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

Newform orbit 29.4.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 8q^{3} + 26q^{4} + 10q^{5} + 34q^{6} + 40q^{7} - 84q^{8} + 33q^{9} - 64q^{10} + 12q^{11} - 224q^{12} + 14q^{13} - 192q^{14} - 74q^{15} + 146q^{16} + 66q^{17} - 108q^{18} + 214q^{19} + 6q^{20} - 98q^{22} + 164q^{23} + 314q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 8x^{4} - 119x^{3} + 1154x^{2} - 3096x + 2592$

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 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
$\operatorname{tr} a_p$	0	8	10	40	12	14	66	214	164	−145	420	378	−1158	−204	248	−554	440	618	1164	−692	−1950	272	512	866	1562

$\operatorname{tr} a_n$ 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)[1]   # 29.4.a.b

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

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