Atlas › Modular forms › Weight 4 › Level 203 › 203.4.a › 203.4.a.b

Newform orbit 203.4.a.b

$q$ -expansion

$q + 4q^{2} + 8q^{3} + 8q^{4} + 2q^{5} + 32q^{6} + 7q^{7} + 37q^{9} + 8q^{10} + 2q^{11} + 64q^{12} - 26q^{13} + 28q^{14} + 16q^{15} - 64q^{16} - 80q^{17} + 148q^{18} + 128q^{19} + 16q^{20} + 56q^{21} + 8q^{22} + O(q^{25})$

Coefficient field

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

$x - 28$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
29	$-1$

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
$\operatorname{tr} a_p$	4	8	2	7	2	−26	−80	128	0	29	160	−274	−36	246	−244	114	−420	188	624	1120	−352	438	−676	−336	−216

$\operatorname{tr} a_n$ 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 203, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.149 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, 7, 29, 203]: [1, 3, 9, 62] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 8 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(203, 4)[1]   # 203.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 203:4 1000