Atlas › Modular forms › Weight 2 › Level 290 › 290.2.a

Newspace 290.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(290))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
290.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$	290a1 rank 1
290.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + q^{3} + 2q^{4} - 2q^{5} - q^{6} + 5q^{7} + O(q^{8})$	+ + −	$+1$
290.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + q^{3} + 2q^{4} + 2q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
290.2.a.d	3	degree 3	$3q + 3q^{2} - q^{3} + 3q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− − −	$+1$
290.2.a.e	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
290.2.a.a	$x + 38$
290.2.a.b	$x^{2} - 69x + 459$
290.2.a.c	$x^{2} - 13x - 507$
290.2.a.d	$x^{3} - 41x^{2} + 21x + 9482$
290.2.a.e	$x^{3} - 33x^{2} - 2265x + 1038$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(290))$	11
Old forms	30
Cusp forms $S_{2}(\Gamma_0(290))$	41
Eisenstein series $E_{2}(\Gamma_0(290))$	7
Modular forms $M_{2}(\Gamma_0(290))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(290)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(145))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.21 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, 2, 5, 10, 29, 58, 145, 290]: [0, 1, 1, 3, 3, 9, 16, 48] (certified at ell = 2147483549)
T = [(3, 1), (7, 3), (11, 5), (13, 7)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 3 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(290, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 290:2 1000

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