Atlas › Modular forms › Weight 2 › Level 209 › 209.2.a › 209.2.a.d

Newform orbit 209.2.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - q^{2} + 2q^{3} + 15q^{4} + 2q^{5} - 2q^{6} + 10q^{7} - 9q^{8} + 11q^{9} - 6q^{10} - 7q^{11} - 16q^{12} - 4q^{13} + 6q^{14} + 12q^{15} + 27q^{16} + 2q^{17} + 9q^{18} + 7q^{19} - 4q^{20} - 14q^{21} + q^{22} + 10q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} - 5x^{6} - 140x^{5} + 656x^{4} + 5492x^{3} - 20358x^{2} - 58557x + 70863$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
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 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	2	2	10	−7	−4	2	7	10	−18	24	0	−12	2	8	2	−10	14	8	10	−6	52	−10	0	−24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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, 11, 19, 209]: [0, 2, 2, 22] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; 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(209, 2)[3]   # 209.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 209:2 1000