Atlas › Modular forms › Weight 2 › Level 553 › 553.2.a › 553.2.a.c

Newform orbit 553.2.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 10x^{10} + 12x^{9} - 129x^{8} - 307x^{7} + 304x^{6} + 1080x^{5} + 31x^{4} - 1064x^{3} - 316x^{2} + 244x + 67$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
79	$+1$

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
$\operatorname{tr} a_p$	−5	−5	−10	−11	−5	−11	−18	−8	−23	16	−11	−10	−16	−5	−56	0	−27	15	2	−2	−29	−11	−48	−18	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 7, 79, 553]: [0, 1, 7, 54] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 4 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(553, 2)[2]   # 553.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 553:2 1000