Atlas › Modular forms › Weight 2 › Level 822 › 822.2.a › 822.2.a.i

Newform orbit 822.2.a.i

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{3} + 7x^{2} - 56x - 16$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
137	$+1$

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	−3	−4	−1	5	−2	−3	8	−17	−6	1	−17	−16	6	−4	−31	−7	7	2	−11	−2	−11	−15	13	−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 0.353 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, 2, 3, 6, 137, 274, 411, 822]: [0, 1, 1, 3, 12, 36, 48, 142] (certified at ell = 2147483249)
T = [(5, 1), (7, 3)]; h of degree 23 = 1 x 23 (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(822, 2)[8]   # 822.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 822:2 1000