Atlas › Modular forms › Weight 2 › Level 893 › 893.2.a › 893.2.a.d

Newform orbit 893.2.a.d

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q + q^{2} + 13q^{3} + 31q^{4} + q^{5} + 2q^{6} + 15q^{7} + 34q^{9} + 5q^{10} + 2q^{11} + 13q^{12} + 13q^{13} + 6q^{14} + q^{15} + 35q^{16} + 12q^{17} + 16q^{18} + 23q^{19} + 3q^{20} + q^{21} - 4q^{22} + 13q^{23} + 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} - 14x^{22} + 6x^{21} + 727x^{20} - 2542x^{19} - 12617x^{18} + 74421x^{17} + 58617x^{16} - 908986x^{15} + 605273x^{14} + 5330573x^{13} - 7848145x^{12} - 14786238x^{11} + 33791207x^{10} + 14100828x^{9} - 67103658x^{8} + 12374221x^{7} + 60779511x^{6} - 29647482x^{5} - 21466791x^{4} + 14359283x^{3} + 2035950x^{2} - 1556484x - 174139$

Atkin–Lehner signs

$p$	$w_p$
19	$-1$
47	$+1$

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
$\operatorname{tr} a_p$	1	13	1	15	2	13	12	23	13	−18	22	8	−16	68	−23	−7	10	28	55	−3	48	−31	47	−6	−18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 19, 47, 893]: [0, 2, 5, 82] (certified at ell = 2147479273)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(893, 2)[3]   # 893.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 893:2 1000