Atlas › Modular forms › Weight 4 › Level 210 › 210.4.a › 210.4.a.f

Newform orbit 210.4.a.f

$q$ -expansion

$q - 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} - 6q^{6} - 7q^{7} - 8q^{8} + 9q^{9} - 10q^{10} + 28q^{11} + 12q^{12} + 54q^{13} + 14q^{14} + 15q^{15} + 16q^{16} - 46q^{17} - 18q^{18} + 12q^{19} + 20q^{20} - 21q^{21} - 56q^{22} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{11} + 3T_{13} + 5T_{19} + 7T_{23} + 9T_{29}$ is a root of

$x - 304$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
5	$-1$
7	$+1$

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
$\operatorname{tr} a_p$	−2	3	5	−7	28	54	−46	12	0	6	296	134	146	556	−448	46	748	−50	−156	−1024	−310	856	−628	−590	−1390

$\operatorname{tr} a_n$ 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 210, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.6 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, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210]: [1, 2, 2, 3, 5, 3, 7, 8, 8, 10, 22, 14, 28, 40, 52, 152] (certified at ell = 2147483629)
T = [(11, 1), (13, 3), (19, 5), (23, 7), (29, 9)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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(210, 4)[5]   # 210.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 210:4 1000