Atlas › Modular forms › Weight 2 › Level 821 › 821.2.a › 821.2.a.a

Newform orbit 821.2.a.a

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q - 3q^{2} - 14q^{3} + 15q^{4} - 11q^{5} - 12q^{6} - 21q^{7} - 6q^{8} + 17q^{9} - 19q^{10} - 22q^{11} - 19q^{12} - 12q^{13} - 6q^{15} - 5q^{16} - q^{17} - 2q^{18} - 68q^{19} - 25q^{20} - 10q^{21} - 10q^{22} - 6q^{23} - 25q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} + 17x^{26} + 73x^{25} - 290x^{24} - 2945x^{23} - 2097x^{22} + 38583x^{21} + 84584x^{20} - 233566x^{19} - 828275x^{18} + 629024x^{17} + 4122941x^{16} - 61801x^{15} - 11850347x^{14} - 3951484x^{13} + 20248079x^{12} + 10044249x^{11} - 20250387x^{10} - 11047719x^{9} + 11369113x^{8} + 5812453x^{7} - 3443032x^{6} - 1470014x^{5} + 522077x^{4} + 162182x^{3} - 32956x^{2} - 5652x + 316$

Atkin–Lehner signs

$p$	$w_p$
821	$+1$

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
$\operatorname{tr} a_p$	−3	−14	−11	−21	−22	−12	−1	−68	−6	−9	−70	−25	−13	−27	6	5	−54	−33	−72	−35	−8	−50	−6	−18	−36

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 821]: [0, 69] (certified at ell = 2147455361)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (phi(ord eps) x dim S_k^new); 7 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(821, 2)[0]   # 821.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 821:2 1000