Atlas › Modular forms › Weight 2 › Level 609 › 609.2.a › 609.2.a.f

Newform orbit 609.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{4} - 184x^{3} + 9378x^{2} - 167256x + 730593$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$-1$
29	$+1$

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	−4	−2	4	0	4	4	16	−6	−4	16	0	8	6	16	0	4	14	8	6	−2	18	4	0	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.421 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, 7, 21, 29, 87, 203, 609]: [0, 1, 1, 4, 3, 12, 22, 84] (certified at ell = 2147483353)
T = [(2, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(609, 2)[5]   # 609.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 609:2 1000