Atlas › Modular forms › Weight 6 › Level 51 › 51.6.a › 51.6.a.b

Newform orbit 51.6.a.b

Trace form

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

$3q + 5q^{2} - 27q^{3} + 37q^{4} - 37q^{5} - 45q^{6} - 176q^{7} - 153q^{8} + 243q^{9} - 860q^{10} - 197q^{11} - 333q^{12} - 1041q^{13} - 1610q^{14} + 333q^{15} - 2959q^{16} - 867q^{17} + 405q^{18} - 55q^{19} - 2148q^{20} + 1584q^{21} + 256q^{22} + 1799q^{23} + 1377q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 32x^{2} - 4031x + 65634$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
17	$+1$

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
$\operatorname{tr} a_p$	5	−27	−37	−176	−197	−1041	−867	−55	1799	−4446	−4438	−17824	−5631	5935	29270	25510	−94374	−137364	−37452	25104	2662	−27738	95750	−9220	−49164

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 17, 51]: [1, 3, 8, 32] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(51, 6)[1]   # 51.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 51:6 1000