Atlas › Modular forms › Weight 2 › Level 559 › 559.2.a

Newspace 559.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(559))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
559.2.a.a	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} + 6q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
559.2.a.b	4	degree 4	$4q - q^{2} + q^{3} + 3q^{4} - 5q^{5} - 11q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$
559.2.a.c	7	degree 7	$7q - 2q^{2} - 3q^{3} + 2q^{4} - q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
559.2.a.d	14	degree 14	$14q + 7q^{2} + q^{3} + 15q^{4} + 12q^{5} - q^{6} + q^{7} + O(q^{8})$	− +	$+1$
559.2.a.e	15	degree 15	$15q + 2q^{2} + q^{3} + 18q^{4} - 2q^{5} + 7q^{6} + q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
559.2.a.a	$x^{3} + 3x^{2} - 45x - 111$
559.2.a.b	$x^{4} - 2x^{3} - 20x^{2} + 24x + 16$
559.2.a.c	$x^{7} + 11x^{6} - 21x^{5} - 387x^{4} + 116x^{3} + 3208x^{2} - 1400x - 4912$
559.2.a.d	$x^{14} - 10x^{13} - 261x^{12} + 2778x^{11} + 24278x^{10} - 291168x^{9} - 929872x^{8} + 14489954x^{7} + 8927703x^{6} - 352350128x^{5} + 273372515x^{4} + 3790500438x^{3} - 5923339868x^{2} - 12916545368x + 25080071216$
559.2.a.e	$x^{15} - 5x^{14} - 397x^{13} + 1765x^{12} + 60034x^{11} - 214542x^{10} - 4569040x^{9} + 11149494x^{8} + 190969235x^{7} - 224037851x^{6} - 4298003117x^{5} - 290397285x^{4} + 44225707084x^{3} + 48788623176x^{2} - 101872914216x - 131870369552$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(559))$	43
Old forms	6
Cusp forms $S_{2}(\Gamma_0(559))$	49
Eisenstein series $E_{2}(\Gamma_0(559))$	3
Modular forms $M_{2}(\Gamma_0(559))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(559)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 13, 43, 559]: [0, 1, 4, 52] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 43 = 1 x 43 (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(559, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 559:2 1000

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