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

Newform orbit 73.6.a.a

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 17q^{2} - 35q^{3} + 191q^{4} - 129q^{5} - 202q^{6} - 294q^{7} - 588q^{8} + 651q^{9} - 233q^{10} - 1171q^{11} - 1227q^{12} - 177q^{13} - 2221q^{14} - 2646q^{15} + 2395q^{16} - 2904q^{17} - 6938q^{18} - 7350q^{19} - 16516q^{20} - 7013q^{21} - 16970q^{22} - 16059q^{23} - 21181q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 52x^{13} - 792x^{12} - 63814x^{11} + 206629x^{10} + 29295414x^{9} - 37695889x^{8} - 5848943222x^{7} + 6163884278x^{6} + 490288333814x^{5} - 348558593935x^{4} - 16667483243784x^{3} + 3314366677392x^{2} + 184128304907988x + 15823968485868$

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
$\operatorname{tr} a_p$	−17	−35	−129	−294	−1171	−177	−2904	−7350	−16059	−7410	−9854	−18516	−80	29070	−64454	−67286	15692	7009	−56256	−105609	−74606	−58407	−260115	−349532	−99315

$\operatorname{tr} a_n$ 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)[0]   # 73.6.a.a

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

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