Atlas › Modular forms › Weight 2 › Level 699 › 699.2.a › 699.2.a.f

Newform orbit 699.2.a.f

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 2q^{2} + 13q^{3} + 16q^{4} + 12q^{5} + 2q^{6} - 2q^{7} + 9q^{8} + 13q^{9} - 8q^{10} + 10q^{11} + 16q^{12} - 3q^{13} + 10q^{14} + 12q^{15} + 26q^{16} + 15q^{17} + 2q^{18} + 23q^{20} - 2q^{21} - 16q^{22} + 33q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

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

$x^{13} - 38x^{12} + 320x^{11} + 4517x^{10} - 85171x^{9} + 193574x^{8} + 3874911x^{7} - 25460981x^{6} + 17698133x^{5} + 183262305x^{4} - 208437886x^{3} - 431740882x^{2} + 183007787x + 4073001$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
233	$+1$

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
$\operatorname{tr} a_p$	2	13	12	−2	10	−3	15	0	33	16	−15	12	−11	−12	39	52	11	4	9	36	−20	−6	−1	2	−23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 233, 699]: [0, 1, 20, 80] (certified at ell = 2147482481)
T = [(2, 1), (5, 3)]; h of degree 39 = 1 x 39 (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(699, 2)[5]   # 699.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 699:2 1000