Atlas › Modular forms › Weight 2 › Level 983 › 983.2.a › 983.2.a.a

Newform orbit 983.2.a.a

Trace form

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

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

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

$x^{28} + 13x^{27} + 10x^{26} - 506x^{25} - 1555x^{24} + 7942x^{23} + 36401x^{22} - 62311x^{21} - 425126x^{20} + 223924x^{19} + 2972438x^{18} + 58930x^{17} - 13315014x^{16} - 3679775x^{15} + 39316046x^{14} + 14305818x^{13} - 76954283x^{12} - 26506612x^{11} + 97838130x^{10} + 26445907x^{9} - 76468566x^{8} - 15351570x^{7} + 33628338x^{6} + 6549460x^{5} - 7661438x^{4} - 1878865x^{3} + 669083x^{2} + 237706x + 15085$

Atkin–Lehner signs

$p$	$w_p$
983	$+1$

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
$\operatorname{tr} a_p$	−7	−6	−7	−25	−10	−28	−24	−13	−9	−14	−16	−58	−24	−43	2	−16	12	−22	−38	2	−124	−16	12	−3	−105

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 983]: [0, 83] (certified at ell = 2147475899)
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(983, 2)[0]   # 983.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 983:2 1000