Atlas › Modular forms › Weight 2 › Level 620 › 620.2.a

Newspace 620.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(620))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
620.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} - 2q^{7} + O(q^{8})$	− +	620c1 rank 1
620.2.a.b	1	$\mathbb{Q}$	$q + q^{5} - 2q^{7} + O(q^{8})$	− +	620b1 rank 1
620.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 4q^{7} + O(q^{8})$	+ −	620a1 rank 1
620.2.a.d	3	degree 3	$3q + 3q^{3} - 3q^{5} + O(q^{8})$	+ +
620.2.a.e	4	degree 4	$4q + 3q^{3} + 4q^{5} + 8q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
620.2.a.a	$x + 9$
620.2.a.b	$x + 6$
620.2.a.c	$x + 11$
620.2.a.d	$x^{3} - 3x^{2} - 129x - 11$
620.2.a.e	$x^{4} - 27x^{3} + 151x^{2} + 443x - 502$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(620))$	10
Old forms	81
Cusp forms $S_{2}(\Gamma_0(620))$	91
Eisenstein series $E_{2}(\Gamma_0(620))$	11
Modular forms $M_{2}(\Gamma_0(620))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(620)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(155))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(310))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.358 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, 4, 5, 10, 20, 31, 62, 124, 155, 310, 620]: [0, 1, 2, 1, 3, 6, 3, 10, 19, 18, 52, 102] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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(620, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 620:2 1000

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