Atlas › Modular forms › Weight 2 › Level 803 › 803.2.a › 803.2.a.f

Newform orbit 803.2.a.f

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 7q^{2} + 3q^{3} + 23q^{4} + 2q^{5} + 16q^{7} + 18q^{8} + 22q^{9} + 8q^{10} + 19q^{11} - 5q^{12} + 30q^{13} - 9q^{14} - 11q^{15} + 31q^{16} + 9q^{17} + 3q^{18} - q^{19} - 9q^{20} + 13q^{21} + 7q^{22} - 2q^{23} - 19q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 10x^{18} - 20x^{17} + 490x^{16} - 770x^{15} - 7558x^{14} + 23934x^{13} + 35422x^{12} - 207958x^{11} + 38679x^{10} + 714386x^{9} - 566200x^{8} - 1047599x^{7} + 1125222x^{6} + 634467x^{5} - 706188x^{4} - 154569x^{3} + 72773x^{2} + 2069x + 3$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
73	$+1$

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	3	2	16	19	30	9	−1	−2	18	5	36	9	33	−28	27	−28	4	21	−7	−19	25	15	−21	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 0.24 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, 11, 73, 803]: [0, 2, 6, 76] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 6 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(803, 2)[5]   # 803.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 803:2 1000