Atlas › Modular forms › Weight 2 › Level 669 › 669.2.a › 669.2.a.e

Newform orbit 669.2.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} + 2x^{2} - 3x - 3$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
223	$+1$

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	−3	0	1	−4	−4	−1	−5	2	−15	−17	13	5	5	−11	9	−18	12	16	−20	−21	−16	−29	−23	22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.335 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, 223, 669]: [0, 1, 19, 76] (certified at ell = 2147482591)
T = [(2, 1), (5, 3)]; h of degree 37 = 1 x 37 (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(669, 2)[4]   # 669.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 669:2 1000