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

Newform orbit 803.2.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 8x^{9} + 2x^{8} - 86x^{7} - 71x^{6} + 261x^{5} + 278x^{4} - 223x^{3} - 316x^{2} - 40x + 25$

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 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
$\operatorname{tr} a_p$	−5	−3	−2	−8	10	−24	−9	−7	−8	−22	−1	−34	7	−26	−6	−19	12	−32	−25	23	10	−3	−13	13	−16

$\operatorname{tr} a_n$ 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)[3]   # 803.2.a.d

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

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