Atlas › Modular forms › Weight 2 › Level 859 › 859.2.a › 859.2.a.a

Newform orbit 859.2.a.a

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q - 10q^{2} - 5q^{3} + 20q^{4} - 21q^{5} - 7q^{6} - 4q^{7} - 27q^{8} + 6q^{9} - 5q^{10} - 24q^{11} - 10q^{12} - 16q^{13} - 24q^{14} - 17q^{15} + 6q^{16} - 24q^{17} - 11q^{18} - 21q^{19} - 25q^{20} - 47q^{21} + 6q^{22} - 19q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} + 15x^{28} + 34x^{27} - 516x^{26} - 2664x^{25} + 5390x^{24} + 55739x^{23} + 8086x^{22} - 588644x^{21} - 625192x^{20} + 3655232x^{19} + 5851324x^{18} - 14271315x^{17} - 28253059x^{16} + 36351296x^{15} + 81947056x^{14} - 61888660x^{13} - 148509533x^{12} + 71245976x^{11} + 167920254x^{10} - 54220410x^{9} - 115411437x^{8} + 25348152x^{7} + 45833264x^{6} - 6495539x^{5} - 9550014x^{4} + 763307x^{3} + 871269x^{2} - 19903x - 21820$

Atkin–Lehner signs

$p$	$w_p$
859	$+1$

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
$\operatorname{tr} a_p$	−10	−5	−21	−4	−24	−16	−24	−21	−19	−97	−5	−21	−50	−20	4	−73	−37	−32	4	−32	11	−5	−9	−35	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 0.272 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, 859]: [0, 72] (certified at ell = 2147483053)
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(859, 2)[0]   # 859.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 859:2 1000