Atlas › Modular forms › Weight 2 › Level 403 › 403.2.a

Newspace 403.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(403))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
403.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - 4q^{3} + 3q^{4} - 6q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
403.2.a.b	6	degree 6	$6q - 2q^{2} - 5q^{3} + 6q^{4} - 9q^{5} + O(q^{8})$	− −	$-1$
403.2.a.c	7	degree 7	$7q + 2q^{2} + 5q^{3} + 8q^{4} + 11q^{5} + 6q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
403.2.a.d	8	degree 8	$8q - 5q^{2} - 3q^{3} + 9q^{4} - 15q^{5} - 4q^{7} + O(q^{8})$	+ +	$-1$
403.2.a.e	8	degree 8	$8q - q^{2} + 7q^{3} + 7q^{4} + 11q^{5} - 2q^{7} + O(q^{8})$	− +	$+1$

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 $T$
403.2.a.a	$x^{2} + 9x + 19$
403.2.a.b	$x^{6} + 17x^{5} + 59x^{4} - 274x^{3} - 1686x^{2} - 1895x - 519$
403.2.a.c	$x^{7} - 17x^{6} + 3x^{5} + 1220x^{4} - 5932x^{3} + 3599x^{2} + 11719x - 6764$
403.2.a.d	$x^{8} + 14x^{7} - 63x^{6} - 1092x^{5} + 405x^{4} + 17913x^{3} + 24750x^{2} - 10186x - 5945$
403.2.a.e	$x^{8} - 20x^{7} + 40x^{6} + 1463x^{5} - 9098x^{4} - 11319x^{3} + 206232x^{2} - 454413x + 215864$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(403))$	31
Old forms	4
Cusp forms $S_{2}(\Gamma_0(403))$	35
Eisenstein series $E_{2}(\Gamma_0(403))$	3
Modular forms $M_{2}(\Gamma_0(403))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(403)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 31, 403]: [0, 1, 3, 38] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 31 = 1 x 31 (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(403, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 403:2 1000

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