Atlas › Modular forms › Weight 4 › Level 73 › 73.4.a › 73.4.a.c

Newform orbit 73.4.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 7q^{2} + 10q^{3} + 47q^{4} + 24q^{5} + 14q^{6} + 42q^{7} + 84q^{8} + 156q^{9} + 11q^{10} + 74q^{11} + 21q^{12} - 2q^{13} + 155q^{14} + 120q^{15} + 275q^{16} + 84q^{17} + 142q^{18} + 2q^{19} + 20q^{20} - 260q^{21} - 570q^{22} + 636q^{23} - 1165q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 17x^{9} - 146x^{8} + 3411x^{7} + 2032x^{6} - 211843x^{5} + 354206x^{4} + 4367265x^{3} - 11777829x^{2} - 11176992x + 26451576$

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
$\operatorname{tr} a_p$	7	10	24	42	74	−2	84	2	636	−150	18	56	412	−182	1528	1300	1136	−394	958	3378	−730	1502	1584	−1580	−642

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.097 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, 20] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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, 4)[2]   # 73.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 73:4 1000