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

Newform orbit 977.2.a.b

Trace form

The 45 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$45q + 4q^{2} + 11q^{3} + 50q^{4} + 7q^{5} + 9q^{6} + 52q^{7} + 9q^{8} + 48q^{9} + 6q^{10} + 5q^{11} + 13q^{12} + 43q^{13} - 6q^{14} + 19q^{15} + 52q^{16} + 12q^{17} - q^{18} + 45q^{19} + 7q^{20} + 11q^{21} + 23q^{22} + 20q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 45. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{45} - 15x^{44} - 58x^{43} + 1907x^{42} - 2628x^{41} - 103193x^{40} + 371508x^{39} + 3065795x^{38} - 16825905x^{37} - 52434343x^{36} + 436438933x^{35} + 436546265x^{34} - 7389588468x^{33} + 1263743756x^{32} + 86180371017x^{31} - 81363164913x^{30} - 709398132393x^{29} + 1091470949780x^{28} + 4149257894417x^{27} - 8639311845724x^{26} - 17050218115339x^{25} + 46058594083602x^{24} + 47150484091226x^{23} - 171912943577296x^{22} - 76231107861469x^{21} + 451841175868129x^{20} + 22312041908433x^{19} - 822447675184403x^{18} + 200989550928649x^{17} + 992411712342427x^{16} - 487105253783233x^{15} - 721858335993547x^{14} + 540011736928757x^{13} + 244972307752156x^{12} - 298680371746719x^{11} + 5154010068510x^{10} + 67735604052263x^{9} - 16538634321426x^{8} - 5048282622886x^{7} + 2573114273316x^{6} - 149394360749x^{5} - 98563983248x^{4} + 22384380766x^{3} - 1641080679x^{2} + 3712900x + 3202207$

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 even 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
$\operatorname{tr} a_p$	4	11	7	52	5	43	12	45	20	−4	54	13	−1	58	44	−12	5	20	54	11	77	54	52	−7	37

$\operatorname{tr} a_n$ 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)[1]   # 977.2.a.b

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

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