Atlas › Modular forms › Weight 2 › Level 762 › 762.2.a › 762.2.a.k

Newform orbit 762.2.a.k

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ , whose minimal polynomial is

$x^{3} + 495x^{2} + 70011x + 2725029$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
127	$+1$

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
$\operatorname{tr} a_p$	−3	−3	4	−6	−2	−3	−3	−13	−3	11	−6	−17	3	−21	−3	0	10	−11	−25	−19	−11	−6	−8	5	10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.108 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, 2, 3, 6, 127, 254, 381, 762]: [0, 1, 1, 3, 11, 34, 44, 132] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 21 = 1 x 21 (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(762, 2)[10]   # 762.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 762:2 1000