Atlas › Modular forms › Weight 2 › Level 973 › 973.2.a › 973.2.a.g

Newform orbit 973.2.a.g

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 7q^{2} - 12q^{3} + 17q^{4} - 19q^{5} + q^{6} + 15q^{7} - 18q^{8} + 17q^{9} + 5q^{10} - 18q^{11} - 14q^{12} - 12q^{13} - 7q^{14} + 4q^{15} + 13q^{16} - 18q^{17} - 15q^{18} - 33q^{20} - 12q^{21} + 7q^{22} - 14q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} + 43x^{14} + 619x^{13} + 1486x^{12} - 46951x^{11} - 416305x^{10} + 63935x^{9} + 14843174x^{8} + 52348747x^{7} - 101926834x^{6} - 916268921x^{5} - 1402130768x^{4} + 886192332x^{3} + 2770627633x^{2} + 420505622x - 334131157$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
139	$-1$

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
$\operatorname{tr} a_p$	−7	−12	−19	15	−18	−12	−18	0	−14	5	−5	5	−37	−14	−46	−14	−47	−13	−25	−19	−18	14	−87	−18	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 0.39 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, 7, 139, 973]: [0, 1, 12, 94] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 69 = 1 x 69 (phi(ord eps) x dim S_k^new); 9 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(973, 2)[6]   # 973.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 973:2 1000