Atlas › Modular forms › Weight 2 › Level 544 › 544.2.a

Newspace 544.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(544))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
544.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	−	544c1 rank 0
544.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + 4q^{5} + 4q^{7} + O(q^{8})$	+	544f1 rank 0
544.2.a.c	1	$\mathbb{Q}$	$q - 2q^{7} + O(q^{8})$	+	544a1 rank 1
544.2.a.d	1	$\mathbb{Q}$	$q + 2q^{7} + O(q^{8})$	+	544d1 rank 0
544.2.a.e	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	−	544b1 rank 0
544.2.a.f	1	$\mathbb{Q}$	$q + 2q^{3} + 4q^{5} - 4q^{7} + O(q^{8})$	+	544e1 rank 0
544.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{5} + O(q^{8})$	+
544.2.a.h	2	$\mathbb{Q}(\sqrt{10})$	$2q - 4q^{5} + O(q^{8})$	+
544.2.a.i	3	degree 3	$3q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	−
544.2.a.j	3	degree 3	$3q + 2q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
544.2.a.a	$x - 300$
544.2.a.b	$x - 42$
544.2.a.c	$x - 70$
544.2.a.d	$x - 114$
544.2.a.e	$x - 4$
544.2.a.f	$x + 146$
544.2.a.g	$x^{2} + 316x - 4804$
544.2.a.h	$x^{2} - 292x + 14556$
544.2.a.i	$x^{3} + 256x^{2} + 17392x + 151136$
544.2.a.j	$x^{3} + 40x^{2} - 12160x + 342688$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(544))$	16
Old forms	49
Cusp forms $S_{2}(\Gamma_0(544))$	65
Eisenstein series $E_{2}(\Gamma_0(544))$	15
Modular forms $M_{2}(\Gamma_0(544))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(544)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(136))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(272))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.713 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, 17, 32, 34, 68, 136, 272, 544]: [0, 1, 2, 3, 4, 2, 6, 6, 12, 22, 40, 76] (certified at ell = 2147483489)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(544, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 544:2 1000

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