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

Newform orbit 857.2.a.b

Trace form

The 43 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$43q + 3q^{2} + 12q^{3} + 55q^{4} + 6q^{5} + 44q^{7} + 3q^{8} + 51q^{9} + 6q^{10} + 9q^{11} + 23q^{12} + 13q^{13} + 32q^{15} + 75q^{16} + 7q^{17} + 4q^{18} + 12q^{19} - 4q^{20} + 3q^{21} + 42q^{22} + 56q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 43. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{43} - 15x^{42} - 48x^{41} + 1746x^{40} - 2901x^{39} - 85853x^{38} + 321881x^{37} + 2309306x^{36} - 12845663x^{35} - 35867387x^{34} + 297602379x^{33} + 280840352x^{32} - 4523710146x^{31} + 297443467x^{30} + 47612063584x^{29} - 33089816508x^{28} - 357161605423x^{27} + 399283266341x^{26} + 1944204405722x^{25} - 2744341388615x^{24} - 7781943143734x^{23} + 12525595185076x^{22} + 23156423965350x^{21} - 39869676848162x^{20} - 51653370058821x^{19} + 90114832012744x^{18} + 86619263898478x^{17} - 144930312408124x^{16} - 108392719077929x^{15} + 164213192312556x^{14} + 98796922834491x^{13} - 128368186058761x^{12} - 62461349008389x^{11} + 67153187779015x^{10} + 25148285584475x^{9} - 22689172581668x^{8} - 5543347066468x^{7} + 4763875165428x^{6} + 464124953038x^{5} - 560295669005x^{4} + 24229209199x^{3} + 26828128784x^{2} - 4466560336x + 178816301$

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 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
$\operatorname{tr} a_p$	3	12	6	44	9	13	7	12	56	3	41	54	−10	84	22	20	−14	12	57	14	29	90	2	−13	33

$\operatorname{tr} a_n$ 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)[1]   # 857.2.a.b

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

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