Atlas › Modular forms › Weight 2 › Level 805 › 805.2.a › 805.2.a.g

Newform orbit 805.2.a.g

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{4} + 167x^{3} + 7636x^{2} + 58537x + 116108$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
7	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	−2	−5	4	4	−8	−8	−6	2	−4	−1	−2	−7	−9	−4	−21	−11	−37	1	−31	−8	25	−19	−7	7	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 5, 7, 23, 35, 115, 161, 805]: [0, 1, 1, 3, 6, 14, 18, 100] (certified at ell = 2147483053)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (17, 9)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 11 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(805, 2)[6]   # 805.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 805:2 1000