Atlas › Modular forms › Weight 6 › Level 74 › 74.6.a › 74.6.a.b

Newform orbit 74.6.a.b

$q$ -expansion

$q + 4q^{2} + 7q^{3} + 16q^{4} - 84q^{5} + 28q^{6} - 139q^{7} + 64q^{8} - 194q^{9} - 336q^{10} - 147q^{11} + 112q^{12} - 280q^{13} - 556q^{14} - 588q^{15} + 256q^{16} + 978q^{17} - 776q^{18} + 1154q^{19} - 1344q^{20} - 973q^{21} - 588q^{22} - 1158q^{23} + 448q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 77$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
37	$-1$

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
$\operatorname{tr} a_p$	4	7	−84	−139	−147	−280	978	1154	−1158	−3198	−5932	1369	10023	−4036	−11631	11193	24660	−13360	−32860	60123	41915	−60898	−80169	−131358	−122872

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 74, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 37, 74]: [1, 2, 17, 49] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(74, 6)[1]   # 74.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 74:6 1000