Atlas › Modular forms › Weight 2 › Level 887 › 887.2.a › 887.2.a.c

Newform orbit 887.2.a.c

Trace form

The 51 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$51q + 4q^{2} + 11q^{3} + 62q^{4} + 8q^{5} - 2q^{6} + 23q^{7} + 6q^{8} + 74q^{9} + 12q^{10} + 9q^{11} + 27q^{12} + 25q^{13} + 12q^{15} + 80q^{16} + 35q^{17} + 12q^{18} + 8q^{19} + 3q^{20} + 6q^{21} + 40q^{22} + 27q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 51. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{51} - 15x^{50} - 81x^{49} + 2225x^{48} - 835x^{47} - 146192x^{46} + 375973x^{45} + 5603505x^{44} - 22530190x^{43} - 138488949x^{42} + 748929102x^{41} + 2287725269x^{40} - 16528911286x^{39} - 24869720163x^{38} + 260048329427x^{37} + 155838360042x^{36} - 3029443105889x^{35} - 93155991129x^{34} + 26757288361315x^{33} - 8901335370406x^{32} - 182111249243735x^{31} + 101006009932402x^{30} + 966275312561115x^{29} - 645850751647636x^{28} - 4028685596439187x^{27} + 2793715971179409x^{26} + 13247491429242222x^{25} - 8543801999847410x^{24} - 34296528816541874x^{23} + 18486458430936808x^{22} + 69293747684382404x^{21} - 27369863525913666x^{20} - 107393042294203556x^{19} + 25119305451301505x^{18} + 124307122604954567x^{17} - 9382306944325198x^{16} - 103579961522833748x^{15} - 6689777820854485x^{14} + 59307703578229026x^{13} + 10400490410117735x^{12} - 22150681162974407x^{11} - 5544060600107521x^{10} + 5159587370057368x^{9} + 1480635695970338x^{8} - 725450892908003x^{7} - 202535586681763x^{6} + 60821329046302x^{5} + 13098288943153x^{4} - 2987281941207x^{3} - 310027357611x^{2} + 67442015067x - 858608529$

Atkin–Lehner signs

$p$	$w_p$
887	$-1$

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	8	23	9	25	35	8	27	13	16	92	16	63	9	40	−15	36	52	−16	99	37	−9	−10	132

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 887]: [0, 75] (certified at ell = 2147476169)
T = [(2, 1), (3, 1)]; h of degree 74 = 1 x 74 (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(887, 2)[2]   # 887.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 887:2 1000