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

Newform orbit 803.2.a.e

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 4q^{3} + 38q^{4} + 4q^{5} + 3q^{7} - 3q^{8} + 26q^{9} + 2q^{10} - 16q^{11} + 25q^{12} + 29q^{13} - 5q^{14} + 7q^{15} + 22q^{16} - 4q^{17} + 14q^{18} + 11q^{19} + 15q^{20} + q^{21} + 17q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 4x^{15} - 64x^{14} + 224x^{13} + 1555x^{12} - 4313x^{11} - 18751x^{10} + 33455x^{9} + 118952x^{8} - 79806x^{7} - 346660x^{6} - 78915x^{5} + 207802x^{4} + 54997x^{3} - 41132x^{2} - 6600x + 2839$

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
$\operatorname{tr} a_p$	0	4	4	3	−16	29	−4	11	17	−14	7	44	−1	15	2	41	22	46	50	5	16	31	−12	−13	22

$\operatorname{tr} a_n$ 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)[4]   # 803.2.a.e

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

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