Atlas › Modular forms › Weight 2 › Level 334 › 334.2.a

Newspace 334.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(334))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
334.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} + q^{7} + O(q^{8})$	− +	$+1$	334a1 rank 0
334.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - q^{3} + 2q^{4} - 2q^{5} + q^{6} + O(q^{8})$	+ +	$-1$
334.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} - 6q^{7} + O(q^{8})$	+ −	$+1$
334.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 4q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
334.2.a.e	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} + O(q^{8})$	+ −	$+1$
334.2.a.f	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} - 3q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
334.2.a.a	$x - 14$
334.2.a.b	$x^{2} + 42x + 121$
334.2.a.c	$x^{2} + 24x - 194$
334.2.a.d	$x^{2} + 108x + 2911$
334.2.a.e	$x^{3} - 76x^{2} + 1483x - 5444$
334.2.a.f	$x^{3} - 56x^{2} - 47x + 26582$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(334))$	13
Old forms	28
Cusp forms $S_{2}(\Gamma_0(334))$	41
Eisenstein series $E_{2}(\Gamma_0(334))$	3
Modular forms $M_{2}(\Gamma_0(334))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(334)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(167))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 167, 334]: [0, 1, 15, 44] (certified at ell = 2147483237)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 13 = 1 x 13 (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(334, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 334:2 1000

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