Atlas › Modular forms › Weight 2 › Level 769 › 769.2.a › 769.2.a.a

Newform orbit 769.2.a.a

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q - 10q^{2} - 7q^{3} + 18q^{4} - 13q^{5} - 7q^{6} - 10q^{7} - 27q^{8} + 10q^{9} - 4q^{10} - 36q^{11} - 9q^{12} - 8q^{13} - 21q^{14} - 38q^{15} + 12q^{16} - 10q^{17} - 14q^{18} - 23q^{19} - 22q^{20} - 20q^{21} + 13q^{22} - 45q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} + 17x^{26} + 70x^{25} - 343x^{24} - 3200x^{23} - 1471x^{22} + 47032x^{21} + 95570x^{20} - 321142x^{19} - 1068362x^{18} + 1000967x^{17} + 5832778x^{16} - 590104x^{15} - 18026634x^{14} - 4441456x^{13} + 33288006x^{12} + 12128778x^{11} - 37043998x^{10} - 13108237x^{9} + 23968887x^{8} + 6592224x^{7} - 8217590x^{6} - 1444034x^{5} + 1261965x^{4} + 112644x^{3} - 63198x^{2} + 283x + 467$

Atkin–Lehner signs

$p$	$w_p$
769	$+1$

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	−7	−13	−10	−36	−8	−10	−23	−45	−36	−47	−5	−37	−9	−53	−19	−65	−5	−7	−131	−3	−38	−19	−61	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 769]: [0, 64] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (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(769, 2)[0]   # 769.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 769:2 1000