Atlas › Modular forms › Weight 2 › Level 366 › 366.2.a

Newspace 366.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(366))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
366.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ + −	$+1$	366e1 rank 0
366.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$	366f1 rank 1
366.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − +	$+1$	366c1 rank 0
366.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	366g1 rank 1
366.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$	366d1 rank 0
366.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$	366b1 rank 0
366.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	366a1 rank 0
366.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
366.2.a.a	$x + 24$
366.2.a.b	$x + 28$
366.2.a.c	$x - 25$
366.2.a.d	$x + 52$
366.2.a.e	$x - 43$
366.2.a.f	$x - 33$
366.2.a.g	$x + 36$
366.2.a.h	$x^{2} - 55x + 548$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(366))$	9
Old forms	50
Cusp forms $S_{2}(\Gamma_0(366))$	59
Eisenstein series $E_{2}(\Gamma_0(366))$	7
Modular forms $M_{2}(\Gamma_0(366))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(366)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(183))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 3, 6, 61, 122, 183, 366]: [0, 1, 1, 3, 5, 17, 22, 66] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 9 = 1 x 9 (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(366, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 366:2 1000

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