Atlas › Modular forms › Weight 2 › Level 267 › 267.2.a › 267.2.a.f

Newform orbit 267.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 4x^{3} - 11x^{2} + 35x - 13$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
89	$-1$

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	−4	3	6	−6	9	2	10	1	−2	−11	23	9	1	−8	7	10	20	−1	−21	24	−1	10	4	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 3, 89, 267]: [0, 1, 8, 32] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(267, 2)[5]   # 267.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 267:2 1000