Atlas › Modular forms › Weight 2 › Level 429 › 429.2.a › 429.2.a.h

Newform orbit 429.2.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{4} + 2x^{3} - 108x^{2} - 150x + 23$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
11	$+1$
13	$-1$

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	−4	0	2	−4	4	−8	18	0	−10	12	−2	2	28	6	−4	16	−10	0	10	−6	−8	−8	6	10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.316 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, 11, 13, 33, 39, 143, 429]: [0, 1, 2, 1, 6, 6, 16, 60] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(429, 2)[7]   # 429.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 429:2 1000