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

Newform orbit 447.2.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} - 12x^{8} + 32x^{7} + 115x^{6} - 773x^{5} + 1443x^{4} - 729x^{3} - 692x^{2} + 737x - 121$

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
$\operatorname{tr} a_p$	4	9	8	1	13	−2	10	−11	11	0	−13	1	17	−8	10	2	25	−13	1	13	5	−22	1	23	16

$\operatorname{tr} a_n$ 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)[2]   # 447.2.a.c

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

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