Atlas › Modular forms › Weight 2 › Level 269 › 269.2.a › 269.2.a.c

Newform orbit 269.2.a.c

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + q^{2} + 5q^{3} + 25q^{4} - q^{5} - 2q^{6} + 11q^{7} + 21q^{9} + 7q^{10} + 16q^{11} - 4q^{12} - q^{13} - 9q^{14} - 5q^{15} + 39q^{16} - 2q^{17} - 8q^{18} + 35q^{19} - 16q^{20} - 3q^{22} + q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 6x^{15} - 43x^{14} + 314x^{13} + 462x^{12} - 5829x^{11} + 2229x^{10} + 45324x^{9} - 56637x^{8} - 137976x^{7} + 239275x^{6} + 160761x^{5} - 347291x^{4} - 80588x^{3} + 167274x^{2} + 27003x + 626$

Atkin–Lehner signs

$p$	$w_p$
269	$-1$

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
$\operatorname{tr} a_p$	1	5	−1	11	16	−1	−2	35	1	2	13	4	−2	51	−9	−29	−13	14	25	−9	4	51	−2	−35	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 269]: [0, 23] (certified at ell = 2147482661)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(269, 2)[2]   # 269.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 269:2 1000