Atlas › Modular forms › Weight 4 › Level 206 › 206.4.a

Newspace 206.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(206))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
206.4.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q - 4q^{2} + 8q^{4} - 14q^{5} - q^{7} + O(q^{8})$	+ −	$-1$
206.4.a.b	2	$\mathbb{Q}(\sqrt{101})$	$2q - 4q^{2} + 2q^{3} + 8q^{4} + 15q^{5} - 4q^{6} - 20q^{7} + O(q^{8})$	+ −	$-1$
206.4.a.c	4	degree 4	$4q + 8q^{2} - 12q^{3} + 16q^{4} - 13q^{5} - 24q^{6} - 25q^{7} + O(q^{8})$	− +	$-1$
206.4.a.d	9	degree 9	$9q - 18q^{2} + 2q^{3} + 36q^{4} + q^{5} - 4q^{6} - 7q^{7} + O(q^{8})$	+ +	$+1$
206.4.a.e	9	degree 9	$9q + 18q^{2} + 6q^{3} + 36q^{4} + 17q^{5} + 12q^{6} + 45q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
206.4.a.a	$x^{2} + 14x + 36$
206.4.a.b	$x^{2} - 17x + 47$
206.4.a.c	$x^{4} + 25x^{3} + 161x^{2} - 164x - 2672$
206.4.a.d	$x^{9} - 3x^{8} - 1035x^{7} + 1338x^{6} + 361209x^{5} - 68333x^{4} - 51686051x^{3} - 46387010x^{2} + 2598245416x + 5748043488$
206.4.a.e	$x^{9} - 23x^{8} - 947x^{7} + 27404x^{6} + 92777x^{5} - 7551817x^{4} + 59423649x^{3} - 35519796x^{2} - 695069064x + 516713160$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(206))$	26
Old forms	50
Cusp forms $S_{4}(\Gamma_0(206))$	76
Eisenstein series $E_{4}(\Gamma_0(206))$	4
Modular forms $M_{4}(\Gamma_0(206))$	80

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(206)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.14 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, 103, 206]: [1, 2, 27, 80] (certified at ell = 2147482501)
T = [(3, 1), (5, 1)]; h of degree 26 = 1 x 26 (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(206, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 206:4 1000

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