Atlas › Modular forms › Weight 2 › Level 737 › 737.2.a › 737.2.a.f

Newform orbit 737.2.a.f

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + q^{2} + 10q^{3} + 19q^{4} + 10q^{5} - 6q^{6} + 20q^{7} + 25q^{9} + 10q^{10} + 17q^{11} + 18q^{12} + q^{13} - 11q^{14} + 7q^{15} + 19q^{16} + 2q^{17} - 24q^{18} + 13q^{19} + 3q^{20} + 5q^{21} + q^{22} + 16q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

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

$x^{17} - 31x^{16} + 130x^{15} + 5263x^{14} - 61600x^{13} - 111232x^{12} + 4862140x^{11} - 15540389x^{10} - 125227417x^{9} + 824489544x^{8} + 317875805x^{7} - 12882535945x^{6} + 21924665530x^{5} + 61000116027x^{4} - 207753488630x^{3} + 59844796152x^{2} + 352210238192x - 315401306708$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
67	$+1$

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	10	10	20	17	1	2	13	16	−5	16	29	−6	19	40	15	−2	−6	−17	2	41	41	2	1	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 67, 737]: [0, 2, 6, 70] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(737, 2)[5]   # 737.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 737:2 1000