Atlas › Modular forms › Weight 2 › Level 253 › 253.2.a › 253.2.a.d

Newform orbit 253.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 3q^{2} + 7q^{3} + 5q^{4} + 3q^{5} + 3q^{6} - q^{7} + 3q^{8} + 9q^{9} - 6q^{10} + 6q^{11} + 11q^{12} - 3q^{13} - 8q^{14} + 10q^{15} - q^{16} + 5q^{17} - 11q^{18} + q^{19} - 7q^{20} - 15q^{21} + 3q^{22} - 6q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 10x^{5} + 25x^{4} + 36x^{3} - 224x^{2} + 265x - 67$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	3	7	3	−1	6	−3	5	1	−6	6	−8	2	−2	10	14	−4	39	−22	8	18	−9	−15	43	17	19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 11, 23, 253]: [0, 2, 3, 26] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (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(253, 2)[3]   # 253.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 253:2 1000