Atlas › Modular forms › Weight 8 › Level 62 › 62.8.a › 62.8.a.a

Newform orbit 62.8.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 24q^{2} - 26q^{3} + 192q^{4} + 152q^{5} + 208q^{6} - 1504q^{7} - 1536q^{8} + 1243q^{9} - 1216q^{10} + 4884q^{11} - 1664q^{12} + 7300q^{13} + 12032q^{14} + 15910q^{15} + 12288q^{16} - 3148q^{17} - 9944q^{18} + 8608q^{19} + 9728q^{20} - 46226q^{21} - 39072q^{22} - 165122q^{23} + 13312q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 126x^{2} - 64789x + 7367766$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
31	$-1$

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
$\operatorname{tr} a_p$	−24	−26	152	−1504	4884	7300	−3148	8608	−165122	−175696	89373	−1009570	−300856	−4810	−234956	1333678	713440	−2247792	−518492	4500560	−4786646	518542	−6233220	4123192	−18045028

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 2, 31, 62]: [1, 3, 19, 58] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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(62, 8)[0]   # 62.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 62:8 1000