Atlas › Modular forms › Weight 10 › Level 39 › 39.10.a › 39.10.a.c

Newform orbit 39.10.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 52q^{2} - 405q^{3} + 1646q^{4} + 2624q^{5} - 4212q^{6} - 4802q^{7} + 18330q^{8} + 32805q^{9} + 30072q^{10} + 2464q^{11} - 133326q^{12} + 142805q^{13} - 324184q^{14} - 212544q^{15} + 916034q^{16} + 913898q^{17} + 341172q^{18} + 420286q^{19} + 5753980q^{20} + 388962q^{21} + 2457672q^{22} + 2639208q^{23} - 1484730q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{5} - 2676x^{4} - 4842903x^{3} + 15269148882x^{2} - 6763232044440x + 156120546332736$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
13	$-1$

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
$\operatorname{tr} a_p$	52	−405	2624	−4802	2464	142805	913898	420286	2639208	−469942	−2059586	7996954	24681228	19074316	−64514568	−146041602	−39022284	−24825590	−261899222	648279804	581297158	587748736	1017573556	3257947192	663122590

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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, 3, 13, 39]: [1, 4, 11, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(39, 10)[2]   # 39.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 39:10 1000