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

Newform orbit 807.2.a.f

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + q^{2} - 15q^{3} + 19q^{4} + 2q^{5} - q^{6} + 7q^{7} + 15q^{9} + 9q^{10} - 18q^{11} - 19q^{12} + 23q^{13} - 2q^{15} + 31q^{16} + 30q^{17} + q^{18} + 11q^{19} + 4q^{20} - 7q^{21} + 16q^{22} + q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{15} - 3x^{14} - 88x^{13} + 243x^{12} + 2881x^{11} - 7361x^{10} - 45687x^{9} + 108430x^{8} + 374108x^{7} - 828740x^{6} - 1544057x^{5} + 3204501x^{4} + 2862737x^{3} - 5519751x^{2} - 1811633x + 3138827$

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 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
$\operatorname{tr} a_p$	1	−15	2	7	−18	23	30	11	1	−8	6	18	12	15	2	24	7	39	19	−20	64	27	23	11	36

$\operatorname{tr} a_n$ 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)[5]   # 807.2.a.f

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

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