Atlas › Modular forms › Weight 2 › Level 777 › 777.2.a › 777.2.a.k

Newform orbit 777.2.a.k

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{41} + 11T_{43} + 12T_{47}$ , whose minimal polynomial is

$x^{5} + 470x^{4} + 48483x^{3} - 2252734x^{2} - 290758320x - 277176166$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$+1$
37	$+1$

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	−5	−3	−5	−5	3	−4	10	−16	−12	−1	−5	2	−10	−16	−11	−9	16	−35	−25	16	−28	−32	19	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.007 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, 7, 21, 37, 111, 259, 777]: [0, 1, 1, 4, 3, 14, 26, 104] (certified at ell = 2147483629)
T = [(2, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (41, 10), (43, 11), (47, 12)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 12 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(777, 2)[10]   # 777.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 777:2 1000