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

Newform orbit 973.2.a.f

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} - 22x^{11} - 32x^{10} + 3601x^{9} - 16074x^{8} - 164292x^{7} + 1323628x^{6} + 388594x^{5} - 26755995x^{4} + 77385915x^{3} - 64554803x^{2} - 12324201x + 25111279$

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 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	6	11	12	4	−2	23	12	3	−18	8	8	11	9	19	−16	39	−6	39	−13	0	0	84	24	−13

$\operatorname{tr} a_n$ 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)[5]   # 973.2.a.f

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

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