Atlas › Modular forms › Weight 8 › Level 49 › 49.8.a

Newspace 49.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(49))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
49.8.a.a	1	$\mathbb{Q}$	$q - 13q^{2} + 41q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$
49.8.a.b	1	$\mathbb{Q}$	$q - 6q^{2} + 42q^{3} - 92q^{4} + 84q^{5} - 252q^{6} + O(q^{8})$
49.8.a.c	2	$\mathbb{Q}(\sqrt{865})$	$2q - 3q^{2} - 94q^{3} + 181q^{4} - 330q^{5} + 1006q^{6} + O(q^{8})$
49.8.a.d	2	$\mathbb{Q}(\sqrt{690})$	$2q + 20q^{2} - 56q^{4} + O(q^{8})$
49.8.a.e	4	degree 4	$4q - 6q^{2} - 28q^{3} + 348q^{4} - 252q^{5} - 1022q^{6} + O(q^{8})$
49.8.a.f	4	degree 4	$4q - 6q^{2} + 28q^{3} + 348q^{4} + 252q^{5} + 1022q^{6} + O(q^{8})$
49.8.a.g	8	degree 8	$8q + 30q^{2} + 458q^{4} + O(q^{8})$

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 $T$
49.8.a.a	$x + 13$
49.8.a.b	$x - 36$
49.8.a.c	$x^{2} + 97x + 406$
49.8.a.d	$x^{2} - 20x - 2660$
49.8.a.e	$x^{4} + 34x^{3} - 4208x^{2} - 4290x + 1786575$
49.8.a.f	$x^{4} - 22x^{3} - 6588x^{2} + 96342x + 9600507$
49.8.a.g	$x^{8} - 30x^{7} - 12839x^{6} + 79720x^{5} + 43881102x^{4} + 854201920x^{3} - 16274303296x^{2} - 526621670400x - 3184201775104$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(49))$	22
Old forms	6
Cusp forms $S_{8}(\Gamma_0(49))$	28
Eisenstein series $E_{8}(\Gamma_0(49))$	8
Modular forms $M_{8}(\Gamma_0(49))$	36

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(49)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 33] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(49, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 49:8 1000

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