Atlas › Modular forms › Weight 4 › Level 49 › 49.4.a

Newspace 49.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(49))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
49.4.a.a	1	$\mathbb{Q}$	$q - 5q^{2} + 17q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$
49.4.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - 7q^{4} - 16q^{5} - 2q^{6} + O(q^{8})$
49.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - 7q^{3} - 4q^{4} - 7q^{5} - 14q^{6} + O(q^{8})$
49.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 7q^{3} - 4q^{4} + 7q^{5} + 14q^{6} + O(q^{8})$
49.4.a.e	4	degree 4	$4q + 2q^{2} + 34q^{4} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
49.4.a.a	$x + 5$
49.4.a.b	$x - 5$
49.4.a.c	$x + 19$
49.4.a.d	$x - 23$
49.4.a.e	$x^{4} - 2x^{3} - 697x^{2} + 4208x + 51016$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(49))$	8
Old forms	2
Cusp forms $S_{4}(\Gamma_0(49))$	10
Eisenstein series $E_{4}(\Gamma_0(49))$	8
Modular forms $M_{4}(\Gamma_0(49))$	18

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(49)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 49]: [1, 3, 15] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 8 = 1 x 8 (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(49, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 49:4 1000

← 48.4.a · 50.4.a →