Atlas › Modular forms › Weight 2 › Level 523 › 523.2.a › 523.2.a.c

Newform orbit 523.2.a.c

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 9q^{2} + 4q^{3} + 31q^{4} + 21q^{5} + 3q^{7} + 27q^{8} + 36q^{9} + 11q^{11} + 14q^{13} + q^{14} + q^{15} + 33q^{16} + 24q^{17} + 14q^{18} - 5q^{19} + 27q^{20} - 8q^{21} - 10q^{22} + 36q^{23} - 20q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 13x^{25} - 14x^{24} + 835x^{23} - 2029x^{22} - 19837x^{21} + 84058x^{20} + 208574x^{19} - 1424220x^{18} - 660108x^{17} + 13080245x^{16} - 6154466x^{15} - 70400782x^{14} + 72403988x^{13} + 223794372x^{12} - 325236457x^{11} - 398974150x^{10} + 767406894x^{9} + 331931261x^{8} - 973707126x^{7} - 11377978x^{6} + 611460916x^{5} - 148070520x^{4} - 142348464x^{3} + 59504064x^{2} - 1724672x - 537728$

Atkin–Lehner signs

$p$	$w_p$
523	$-1$

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
$\operatorname{tr} a_p$	9	4	21	3	11	14	24	−5	36	42	−8	20	−1	2	25	119	−4	−4	3	−7	−19	−16	−12	7	−24

$\operatorname{tr} a_n$ 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, 523]: [0, 44] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (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(523, 2)[2]   # 523.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 523:2 1000