Atlas › Modular forms › Weight 2 › Level 626 › 626.2.a

Newspace 626.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(626))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
626.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + O(q^{8})$	+ +	$-1$	626a1 rank 1
626.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$	626b1 rank 0
626.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{4} - 5q^{7} + O(q^{8})$	+ +	$-1$
626.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
626.2.a.e	9	degree 9	$9q - 9q^{2} + q^{3} + 9q^{4} - 2q^{5} - q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
626.2.a.f	12	degree 12	$12q + 12q^{2} + 4q^{3} + 12q^{4} + 6q^{5} + 4q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
626.2.a.a	$x$
626.2.a.b	$x - 25$
626.2.a.c	$x^{2} + 60x + 895$
626.2.a.d	$x^{2} + 58x + 521$
626.2.a.e	$x^{9} - 67x^{8} - 2712x^{7} + 210478x^{6} + 2542307x^{5} - 213376795x^{4} - 1339954117x^{3} + 69456620819x^{2} + 552326684881x - 987872773531$
626.2.a.f	$x^{12} - 110x^{11} + 2155x^{10} + 137382x^{9} - 5288339x^{8} - 17553804x^{7} + 2525816784x^{6} - 14526186982x^{5} - 313368985412x^{4} + 1904004246270x^{3} + 10848335691769x^{2} - 52149770837876x + 50484818112634$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(626))$	27
Old forms	50
Cusp forms $S_{2}(\Gamma_0(626))$	77
Eisenstein series $E_{2}(\Gamma_0(626))$	3
Modular forms $M_{2}(\Gamma_0(626))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(626)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(313))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.35 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, 313, 626]: [0, 1, 26, 80] (certified at ell = 2147482273)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 27 = 1 x 27 (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(626, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 626:2 1000

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