Atlas › Modular forms › Weight 4 › Level 29 › 29.4.a › 29.4.a.a

Newform orbit 29.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{2} - 10q^{3} - 10q^{4} - 10q^{5} - 2q^{6} - 16q^{7} + 18q^{8} + 32q^{9} + 26q^{10} - 26q^{11} + 74q^{12} - 26q^{13} + 56q^{14} + 2q^{15} + 18q^{16} + 60q^{17} + 88q^{18} - 220q^{19} + 18q^{20} - 40q^{21} - 122q^{22} + 52q^{23} + 42q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 12x + 28$

Atkin–Lehner signs

$p$	$w_p$
29	$-1$

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	−10	−10	−16	−26	−26	60	−220	52	58	−294	312	40	−322	−130	1002	−900	−948	320	−660	648	258	1212	760	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.095 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 engine's checks

dim M^+ at levels [1, 29]: [1, 9] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; 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(29, 4)[0]   # 29.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 29:4 1000