Atlas › Modular forms › Weight 4 › Level 77 › 77.4.a › 77.4.a.c

Newform orbit 77.4.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} - 12q^{3} + 22q^{4} - 18q^{5} + 2q^{6} - 28q^{7} - 60q^{8} + 66q^{9} - 92q^{10} + 44q^{11} - 186q^{12} - 134q^{13} + 28q^{14} - 62q^{15} - 6q^{16} - 74q^{17} - 256q^{18} - 164q^{19} + 116q^{20} + 84q^{21} - 44q^{22} + 194q^{23} + 570q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 16x^{3} + 12x^{2} - 100x + 63$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
11	$-1$

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
$\operatorname{tr} a_p$	−4	−12	−18	−28	44	−134	−74	−164	194	−108	−412	286	−18	−496	62	−828	−1224	−350	−1498	2326	−1630	−1020	−1920	1550	−2202

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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 engine's checks

dim M^+ at levels [1, 7, 11, 77]: [1, 3, 4, 26] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(77, 4)[2]   # 77.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 77:4 1000