Atlas › Modular forms › Weight 2 › Level 726 › 726.2.a

Newspace 726.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(726))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
726.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ +	726b1 rank 0
726.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} + O(q^{8})$	+ +	726a1 rank 1
726.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ +	726c1 rank 0
726.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ −	726e1 rank 1
726.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ −	726d1 rank 1
726.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− +	726g1 rank 1
726.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + O(q^{8})$	− +	726f1 rank 0
726.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− −	726i1 rank 0
726.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - 2q^{7} + O(q^{8})$	− −	726h1 rank 0
726.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 2q^{6} - 7q^{7} + O(q^{8})$	+ +
726.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 5q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ −
726.2.a.l	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 7q^{7} + O(q^{8})$	− +
726.2.a.m	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 5q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
726.2.a.a	$x - 286$
726.2.a.b	$x + 70$
726.2.a.c	$x + 92$
726.2.a.d	$x + 280$
726.2.a.e	$x + 154$
726.2.a.f	$x + 42$
726.2.a.g	$x - 314$
726.2.a.h	$x - 30$
726.2.a.i	$x + 200$
726.2.a.j	$x^{2} - 216x + 4444$
726.2.a.k	$x^{2} - 2x - 23804$
726.2.a.l	$x^{2} - 118x - 164$
726.2.a.m	$x^{2} - 80x - 2900$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(726))$	17
Old forms	92
Cusp forms $S_{2}(\Gamma_0(726))$	109
Eisenstein series $E_{2}(\Gamma_0(726))$	47
Modular forms $M_{2}(\Gamma_0(726))$	156

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(726)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(242))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(363))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.52 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, 11, 22, 33, 66, 121, 242, 363, 726]: [0, 1, 1, 3, 2, 5, 6, 16, 12, 35, 46, 136] (certified at ell = 2147482921)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 17 = 1 x 17 (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(726, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 726:2 1000

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