Atlas › Modular forms › Weight 2 › Level 369 › 369.2.a

Newspace 369.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(369))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
369.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	−	369a1 rank 1
369.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	+	369b1 rank 0
369.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{5} - 4q^{7} + O(q^{8})$	−
369.2.a.d	3	degree 3	$3q - 2q^{2} + 2q^{4} - 4q^{5} + O(q^{8})$	+
369.2.a.e	3	degree 3	$3q - q^{2} + 3q^{4} - 4q^{5} + 2q^{7} + O(q^{8})$	+
369.2.a.f	3	degree 3	$3q + q^{2} + 5q^{4} + 2q^{5} + 6q^{7} + O(q^{8})$	+
369.2.a.g	3	degree 3	$3q + 2q^{2} + 2q^{4} + 4q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
369.2.a.a	$x - 6$
369.2.a.b	$x - 14$
369.2.a.c	$x^{2} + 12x + 28$
369.2.a.d	$x^{3} + 14x^{2} + 52x + 40$
369.2.a.e	$x^{3} + 13x^{2} - 28x - 332$
369.2.a.f	$x^{3} - 7x^{2} - 5x + 43$
369.2.a.g	$x^{3} - 14x^{2} + 52x - 40$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(369))$	16
Old forms	23
Cusp forms $S_{2}(\Gamma_0(369))$	39
Eisenstein series $E_{2}(\Gamma_0(369))$	7
Modular forms $M_{2}(\Gamma_0(369))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(369)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 9, 41, 123, 369]: [0, 1, 2, 4, 16, 44] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 16 = 1 x 16 (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(369, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 369:2 1000

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