Atlas › Modular forms › Weight 10 › Level 38 › 38.10.a › 38.10.a.c

Newform orbit 38.10.a.c

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 48q^{2} + 3q^{3} + 768q^{4} + 486q^{5} - 48q^{6} - 13317q^{7} - 12288q^{8} + 22896q^{9} - 7776q^{10} + 55968q^{11} + 768q^{12} + 158181q^{13} + 213072q^{14} + 135660q^{15} + 196608q^{16} - 629091q^{17} - 366336q^{18} - 390963q^{19} + 124416q^{20} - 873405q^{21} - 895488q^{22} - 924627q^{23} - 12288q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} - 489x^{2} - 584001x - 100504663$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
19	$+1$

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
$\operatorname{tr} a_p$	−48	3	486	−13317	55968	158181	−629091	−390963	−924627	−9839019	1364628	−2289090	−12899580	−22378638	58896366	8770629	16426299	−126843780	−288075309	78122274	−557941845	−320222022	430491462	437689644	−384952146

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 2, 19, 38]: [1, 3, 16, 47] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(38, 10)[2]   # 38.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 38:10 1000