Atlas › Modular forms › Weight 6 › Level 46 › 46.6.a › 46.6.a.d

Newform orbit 46.6.a.d

Trace form

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

$3q + 12q^{2} + 14q^{3} + 48q^{4} + 82q^{5} + 56q^{6} + 172q^{7} + 192q^{8} + 217q^{9} + 328q^{10} + 520q^{11} + 224q^{12} + 700q^{13} + 688q^{14} + 256q^{15} + 768q^{16} + 70q^{17} + 868q^{18} - 620q^{19} + 1312q^{20} - 2340q^{21} + 2080q^{22} - 1587q^{23} + 896q^{24} + O(q^{25})$

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

$x^{3} - 96x^{2} + 365x - 78$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
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
$\operatorname{tr} a_p$	12	14	82	172	520	700	70	−620	−1587	−4912	−2718	−15102	−11888	−2896	5522	9318	62924	43814	96744	56298	−25156	1444	−5868	−63414	−167442

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 23, 46]: [1, 2, 11, 32] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(46, 6)[3]   # 46.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 46:6 1000