Atlas › Modular forms › Weight 2 › Level 303 › 303.2.a

Newspace 303.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(303))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
303.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	303b1 rank 1
303.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 3q^{5} + O(q^{8})$	− −	$-1$	303a1 rank 1
303.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	+ +	$-1$
303.2.a.d	6	degree 6	$6q + q^{2} + 6q^{3} + 3q^{4} + 6q^{5} + q^{6} + O(q^{8})$	− +	$+1$
303.2.a.e	7	degree 7	$7q - 7q^{3} + 10q^{4} + 6q^{5} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
303.2.a.a	$x + 5$
303.2.a.b	$x + 9$
303.2.a.c	$x^{2} + 6x + 1$
303.2.a.d	$x^{6} - 19x^{5} + 32x^{4} + 962x^{3} - 3944x^{2} - 409x + 11241$
303.2.a.e	$x^{7} - 18x^{6} - 171x^{5} + 4302x^{4} - 5210x^{3} - 231371x^{2} + 1102212x - 738193$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(303))$	17
Old forms	16
Cusp forms $S_{2}(\Gamma_0(303))$	33
Eisenstein series $E_{2}(\Gamma_0(303))$	3
Modular forms $M_{2}(\Gamma_0(303))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(303)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 101, 303]: [0, 1, 9, 36] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(303, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 303:2 1000

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