Atlas › Modular forms › Weight 2 › Level 929 › 929.2.a › 929.2.a.b

Newform orbit 929.2.a.b

Trace form

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

$28q - q^{2} - 5q^{3} + 13q^{4} - 7q^{5} - 7q^{6} - 32q^{7} - 3q^{8} + 11q^{9} - 18q^{10} - 20q^{11} - 27q^{12} - 20q^{13} - 22q^{14} - 38q^{15} - 21q^{16} - 9q^{17} - 6q^{18} - 41q^{19} - 22q^{20} - 4q^{21} - 26q^{22} - 23q^{23} - 10q^{24} + O(q^{25})$

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

$x^{28} + 6x^{27} - 57x^{26} - 353x^{25} + 1431x^{24} + 8720x^{23} - 22110x^{22} - 120242x^{21} + 239682x^{20} + 1020060x^{19} - 1882048x^{18} - 5465500x^{17} + 10442420x^{16} + 18072375x^{15} - 39020930x^{14} - 33691597x^{13} + 92665084x^{12} + 25845870x^{11} - 130677463x^{10} + 11983188x^{9} + 101440308x^{8} - 33712932x^{7} - 37713277x^{6} + 20062121x^{5} + 3689498x^{4} - 3835865x^{3} + 735427x^{2} - 51799x + 1147$

Atkin–Lehner signs

$p$	$w_p$
929	$+1$

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
$\operatorname{tr} a_p$	−1	−5	−7	−32	−20	−20	−9	−41	−23	−28	−66	−12	−9	−23	5	−10	−22	1	−61	−52	−8	−118	32	−29	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 929]: [0, 78] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 77 = 1 x 77 (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(929, 2)[1]   # 929.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 929:2 1000