Atlas › Modular forms › Weight 2 › Level 71 › 71.2.a › 71.2.a.a

Newform orbit 71.2.a.a

Trace form

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

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

Coefficient field

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

$x^{3} - 2x^{2} - 16x + 24$

Atkin–Lehner signs

$p$	$w_p$
71	$-1$

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	1	5	2	0	−6	−2	1	−12	11	12	−15	−2	13	4	−18	−4	16	−12	3	27	−3	−19	1	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 0.078 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, 71]: [0, 7] (certified at ell = 2147483171)
T = [(2, 1), (3, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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(71, 2)[0]   # 71.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 71:2 1000