Atlas › Modular forms › Weight 2 › Level 557 › 557.2.a

Newspace 557.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(557))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
557.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + 2q^{7} + O(q^{8})$	+	$-1$	557a1 rank 1
557.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} + 5q^{7} + O(q^{8})$	−	$+1$	557b1 rank 0
557.2.a.c	18	degree 18	$18q - 6q^{2} - 15q^{3} + 12q^{4} - 6q^{5} - 2q^{6} - 23q^{7} + O(q^{8})$	+	$-1$
557.2.a.d	26	degree 26	$26q + q^{2} + 16q^{3} + 29q^{4} + 4q^{5} - q^{6} + 14q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
557.2.a.a	$x$
557.2.a.b	$x - 4$
557.2.a.c	$x^{18} + 21x^{17} + 161x^{16} + 433x^{15} - 857x^{14} - 7431x^{13} - 10117x^{12} + 23262x^{11} + 68245x^{10} + 3356x^{9} - 114160x^{8} - 39677x^{7} + 85091x^{6} + 22945x^{5} - 25517x^{4} - 4354x^{3} + 2316x^{2} + 271x - 28$
557.2.a.d	$x^{26} - 17x^{25} + 50x^{24} + 670x^{23} - 4482x^{22} - 4800x^{21} + 102201x^{20} - 116862x^{19} - 1077287x^{18} + 2491481x^{17} + 5855330x^{16} - 20595678x^{15} - 15583224x^{14} + 92797134x^{13} + 10622593x^{12} - 247238093x^{11} + 40044973x^{10} + 395051638x^{9} - 92298701x^{8} - 367371116x^{7} + 61706003x^{6} + 180478376x^{5} - 6166086x^{4} - 35270479x^{3} - 3023906x^{2} + 534244x + 47636$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(557))$	46
Old forms	0
Cusp forms $S_{2}(\Gamma_0(557))$	46
Eisenstein series $E_{2}(\Gamma_0(557))$	1
Modular forms $M_{2}(\Gamma_0(557))$	47

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 557]: [0, 47] (certified at ell = 2147475497)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 5 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(557, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 557:2 1000

← 556.2.a · 558.2.a →