Atlas › Modular forms › Weight 4 › Level 123 › 123.4.a

Newspace 123.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(123))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
123.4.a.a	4	degree 4	$4q - 4q^{2} + 12q^{3} + 6q^{4} - 33q^{5} - 12q^{6} - 28q^{7} + O(q^{8})$	− +	$-1$
123.4.a.b	4	degree 4	$4q - 12q^{3} + 10q^{4} - 9q^{5} - 14q^{7} + O(q^{8})$	+ −	$-1$
123.4.a.c	6	degree 6	$6q + 2q^{2} - 18q^{3} + 30q^{4} + 21q^{5} - 6q^{6} + 28q^{7} + O(q^{8})$	+ +	$+1$
123.4.a.d	6	degree 6	$6q + 6q^{2} + 18q^{3} + 26q^{4} + 37q^{5} + 18q^{6} + 14q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
123.4.a.a	$x^{4} + 37x^{3} + 420x^{2} + 1462x + 268$
123.4.a.b	$x^{4} + 9x^{3} - 90x^{2} + 50x + 300$
123.4.a.c	$x^{6} - 23x^{5} - 308x^{4} + 9514x^{3} - 25280x^{2} - 353516x + 661608$
123.4.a.d	$x^{6} - 43x^{5} + 346x^{4} + 5790x^{3} - 97580x^{2} + 384092x - 33784$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(123))$	20
Old forms	20
Cusp forms $S_{4}(\Gamma_0(123))$	40
Eisenstein series $E_{4}(\Gamma_0(123))$	4
Modular forms $M_{4}(\Gamma_0(123))$	44

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(123)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.112 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, 3, 41, 123]: [1, 2, 12, 44] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 4 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(123, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 123:4 1000

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