Atlas › Modular forms › Weight 2 › Level 529 › 529.2.a

Newspace 529.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(529))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
529.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} + 2q^{5} - 5q^{6} - 2q^{7} + O(q^{8})$
529.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} + 6q^{6} - 6q^{7} + O(q^{8})$
529.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} + 6q^{6} + 6q^{7} + O(q^{8})$
529.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$
529.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$
529.2.a.f	3	degree 3	$3q + 6q^{4} + 3q^{6} + O(q^{8})$	$\mathbb{Q}(\sqrt{-23})$
529.2.a.g	4	degree 4	$4q - 4q^{2} - 4q^{3} - 4q^{4} + 4q^{6} + O(q^{8})$
529.2.a.h	4	degree 4	$4q - 2q^{2} - 4q^{3} + 6q^{4} + 2q^{6} + O(q^{8})$
529.2.a.i	5	degree 5	$5q + 2q^{2} + 2q^{3} + 4q^{4} - 7q^{5} - 8q^{6} - 8q^{7} + O(q^{8})$
529.2.a.j	5	degree 5	$5q + 2q^{2} + 2q^{3} + 4q^{4} + 7q^{5} - 8q^{6} + 8q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ 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 $T$
529.2.a.a	$x^{2} - 49x + 449$
529.2.a.b	$x^{2} + 90x + 1878$
529.2.a.c	$x^{2} - 102x + 2598$
529.2.a.d	$x^{2} + 36x - 188$
529.2.a.e	$x^{2} - 40x + 272$
529.2.a.f	$x^{3} - 96x + 123$
529.2.a.g	$x^{4} + 16x^{3} - 1162x^{2} - 19852x + 37054$
529.2.a.h	$x^{4} + 14x^{3} - 1515x^{2} - 11806x + 584081$
529.2.a.i	$x^{5} + 200x^{4} + 15648x^{3} + 599201x^{2} + 11239584x + 82717799$
529.2.a.j	$x^{5} - 216x^{4} + 18350x^{3} - 768289x^{2} + 15881754x - 129851479$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(529))$	31
Old forms	4
Cusp forms $S_{2}(\Gamma_0(529))$	35
Eisenstein series $E_{2}(\Gamma_0(529))$	23
Modular forms $M_{2}(\Gamma_0(529))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(529)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 23, 529]: [0, 3, 47] (certified at ell = 2147474627)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 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(529, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 529:2 1000

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