Atlas › Modular forms › Weight 2 › Level 277 › 277.2.a › 277.2.a.b

Newform orbit 277.2.a.b

Trace form

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

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

Coefficient field

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

$x^{3} - 5x^{2} + 5x + 1$

Atkin–Lehner signs

$p$	$w_p$
277	$-1$

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
$\operatorname{tr} a_p$	−1	6	4	4	11	1	−4	−10	−8	3	−11	12	9	−1	10	−16	8	−20	8	10	−4	0	12	7	32

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.171 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, 277]: [0, 23] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(277, 2)[1]   # 277.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 277:2 1000