Atlas › Modular forms › Weight 2 › Level 977 › 977.2.a › 977.2.a.a

Newform orbit 977.2.a.a

Trace form

The 36 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$36q - 7q^{2} - 13q^{3} + 27q^{4} - 9q^{5} - 9q^{6} - 48q^{7} - 18q^{8} + 31q^{9} - 18q^{10} - 7q^{11} - 29q^{12} - 41q^{13} + 2q^{14} - 25q^{15} + 13q^{16} - 18q^{17} - 32q^{18} - 43q^{19} - 17q^{20} - q^{21} - 21q^{22} - 26q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 36. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{36} + 20x^{35} + 90x^{34} - 705x^{33} - 6884x^{32} + 1234x^{31} + 177266x^{30} + 322792x^{29} - 2398729x^{28} - 7502773x^{27} + 18986388x^{26} + 88015076x^{25} - 86895248x^{24} - 650469144x^{23} + 176370496x^{22} + 3279167627x^{21} + 326756487x^{20} - 11738715979x^{19} - 3300949531x^{18} + 30503830560x^{17} + 10376051957x^{16} - 58062118793x^{15} - 18234977617x^{14} + 80627719655x^{13} + 18718336313x^{12} - 80115706472x^{11} - 9758383867x^{10} + 54845297239x^{9} + 449046195x^{8} - 24351491779x^{7} + 2137774075x^{6} + 6430377324x^{5} - 975455369x^{4} - 899852510x^{3} + 165152072x^{2} + 51142851x - 9604225$

Atkin–Lehner signs

$p$	$w_p$
977	$+1$

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	−13	−9	−48	−7	−41	−18	−43	−26	−2	−66	−19	−3	−58	−68	−8	−3	−22	−54	−9	−81	−36	−32	19	−49

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 977]: [0, 82] (certified at ell = 2147475233)
T = [(2, 1), (3, 1)]; h of degree 81 = 1 x 81 (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(977, 2)[0]   # 977.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 977:2 1000