Atlas › Modular forms › Weight 2 › Level 537 › 537.2.a › 537.2.a.h

Newform orbit 537.2.a.h

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{6} + 112x^{5} + 3547x^{4} - 8472x^{3} - 2125369x^{2} - 33175460x - 154061275$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
179	$-1$

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
$\operatorname{tr} a_p$	−4	6	−10	−10	−4	−6	−22	−2	−24	−18	0	2	−8	−10	−10	−6	6	−4	−10	−2	6	−24	2	10	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.316 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, 179, 537]: [0, 1, 16, 62] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 29 = 1 x 29 (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(537, 2)[7]   # 537.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 537:2 1000