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

Newform orbit 73.4.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 12q^{2} - 6q^{3} + 22q^{4} - 32q^{5} + 14q^{6} - 22q^{7} - 87q^{8} + 29q^{9} - 67q^{10} - 152q^{11} - 211q^{12} - 46q^{13} + 33q^{14} - 252q^{15} + 122q^{16} - 278q^{17} - 401q^{18} + 82q^{19} + 156q^{20} - 264q^{21} + 532q^{22} - 420q^{23} + 927q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 18x^{6} - 713x^{4} - 1303x^{3} + 5484x^{2} + 10582x - 2405$

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 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
$\operatorname{tr} a_p$	−12	−6	−32	−22	−152	−46	−278	82	−420	−334	54	174	−338	180	−728	−464	−1354	−312	−1562	−2462	511	1038	−1626	878	−544

$\operatorname{tr} a_n$ 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)[1]   # 73.4.a.b

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

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