Atlas › Modular forms › Weight 2 › Level 449 › 449.2.a

Newspace 449.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(449))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
449.2.a.a	14	degree 14	$14q - 3q^{2} - 5q^{3} + 7q^{4} - 7q^{5} - 12q^{6} - 22q^{7} + O(q^{8})$	+	$-1$
449.2.a.b	23	degree 23	$23q + 3q^{3} + 30q^{4} + 3q^{5} + 6q^{6} + 26q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
449.2.a.a	$x^{14} + 8x^{13} + 3x^{12} - 103x^{11} - 150x^{10} + 399x^{9} + 595x^{8} - 771x^{7} - 714x^{6} + 796x^{5} + 174x^{4} - 247x^{3} - 2x^{2} + 18x + 1$
449.2.a.b	$x^{23} - 3x^{22} - 85x^{21} + 266x^{20} + 2860x^{19} - 9102x^{18} - 50365x^{17} + 158186x^{16} + 527071x^{15} - 1580022x^{14} - 3483483x^{13} + 9616835x^{12} + 14803993x^{11} - 36439399x^{10} - 39689209x^{9} + 85376416x^{8} + 62919746x^{7} - 118886914x^{6} - 50446161x^{5} + 88782974x^{4} + 11218709x^{3} - 26594058x^{2} + 4210915x + 22925$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(449))$	37
Old forms	0
Cusp forms $S_{2}(\Gamma_0(449))$	37
Eisenstein series $E_{2}(\Gamma_0(449))$	1
Modular forms $M_{2}(\Gamma_0(449))$	38

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 449]: [0, 38] (certified at ell = 2147479937)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(449, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 449:2 1000

← 448.2.a · 450.2.a →