Atlas › Modular forms › Weight 2 › Level 447 › 447.2.a › 447.2.a.d

Newform orbit 447.2.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} - 7x^{9} - 42x^{8} + 353x^{7} + 362x^{6} - 5134x^{5} - 170x^{4} + 28405x^{3} - 3617x^{2} - 48254x + 3115$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
149	$-1$

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 even 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	−10	4	9	−7	4	10	9	9	8	15	31	−5	8	2	4	−59	1	5	−11	51	−6	−21	−5	32

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 3, 149, 447]: [0, 1, 13, 52] (certified at ell = 2147481629)
T = [(2, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(447, 2)[3]   # 447.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 447:2 1000