Atlas › Modular forms › Weight 2 › Level 877 › 877.2.a › 877.2.a.c

Newform orbit 877.2.a.c

Trace form

The 38 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$38q + 7q^{2} + 22q^{3} + 39q^{4} + 21q^{5} + 7q^{6} + 15q^{7} + 15q^{8} + 44q^{9} + q^{10} + 37q^{11} + 36q^{12} + 3q^{13} + 7q^{14} + 13q^{15} + 33q^{16} + 29q^{17} + 5q^{18} + 16q^{19} + 41q^{20} + q^{21} - 10q^{22} + 50q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 38. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{38} - 73x^{37} + 1875x^{36} - 9536x^{35} - 468577x^{34} + 8847973x^{33} - 4195877x^{32} - 1399884399x^{31} + 11606807742x^{30} + 78415041539x^{29} - 1569052910306x^{28} + 1775212778014x^{27} + 101619329508711x^{26} - 521475934960091x^{25} - 3437496463626383x^{24} + 34945736631799296x^{23} + 35107052972407973x^{22} - 1301316953863533142x^{21} + 1974646563280128210x^{20} + 30014080540214388732x^{19} - 100908266591519162575x^{18} - 423402033876336094356x^{17} + 2335254793458096959145x^{16} + 3114600735840473510080x^{15} - 32397525502881499747157x^{14} - 480809322727526630310x^{13} + 282051802068638894609672x^{12} - 197329935105270671682691x^{11} - 1520118253701132103915608x^{10} + 1628003796915493201473997x^{9} + 4811194799192841134424975x^{8} - 5699900713271527141910182x^{7} - 7915506754295583024331230x^{6} + 8741025444603962198248333x^{5} + 4766260220936457385284088x^{4} - 5333841745602798912139034x^{3} - 572360126776219505075320x^{2} + 1091007111842176906373286x - 181869867446789552723027$

Atkin–Lehner signs

$p$	$w_p$
877	$-1$

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
$\operatorname{tr} a_p$	7	22	21	15	37	3	29	16	50	11	7	6	19	26	74	31	73	−4	24	23	3	−27	163	20	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 877]: [0, 73] (certified at ell = 2147478961)
T = [(2, 1), (3, 3)]; h of degree 72 = 1 x 72 (phi(ord eps) x dim S_k^new); 10 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(877, 2)[2]   # 877.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 877:2 1000