Atlas › Modular forms › Weight 2 › Level 309 › 309.2.a

Newspace 309.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(309))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
309.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	309a1 rank 1
309.2.a.b	3	degree 3	$3q + q^{2} - 3q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
309.2.a.c	5	degree 5	$5q - 2q^{2} - 5q^{3} + 2q^{4} - 5q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
309.2.a.d	8	degree 8	$8q - q^{2} + 8q^{3} + 11q^{4} - q^{5} - q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
309.2.a.a	$x + 2$
309.2.a.b	$x^{3} - 2x^{2} - 12x + 8$
309.2.a.c	$x^{5} + 7x^{4} + x^{3} - 67x^{2} - 117x - 36$
309.2.a.d	$x^{8} + 2x^{7} - 42x^{6} - 8x^{5} + 430x^{4} - 271x^{3} - 948x^{2} + 688x - 64$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(309))$	17
Old forms	16
Cusp forms $S_{2}(\Gamma_0(309))$	33
Eisenstein series $E_{2}(\Gamma_0(309))$	3
Modular forms $M_{2}(\Gamma_0(309))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(309)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 3, 103, 309]: [0, 1, 9, 36] (certified at ell = 2147482501)
T = [(2, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(309, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 309:2 1000

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