Atlas › Modular forms › Weight 2 › Level 494 › 494.2.a › 494.2.a.g

Newform orbit 494.2.a.g

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{3} - 3x^{2} - 2097x + 9099$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	$+1$
19	$+1$

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
$\operatorname{tr} a_p$	3	5	5	−2	−1	−3	−5	−3	−1	5	9	−12	3	4	8	1	6	−12	16	−3	−8	12	5	−21	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 13, 19, 26, 38, 247, 494]: [0, 1, 1, 2, 5, 7, 24, 74] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 17 = 1 x 17 (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(494, 2)[6]   # 494.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 494:2 1000