Atlas › Modular forms › Weight 4 › Level 73 › 73.4.a › 73.4.a.a

Newform orbit 73.4.a.a

$q$ -expansion

$q + 3q^{2} - 8q^{3} + q^{4} + 6q^{5} - 24q^{6} - 34q^{7} - 21q^{8} + 37q^{9} + 18q^{10} + 6q^{11} - 8q^{12} - 34q^{13} - 102q^{14} - 48q^{15} - 71q^{16} + 90q^{17} + 111q^{18} - 16q^{19} + 6q^{20} + 272q^{21} + 18q^{22} + 60q^{23} + 168q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 5$

Atkin–Lehner signs

$p$	$w_p$
73	$-1$

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
$\operatorname{tr} a_p$	3	−8	6	−34	6	−34	90	−16	60	102	−214	−286	150	−322	−534	−474	786	−574	−16	192	73	−988	1242	−6	614

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 73, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.097 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, 73]: [1, 20] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (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(73, 4)[0]   # 73.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 73:4 1000