Atlas › Modular forms › Weight 6 › Level 29 › 29.6.a › 29.6.a.a

Newform orbit 29.6.a.a

Trace form

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

$4q - 28q^{3} + 10q^{4} - 68q^{5} - 194q^{6} - 208q^{7} - 504q^{8} - 280q^{9} - 788q^{10} - 124q^{11} + 20q^{12} - 460q^{13} + 768q^{14} + 932q^{15} - 414q^{16} + 184q^{17} + 3208q^{18} - 2392q^{19} + 2822q^{20} + 992q^{21} + 5538q^{22} - 1192q^{23} + 6786q^{24} + O(q^{25})$

Coefficient field

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

$x^{4} + 28x^{3} + 171x^{2} - 980x - 8704$

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
$\operatorname{tr} a_p$	0	−28	−68	−208	−124	−460	184	−2392	−1192	−3364	−19212	−10928	−1120	−21420	23772	8860	−10840	49448	−7840	−48744	−74992	−106076	62888	107568	−49520

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 13] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (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(29, 6)[0]   # 29.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 29:6 1000