Atlas › Modular forms › Weight 2 › Level 521 › 521.2.a › 521.2.a.b

Newform orbit 521.2.a.b

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q + q^{2} + 3q^{3} + 43q^{4} - q^{5} + 7q^{6} + 26q^{7} + 36q^{9} + 8q^{10} + 14q^{11} - 3q^{12} + 10q^{13} - 9q^{14} + 12q^{15} + 67q^{16} - 3q^{17} - 8q^{18} + 22q^{19} - 21q^{20} - 2q^{21} + 15q^{22} + 24q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} - 4x^{28} - 111x^{27} + 461x^{26} + 5095x^{25} - 21773x^{24} - 128913x^{23} + 561875x^{22} + 2021212x^{21} - 8867281x^{20} - 20951966x^{19} + 90454484x^{18} + 149612772x^{17} - 612623050x^{16} - 754460763x^{15} + 2775715805x^{14} + 2708607364x^{13} - 8341358989x^{12} - 6844085520x^{11} + 16199747646x^{10} + 11784432836x^{9} - 19309001614x^{8} - 13046598374x^{7} + 12762977266x^{6} + 8373324855x^{5} - 3697702544x^{4} - 2500068931x^{3} + 155360803x^{2} + 150580769x + 582871$

Atkin–Lehner signs

$p$	$w_p$
521	$-1$

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
$\operatorname{tr} a_p$	1	3	−1	26	14	10	−3	22	24	12	49	5	−11	42	8	−11	−12	20	13	21	−6	150	−12	−35	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.2 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, 521]: [0, 44] (certified at ell = 2147482481)
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(521, 2)[1]   # 521.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 521:2 1000