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

Newform orbit 859.2.a.b

Trace form

The 42 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$42q + 10q^{2} + q^{3} + 48q^{4} + 21q^{5} + q^{6} + 2q^{7} + 27q^{8} + 63q^{9} + 3q^{10} + 22q^{11} - 8q^{12} + 14q^{13} + 26q^{14} + 7q^{15} + 52q^{16} + 30q^{17} + 19q^{18} + q^{19} + 29q^{20} + 49q^{21} + 2q^{22} + 9q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 42. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{42} - 11x^{41} - 101x^{40} + 1493x^{39} + 3175x^{38} - 87871x^{37} + 21114x^{36} + 2965913x^{35} - 4291763x^{34} - 64195522x^{33} + 141206361x^{32} + 947965270x^{31} - 2584851664x^{30} - 9973197039x^{29} + 30984291365x^{28} + 77534652253x^{27} - 259133038819x^{26} - 460480238670x^{25} + 1560466864259x^{24} + 2146823606298x^{23} - 6866489938230x^{22} - 7962840175250x^{21} + 22144517921528x^{20} + 23322353792882x^{19} - 52025456485951x^{18} - 52487265483775x^{17} + 87950061960604x^{16} + 87328619255832x^{15} - 105473690650035x^{14} - 103041899891215x^{13} + 88915492740075x^{12} + 82758948837835x^{11} - 52866960396020x^{10} - 43326353157739x^{9} + 22378858105551x^{8} + 13838458968800x^{7} - 6635770705258x^{6} - 2309093523052x^{5} + 1243251376360x^{4} + 95369447776x^{3} - 106598773344x^{2} + 15343523712x - 652551296$

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 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
$\operatorname{tr} a_p$	10	1	21	2	22	14	30	1	9	107	−3	15	44	−2	14	89	29	42	−14	18	−11	33	5	63	−1

$\operatorname{tr} a_n$ 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)[1]   # 859.2.a.b

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

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