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

Newform orbit 209.2.a.c

Trace form

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

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

Coefficient field

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

$x^{5} - 5x^{4} - 75x^{3} + 559x^{2} - 991x + 125$

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
$\operatorname{tr} a_p$	2	1	−5	6	5	4	−4	−5	3	10	11	1	2	20	−20	−14	3	−10	9	23	0	44	−14	−27	15

$\operatorname{tr} a_n$ 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)[2]   # 209.2.a.c

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

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