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

Newform orbit 269.2.a.b

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 6x^{4} + 7x^{3} - 6x^{2} - 10x - 3$

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 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
$\operatorname{tr} a_p$	−1	−5	−4	−5	−9	−5	6	−25	2	−2	1	−9	−3	−28	24	−8	1	−15	−2	5	20	−33	−20	0	9

$\operatorname{tr} a_n$ 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)[1]   # 269.2.a.b

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

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