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

Newform orbit 473.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + q^{2} - 3q^{3} + 3q^{4} - 6q^{5} + 3q^{6} - 15q^{7} + 8q^{9} - 10q^{10} - 5q^{11} - 27q^{12} - q^{13} - 9q^{14} - 3q^{15} + 15q^{16} + 5q^{17} - 9q^{18} - 3q^{19} - 8q^{20} + 17q^{21} - q^{22} - 4q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 2x^{4} - 19x^{3} + 8x^{2} + 10x - 1$

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 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
$\operatorname{tr} a_p$	1	−3	−6	−15	−5	−1	5	−3	−4	4	−18	−17	−8	−5	−1	11	−29	30	−29	−1	2	−24	9	−18	−30

$\operatorname{tr} a_n$ 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)[4]   # 473.2.a.e

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

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