Atlas › Modular forms › Weight 2 › Level 902 › 902.2.a › 902.2.a.f

Newform orbit 902.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{3} + 44x^{2} + 28x - 2591$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
11	$-1$
41	$-1$

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	−1	−3	−3	3	−4	1	2	4	2	−11	−16	3	4	−6	−5	8	1	−22	−9	−18	−10	−4	23	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.468 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, 11, 22, 41, 82, 451, 902]: [0, 1, 2, 5, 4, 12, 44, 130] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 9 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(902, 2)[5]   # 902.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 902:2 1000