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

Newform orbit 473.2.a.d

Trace form

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

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

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

$x^{5} + 4x^{4} - 9x^{3} - 22x^{2} + 50x - 23$

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
$\operatorname{tr} a_p$	−3	−1	−4	−9	5	−9	−13	−7	8	10	−22	−13	10	5	−37	11	−13	−8	−21	−3	2	−2	−11	14	−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)[3]   # 473.2.a.d

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

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