Atlas › Modular forms › Weight 6 › Level 49 › 49.6.a

Newspace 49.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(49))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
49.6.a.a	1	$\mathbb{Q}$	$q - 10q^{2} + 14q^{3} + 68q^{4} + 56q^{5} - 140q^{6} + O(q^{8})$
49.6.a.b	1	$\mathbb{Q}$	$q + 11q^{2} + 89q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$
49.6.a.c	2	$\mathbb{Q}(\sqrt{39})$	$2q - 4q^{2} - 56q^{4} + O(q^{8})$
49.6.a.d	2	$\mathbb{Q}(\sqrt{37})$	$2q + 2q^{2} - 8q^{3} + 12q^{4} - 38q^{5} - 82q^{6} + O(q^{8})$
49.6.a.e	2	$\mathbb{Q}(\sqrt{37})$	$2q + 2q^{2} + 8q^{3} + 12q^{4} + 38q^{5} + 82q^{6} + O(q^{8})$
49.6.a.f	2	$\mathbb{Q}(\sqrt{57})$	$2q + 9q^{2} + 6q^{3} + 5q^{4} + 18q^{5} + 198q^{6} + O(q^{8})$
49.6.a.g	4	degree 4	$4q - 10q^{2} + 10q^{4} + O(q^{8})$

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 $T$
49.6.a.a	$x - 32$
49.6.a.b	$x - 11$
49.6.a.c	$x^{2} + 4x - 5612$
49.6.a.d	$x^{2} + 22x - 27$
49.6.a.e	$x^{2} - 26x - 423$
49.6.a.f	$x^{2} - 27x - 4962$
49.6.a.g	$x^{4} + 10x^{3} - 5383x^{2} + 21776x + 1858462$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(49))$	14
Old forms	6
Cusp forms $S_{6}(\Gamma_0(49))$	20
Eisenstein series $E_{6}(\Gamma_0(49))$	8
Modular forms $M_{6}(\Gamma_0(49))$	28

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(49)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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, 5, 25] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 14 = 1 x 14 (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(49, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 49:6 1000

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