Atlas › Modular forms › Weight 2 › Level 857 › 857.2.a › 857.2.a.a

Newform orbit 857.2.a.a

Trace form

The 28 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$28q - 4q^{2} - 12q^{3} + 14q^{4} - 6q^{5} - 6q^{6} - 38q^{7} - 12q^{8} + 16q^{9} - 10q^{10} - 7q^{11} - 19q^{12} - 17q^{13} - 28q^{15} - 6q^{16} - 11q^{17} - 7q^{18} - 12q^{19} - 16q^{20} + 7q^{21} - 40q^{22} - 48q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 28. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{28} + 16x^{27} + 49x^{26} - 472x^{25} - 3057x^{24} + 2812x^{23} + 56742x^{22} + 49899x^{21} - 529735x^{20} - 957506x^{19} + 2812424x^{18} + 7390483x^{17} - 8408608x^{16} - 32730605x^{15} + 10560123x^{14} + 89744142x^{13} + 14583964x^{12} - 151961020x^{11} - 79128792x^{10} + 146999099x^{9} + 129287832x^{8} - 59889842x^{7} - 97438842x^{6} - 11761675x^{5} + 26335136x^{4} + 13087401x^{3} + 1791728x^{2} - 62695x - 11525$

Atkin–Lehner signs

$p$	$w_p$
857	$+1$

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	−6	−38	−7	−17	−11	−12	−48	−3	−35	−50	8	−64	−24	−10	−6	−26	−71	−34	−17	−82	6	33	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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, 857]: [0, 72] (certified at ell = 2147478017)
T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (phi(ord eps) x dim S_k^new); 7 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(857, 2)[0]   # 857.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 857:2 1000