Atlas › Modular forms › Weight 2 › Level 257 › 257.2.a › 257.2.a.b

Newform orbit 257.2.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 2q^{2} + 3q^{3} + 18q^{4} - q^{5} - 4q^{6} + 24q^{7} + 13q^{9} + 2q^{11} - q^{12} + 12q^{13} - 2q^{14} + 6q^{15} + 26q^{16} - 4q^{17} + 2q^{18} + 9q^{19} - 15q^{20} - 5q^{21} + 10q^{22} + 20q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 5x^{13} - 34x^{12} + 192x^{11} + 352x^{10} - 2662x^{9} - 398x^{8} + 15681x^{7} - 11622x^{6} - 32938x^{5} + 45747x^{4} + 1133x^{3} - 21671x^{2} + 4763x + 1373$

Atkin–Lehner signs

$p$	$w_p$
257	$-1$

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
$\operatorname{tr} a_p$	2	3	−1	24	2	12	−4	9	20	−3	3	5	−9	48	2	−9	−28	−10	33	−15	3	47	18	−15	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.168 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, 257]: [0, 22] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (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(257, 2)[1]   # 257.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 257:2 1000