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

Newform orbit 277.2.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 6q^{2} - 10q^{3} + 10q^{4} - 12q^{5} + 3q^{6} - 2q^{7} - 9q^{8} + 11q^{9} - 4q^{10} - 14q^{11} - 23q^{12} - 2q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 19q^{17} - 7q^{18} + q^{19} - 24q^{20} - 2q^{21} - 9q^{22} - 30q^{23} + 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 16x^{8} + 92x^{7} + 219x^{6} + 145x^{5} - 149x^{4} - 115x^{3} + 51x^{2} + 14x - 5$

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 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
$\operatorname{tr} a_p$	−6	−10	−12	−2	−14	−2	−19	1	−30	−8	0	1	−13	−14	−10	−21	−45	6	5	−17	11	24	−53	12	−7

$\operatorname{tr} a_n$ 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)[2]   # 277.2.a.c

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

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