Atlas › Modular forms › Weight 2 › Level 623 › 623.2.a

Newspace 623.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(623))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
623.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	− −	$-1$	623a1 rank 1
623.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +	$-1$
623.2.a.c	4	degree 4	$4q + q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
623.2.a.d	5	degree 5	$5q - 2q^{2} - 3q^{3} + 2q^{4} - 5q^{5} - 5q^{6} + 5q^{7} + O(q^{8})$	− −	$-1$
623.2.a.e	16	degree 16	$16q + 6q^{3} + 16q^{4} + 6q^{5} - 2q^{6} + 16q^{7} + O(q^{8})$	− +	$+1$
623.2.a.f	17	degree 17	$17q + 3q^{2} + 4q^{3} + 23q^{4} + 6q^{5} - 2q^{6} - 17q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
623.2.a.a	$x$
623.2.a.b	$x^{2} + 2x - 4$
623.2.a.c	$x^{4} + x^{3} - 10x^{2} + 8x - 1$
623.2.a.d	$x^{5} + 5x^{4} + 2x^{3} - 18x^{2} - 25x - 8$
623.2.a.e	$x^{16} - 6x^{15} - 39x^{14} + 272x^{13} + 424x^{12} - 4262x^{11} - 494x^{10} + 28255x^{9} - 7783x^{8} - 91464x^{7} + 23586x^{6} + 146128x^{5} - 3955x^{4} - 95881x^{3} - 25789x^{2} + 3954x + 1268$
623.2.a.f	$x^{17} - 7x^{16} - 43x^{15} + 357x^{14} + 600x^{13} - 6714x^{12} - 3362x^{11} + 61015x^{10} + 10850x^{9} - 295347x^{8} - 50500x^{7} + 756222x^{6} + 205519x^{5} - 908000x^{4} - 314834x^{3} + 369917x^{2} + 88888x - 29932$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(623))$	45
Old forms	14
Cusp forms $S_{2}(\Gamma_0(623))$	59
Eisenstein series $E_{2}(\Gamma_0(623))$	3
Modular forms $M_{2}(\Gamma_0(623))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(623)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 7, 89, 623]: [0, 1, 8, 62] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(623, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 623:2 1000

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