Atlas › Modular forms › Weight 2 › Level 622 › 622.2.a

Newspace 622.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(622))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
622.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 4q^{5} + q^{7} + O(q^{8})$	− −	$-1$	622a1 rank 1
622.2.a.b	4	degree 4	$4q + 4q^{2} - 5q^{3} + 4q^{4} - 3q^{5} - 5q^{6} - 11q^{7} + O(q^{8})$	− −	$-1$
622.2.a.c	5	degree 5	$5q - 5q^{2} - q^{3} + 5q^{4} - 5q^{5} + q^{6} + O(q^{8})$	+ +	$-1$
622.2.a.d	7	degree 7	$7q + 7q^{2} + 3q^{3} + 7q^{4} + 5q^{5} + 3q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
622.2.a.e	8	degree 8	$8q - 8q^{2} + 3q^{3} + 8q^{4} + 9q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
622.2.a.a	$x + 4$
622.2.a.b	$x^{4} + 8x^{3} + 20x^{2} + 15x - 1$
622.2.a.c	$x^{5} + 6x^{4} + 2x^{3} - 25x^{2} + x + 16$
622.2.a.d	$x^{7} - 8x^{6} + 143x^{4} - 413x^{3} + 432x^{2} - 168x + 16$
622.2.a.e	$x^{8} - 12x^{7} + 18x^{6} + 227x^{5} - 713x^{4} - 386x^{3} + 2024x^{2} - 48x - 608$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(622))$	25
Old forms	52
Cusp forms $S_{2}(\Gamma_0(622))$	77
Eisenstein series $E_{2}(\Gamma_0(622))$	3
Modular forms $M_{2}(\Gamma_0(622))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(622)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(311))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 311, 622]: [0, 1, 27, 80] (certified at ell = 2147479121)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 3 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(622, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 622:2 1000

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