Atlas › Modular forms › Weight 4 › Level 53 › 53.4.a › 53.4.a.b

Newform orbit 53.4.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} - 15q^{3} + 16q^{4} - q^{5} - 27q^{6} - 46q^{7} - 66q^{8} + 27q^{9} - 109q^{10} - 63q^{11} - 61q^{12} - 113q^{13} + 164q^{14} + 74q^{15} + 28q^{16} + 142q^{17} + 375q^{18} - 145q^{19} + 207q^{20} + 54q^{21} + 41q^{22} - 348q^{23} + 465q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 19x^{3} + 116x^{2} + 240x + 100$

Atkin–Lehner signs

$p$	$w_p$
53	$-1$

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	−15	−1	−46	−63	−113	142	−145	−348	437	−283	29	384	−911	−566	212	−271	604	−254	587	−74	−998	−835	279	506

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 53]: [1, 15] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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(53, 4)[1]   # 53.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 53:4 1000