Atlas › Modular forms › Weight 2 › Level 523 › 523.2.a

Newspace 523.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(523))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
523.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} - 3q^{3} + 3q^{4} + 7q^{6} - 2q^{7} + O(q^{8})$	+	$-1$
523.2.a.b	15	degree 15	$15q - 6q^{2} - 3q^{3} + 10q^{4} - 19q^{5} - 7q^{6} - q^{7} + O(q^{8})$	+	$-1$
523.2.a.c	26	degree 26	$26q + 9q^{2} + 4q^{3} + 31q^{4} + 21q^{5} + 3q^{7} + O(q^{8})$	−	$+1$

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 $T$
523.2.a.a	$x^{2} + 6x + 4$
523.2.a.b	$x^{15} + 9x^{14} + 6x^{13} - 139x^{12} - 254x^{11} + 830x^{10} + 1806x^{9} - 2593x^{8} - 5466x^{7} + 4812x^{6} + 7695x^{5} - 5131x^{4} - 4404x^{3} + 2222x^{2} + 639x - 34$
523.2.a.c	$x^{26} - 13x^{25} - 14x^{24} + 835x^{23} - 2029x^{22} - 19837x^{21} + 84058x^{20} + 208574x^{19} - 1424220x^{18} - 660108x^{17} + 13080245x^{16} - 6154466x^{15} - 70400782x^{14} + 72403988x^{13} + 223794372x^{12} - 325236457x^{11} - 398974150x^{10} + 767406894x^{9} + 331931261x^{8} - 973707126x^{7} - 11377978x^{6} + 611460916x^{5} - 148070520x^{4} - 142348464x^{3} + 59504064x^{2} - 1724672x - 537728$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(523))$	43
Old forms	0
Cusp forms $S_{2}(\Gamma_0(523))$	43
Eisenstein series $E_{2}(\Gamma_0(523))$	1
Modular forms $M_{2}(\Gamma_0(523))$	44

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 523]: [0, 44] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (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(523, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 523:2 1000

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