Atlas › Modular forms › Weight 2 › Level 991 › 991.2.a › 991.2.a.a

Newform orbit 991.2.a.a

Trace form

The 33 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$33q - 5q^{2} - 6q^{3} + 25q^{4} - 24q^{5} - 18q^{6} - 9q^{7} - 12q^{8} + 5q^{9} - 11q^{10} - 24q^{11} - 18q^{12} - 12q^{13} - 31q^{14} - 11q^{15} + 9q^{16} - 60q^{17} - 6q^{18} - 18q^{19} - 43q^{20} - 19q^{21} - 3q^{22} - 24q^{23} - 39q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 33. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{33} + 11x^{32} - 19x^{31} - 575x^{30} - 617x^{29} + 13030x^{28} + 26635x^{27} - 171254x^{26} - 431965x^{25} + 1478677x^{24} + 4024780x^{23} - 9035164x^{22} - 24024686x^{21} + 40979360x^{20} + 95361911x^{19} - 140820851x^{18} - 251587355x^{17} + 362269549x^{16} + 424348718x^{15} - 668217428x^{14} - 408179442x^{13} + 824221042x^{12} + 138808147x^{11} - 614290178x^{10} + 95530256x^{9} + 234053309x^{8} - 95077656x^{7} - 28851200x^{6} + 22719248x^{5} - 2769846x^{4} - 445600x^{3} + 63061x^{2} + 2431x - 169$

Atkin–Lehner signs

$p$	$w_p$
991	$+1$

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
$\operatorname{tr} a_p$	−5	−6	−24	−9	−24	−12	−60	−18	−24	−53	−18	4	−109	−1	−14	−26	−28	−35	28	−28	−14	−29	−36	−117	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 991]: [0, 83] (certified at ell = 2147476321)
T = [(2, 1), (3, 1)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 8 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(991, 2)[0]   # 991.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 991:2 1000