Atlas › Modular forms › Weight 6 › Level 73 › 73.6.a › 73.6.a.b

Newform orbit 73.6.a.b

Trace form

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

$16q + 15q^{2} + 37q^{3} + 287q^{4} + 121q^{5} - 58q^{6} + 392q^{7} + 948q^{8} + 1461q^{9} + 367q^{10} + 1249q^{11} + 1653q^{12} + 837q^{13} + 915q^{14} + 3654q^{15} + 5979q^{16} + 1720q^{17} + 838q^{18} + 1654q^{19} + 14370q^{20} + 9989q^{21} + 17666q^{22} + 14733q^{23} + 19479q^{24} + O(q^{25})$

Coefficient field

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

$x^{16} - 52x^{15} - 1664x^{14} + 115686x^{13} + 740777x^{12} - 101034638x^{11} + 128065667x^{10} + 45024704978x^{9} - 197074281950x^{8} - 11049054393398x^{7} + 61820829381933x^{6} + 1477225985293620x^{5} - 8576138184606820x^{4} - 96921398753645940x^{3} + 520354175701890264x^{2} + 2249590278177577296x - 10051589695790698464$

Atkin–Lehner signs

$p$	$w_p$
73	$-1$

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
$\operatorname{tr} a_p$	15	37	121	392	1249	837	1720	1654	14733	5918	5300	−13760	16636	−1120	48280	27238	17678	−34519	6784	214807	85264	46665	236109	180344	200417

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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 engine's checks

dim M^+ at levels [1, 73]: [1, 32] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 6 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(73, 6)[1]   # 73.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 73:6 1000