Atlas › Modular forms › Weight 2 › Level 807 › 807.2.a › 807.2.a.d

Newform orbit 807.2.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 5q^{2} + 8q^{3} + 5q^{4} - 6q^{5} - 5q^{6} - 5q^{7} - 12q^{8} + 8q^{9} - 5q^{10} - 20q^{11} + 5q^{12} - 13q^{13} - 6q^{14} - 6q^{15} + 3q^{16} - 28q^{17} - 5q^{18} - 7q^{19} - 5q^{21} + 10q^{22} - 21q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} + 11x^{7} + 34x^{6} - 19x^{5} - 247x^{4} - 243x^{3} + 313x^{2} + 401x - 73$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
269	$-1$

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	8	−6	−5	−20	−13	−28	−7	−21	−10	6	−6	−22	−1	−16	−12	3	−5	7	8	−28	7	−7	−11	−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 0.345 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, 3, 269, 807]: [0, 1, 23, 92] (certified at ell = 2147482661)
T = [(2, 1), (5, 1)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 5 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(807, 2)[3]   # 807.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 807:2 1000