Atlas › Modular forms › Weight 2 › Level 319 › 319.2.a

Newspace 319.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(319))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
319.2.a.a	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 2q^{4} + q^{5} - 6q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	319a1 rank 0
319.2.a.b	3	degree 3	$3q - 6q^{5} - 6q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
319.2.a.c	4	degree 4	$4q - 2q^{2} - 3q^{3} + 2q^{4} - 5q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
319.2.a.d	7	degree 7	$7q + 3q^{2} + 3q^{4} + 4q^{5} + q^{7} + O(q^{8})$	− +	$+1$
319.2.a.e	8	degree 8	$8q + 4q^{3} + 10q^{4} + 10q^{5} + 14q^{6} - 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
319.2.a.a	$x + 7$
319.2.a.b	$x^{3} - 12x + 8$
319.2.a.c	$x^{4} + 11x^{3} - 228x - 472$
319.2.a.d	$x^{7} - 3x^{6} - 157x^{5} + 363x^{4} + 6373x^{3} - 5846x^{2} - 72612x - 75800$
319.2.a.e	$x^{8} - 12x^{7} - 154x^{6} + 2372x^{5} + 722x^{4} - 105053x^{3} + 301902x^{2} + 648040x - 2564168$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(319))$	23
Old forms	6
Cusp forms $S_{2}(\Gamma_0(319))$	29
Eisenstein series $E_{2}(\Gamma_0(319))$	3
Modular forms $M_{2}(\Gamma_0(319))$	32

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(319)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

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, 11, 29, 319]: [0, 2, 3, 32] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (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(319, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 319:2 1000

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