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

Newform orbit 737.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} - 20x^{14} - 106x^{13} + 4207x^{12} - 7555x^{11} - 297401x^{10} + 1154057x^{9} + 9638078x^{8} - 46565611x^{7} - 159971067x^{6} + 813310990x^{5} + 1495510897x^{4} - 6241732493x^{3} - 8315944528x^{2} + 15508607612x + 20025303888$

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
$\operatorname{tr} a_p$	2	6	−4	24	−15	19	5	14	3	4	42	−8	0	47	9	−15	−15	32	15	−2	16	27	8	−16	31

$\operatorname{tr} a_n$ 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)[4]   # 737.2.a.e

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

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