Atlas › Modular forms › Weight 6 › Level 69 › 69.6.a › 69.6.a.c

Newform orbit 69.6.a.c

Trace form

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

$4q + 4q^{2} - 36q^{3} - 46q^{4} - 122q^{5} - 36q^{6} + 62q^{7} + 72q^{8} + 324q^{9} + 642q^{10} + 32q^{11} + 414q^{12} + 1364q^{13} + 2754q^{14} + 1098q^{15} + 18q^{16} + 278q^{17} + 324q^{18} + 2862q^{19} + 3830q^{20} - 558q^{21} + 3176q^{22} + 2116q^{23} - 648q^{24} + O(q^{25})$

Coefficient field

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

$x^{4} + 118x^{3} - 2943x^{2} - 393660x + 7479540$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
23	$-1$

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
$\operatorname{tr} a_p$	4	−36	−122	62	32	1364	278	2862	2116	−5180	−1788	3348	−17664	25398	−26040	−32006	−61136	35844	73458	24432	122512	90170	28592	−27926	16580

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 3, 23, 69]: [1, 3, 11, 42] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(69, 6)[2]   # 69.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 69:6 1000