Atlas › Modular forms › Weight 2 › Level 473 › 473.2.a › 473.2.a.f

Newform orbit 473.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 9x^{8} + 16x^{7} + 43x^{6} - 93x^{5} - 117x^{4} + 125x^{3} + 189x^{2} + 64x + 4$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
43	$+1$

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	5	0	19	9	11	3	17	0	18	14	15	4	−9	−9	−13	−13	16	3	1	32	4	31	−24	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 11, 43, 473]: [0, 2, 4, 46] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (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(473, 2)[5]   # 473.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 473:2 1000