Atlas › Modular forms › Weight 2 › Level 394 › 394.2.a

Newspace 394.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(394))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
394.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 5q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
394.2.a.b	2	$\mathbb{Q}(\sqrt{21})$	$2q + 2q^{2} - q^{3} + 2q^{4} - q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
394.2.a.c	2	$\mathbb{Q}(\sqrt{29})$	$2q + 2q^{2} + 2q^{4} + 3q^{5} + 4q^{7} + O(q^{8})$	− +	$+1$
394.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{4} + 5q^{5} - 6q^{7} + O(q^{8})$	− +	$+1$
394.2.a.e	4	degree 4	$4q - 4q^{2} - 3q^{3} + 4q^{4} - 5q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$
394.2.a.f	4	degree 4	$4q - 4q^{2} + 2q^{3} + 4q^{4} + 2q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
394.2.a.a	$x^{2} + 7x + 11$
394.2.a.b	$x^{2} + x - 5$
394.2.a.c	$x^{2} - 3x - 5$
394.2.a.d	$x^{2} - 5x + 5$
394.2.a.e	$x^{4} + 8x^{3} + 15x^{2} + 6x - 1$
394.2.a.f	$x^{4} - 4x^{3} - 11x^{2} + 64x - 67$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(394))$	16
Old forms	32
Cusp forms $S_{2}(\Gamma_0(394))$	48
Eisenstein series $E_{2}(\Gamma_0(394))$	3
Modular forms $M_{2}(\Gamma_0(394))$	51

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(394)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(197))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 197, 394]: [0, 1, 17, 51] (certified at ell = 2147479489)
T = [(3, 1), (5, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 2 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(394, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 394:2 1000

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