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

Newform orbit 449.2.a.a

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 3q^{2} - 5q^{3} + 7q^{4} - 7q^{5} - 12q^{6} - 22q^{7} - 6q^{8} + 5q^{9} - 15q^{10} - 14q^{11} - 7q^{12} - 10q^{13} - 2q^{14} - 18q^{15} - 7q^{16} - 6q^{17} + q^{18} - 39q^{19} - 5q^{20} - 7q^{21} - q^{22} - 6q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 8x^{13} + 3x^{12} - 103x^{11} - 150x^{10} + 399x^{9} + 595x^{8} - 771x^{7} - 714x^{6} + 796x^{5} + 174x^{4} - 247x^{3} - 2x^{2} + 18x + 1$

Atkin–Lehner signs

$p$	$w_p$
449	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−3	−5	−7	−22	−14	−10	−6	−39	−6	−3	−57	−4	−14	−10	1	20	−15	−40	5	−10	−3	−58	22	−8	−18

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

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)[0]   # 449.2.a.a

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

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