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

Newform orbit 737.2.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} + 14x^{11} - 113x^{10} - 1915x^{9} + 3477x^{8} + 78892x^{7} - 73317x^{6} - 1158127x^{5} + 1372719x^{4} + 2741282x^{3} - 236656x^{2} - 449796x + 54420$

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 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
$\operatorname{tr} a_p$	−2	−4	4	−20	−12	−9	−16	−17	−12	1	−18	−11	−6	−25	−12	21	6	−4	−12	−2	−39	−25	6	33	−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)[3]   # 737.2.a.d

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

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