Atlas › Modular forms › Weight 2 › Level 741 › 741.2.a › 741.2.a.j

Newform orbit 741.2.a.j

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{5} + 4x^{4} - 159x^{3} - 591x^{2} + 6004x + 19757$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
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 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
$\operatorname{tr} a_p$	−1	−5	−1	1	−13	−5	−1	−5	−2	−10	−8	−8	−6	3	3	−8	−20	−19	−14	−16	21	−18	−38	−26	18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 3, 13, 19, 39, 57, 247, 741]: [0, 1, 1, 2, 6, 8, 24, 96] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 35 = 1 x 35 (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(741, 2)[9]   # 741.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 741:2 1000