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

Newform orbit 973.2.a.i

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{16} + 30x^{15} + 164x^{14} - 2971x^{13} - 31409x^{12} + 74952x^{11} + 1616124x^{10} + 153607x^{9} - 39051503x^{8} - 24734705x^{7} + 492853475x^{6} + 212562575x^{5} - 3003374010x^{4} + 27359165x^{3} + 5910988649x^{2} - 816857081x - 74824127$

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
$\operatorname{tr} a_p$	−9	−7	5	−16	−18	−4	−11	−11	−20	−24	3	−14	−1	−18	−34	−34	−13	12	−38	−50	10	−30	−79	14	−11

$\operatorname{tr} a_n$ 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)[8]   # 973.2.a.i

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

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