Atlas › Modular forms › Weight 2 › Level 314 › 314.2.a

Newspace 314.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(314))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
314.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{7} + O(q^{8})$	+ +	$-1$	314a1 rank 1
314.2.a.b	6	degree 6	$6q - 6q^{2} + 3q^{3} + 6q^{4} + q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
314.2.a.c	7	degree 7	$7q + 7q^{2} - q^{3} + 7q^{4} + 3q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

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 $T$
314.2.a.a	$x$
314.2.a.b	$x^{6} - 4x^{5} - 39x^{4} + 184x^{3} + 104x^{2} - 710x - 607$
314.2.a.c	$x^{7} - 2x^{6} - 37x^{5} + 146x^{4} - 136x^{3} - 30x^{2} + 47x + 12$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(314))$	14
Old forms	24
Cusp forms $S_{2}(\Gamma_0(314))$	38
Eisenstein series $E_{2}(\Gamma_0(314))$	3
Modular forms $M_{2}(\Gamma_0(314))$	41

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(314)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(157))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.183 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, 157, 314]: [0, 1, 13, 41] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(314, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 314:2 1000

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