142x^{5} - 24353x^{4} - 1726788x^{3} + 171661167x^{2} + 579320198x - 20790308239$
637.2.a.n	$x^{6} - 42x^{5} - 42497x^{4} - 445300x^{3} + 444743551x^{2} + 22931278542x + 209663210817$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(637))$	41
Old forms	16
Cusp forms $S_{2}(\Gamma_0(637))$	57
Eisenstein series $E_{2}(\Gamma_0(637))$	15
Modular forms $M_{2}(\Gamma_0(637))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(637)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.681 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, 7, 13, 49, 91, 637]: [0, 1, 1, 5, 10, 66] (certified at ell = 2147483353)
T = [(2, 1), (3, 2), (5, 3), (11, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 13 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(637, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 637:2 1000

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