Atlas › Modular forms › Weight 2 › Level 919 › 919.2.a › 919.2.a.b

Newform orbit 919.2.a.b

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q - 6q^{2} - 9q^{3} + 24q^{4} - 17q^{5} - 9q^{6} - 7q^{7} - 15q^{8} - 6q^{9} - 6q^{10} - 13q^{11} - 18q^{12} - 15q^{13} - 25q^{14} - 17q^{15} - 2q^{16} - 47q^{17} + 3q^{18} - 8q^{19} - 43q^{20} + 3q^{21} - 4q^{22} - 12q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} + 15x^{26} + 45x^{25} - 364x^{24} - 2299x^{23} + 1543x^{22} + 35787x^{21} + 33699x^{20} - 276422x^{19} - 491502x^{18} + 1188876x^{17} + 2958932x^{16} - 2889930x^{15} - 9919066x^{14} + 3643777x^{13} + 19763270x^{12} - 1470956x^{11} - 23343722x^{10} - 1203764x^{9} + 15601855x^{8} + 1103279x^{7} - 5414747x^{6} - 191953x^{5} + 845279x^{4} - 6010x^{3} - 44508x^{2} + 423x + 421$

Atkin–Lehner signs

$p$	$w_p$
919	$+1$

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
$\operatorname{tr} a_p$	−6	−9	−17	−7	−13	−15	−47	−8	−12	−41	−12	1	−80	0	−2	−34	−72	−18	13	−34	−9	−41	−25	−133	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.384 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, 919]: [0, 77] (certified at ell = 2147470057)
T = [(2, 1), (3, 1)]; h of degree 76 = 1 x 76 (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(919, 2)[1]   # 919.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 919:2 1000