Atlas › Modular forms › Weight 2 › Level 129 › 129.2.a › 129.2.a.d

Newform orbit 129.2.a.d

Trace form

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

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

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

$x^{3} + 14x^{2} + 40x + 16$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
43	$+1$

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	3	−4	4	1	9	1	−4	11	2	−5	0	−15	−3	−2	−5	8	−16	−11	22	−16	24	−7	−38	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.171 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, 43, 129]: [0, 1, 4, 16] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 7 = 1 x 7 (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(129, 2)[3]   # 129.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 129:2 1000