Atlas › Modular forms › Weight 4 › Level 205 › 205.4.a

Newspace 205.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(205))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
205.4.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - 7q^{4} + 5q^{5} - 2q^{6} + 8q^{7} + O(q^{8})$	− +	$-1$
205.4.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - 7q^{4} - 5q^{5} + 2q^{6} + 26q^{7} + O(q^{8})$	+ −	$-1$
205.4.a.c	8	degree 8	$8q - 7q^{2} - 22q^{3} + 31q^{4} + 40q^{5} - 9q^{6} - 60q^{7} + O(q^{8})$	− +	$-1$
205.4.a.d	8	degree 8	$8q + q^{2} - 12q^{3} + 35q^{4} - 40q^{5} - 33q^{6} - 8q^{7} + O(q^{8})$	+ −	$-1$
205.4.a.e	11	degree 11	$11q + 4q^{2} + 22q^{3} + 48q^{4} + 55q^{5} + 25q^{6} + 74q^{7} + O(q^{8})$	− −	$+1$
205.4.a.f	11	degree 11	$11q + 6q^{2} + 8q^{3} + 52q^{4} - 55q^{5} + 5q^{6} - 52q^{7} + O(q^{8})$	+ +	$+1$

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 $T$
205.4.a.a	$x - 1$
205.4.a.b	$x - 3$
205.4.a.c	$x^{8} + 29x^{7} + 236x^{6} - 121x^{5} - 7192x^{4} - 13621x^{3} + 2136x^{2} + 1361x - 109$
205.4.a.d	$x^{8} + 11x^{7} - 102x^{6} - 1111x^{5} + 3484x^{4} + 32189x^{3} - 53306x^{2} - 279665x + 358275$
205.4.a.e	$x^{11} - 26x^{10} + 25x^{9} + 4075x^{8} - 24063x^{7} - 179595x^{6} + 1646263x^{5} + 1210265x^{4} - 33274754x^{3} + 29958573x^{2} + 210852720x - 294644300$
205.4.a.f	$x^{11} - 14x^{10} - 197x^{9} + 2643x^{8} + 14873x^{7} - 160333x^{6} - 506631x^{5} + 3377597x^{4} + 4923938x^{3} - 22248105x^{2} - 4077808x + 312372$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(205))$	40
Old forms	22
Cusp forms $S_{4}(\Gamma_0(205))$	62
Eisenstein series $E_{4}(\Gamma_0(205))$	4
Modular forms $M_{4}(\Gamma_0(205))$	66

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(205)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 5, 41, 205]: [1, 3, 12, 66] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (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(205, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 205:4 1000

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