Atlas › Modular forms › Weight 4 › Level 154 › 154.4.a › 154.4.a.f

Newform orbit 154.4.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} - 5q^{3} + 8q^{4} - 7q^{5} + 10q^{6} + 14q^{7} - 16q^{8} + 27q^{9} + 14q^{10} - 22q^{11} - 20q^{12} + 46q^{13} - 28q^{14} - 51q^{15} + 32q^{16} - 88q^{17} - 54q^{18} - 46q^{19} - 28q^{20} - 35q^{21} + 44q^{22} - 115q^{23} + 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{137})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{2} + 826x + 98096$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$-1$
11	$+1$

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
$\operatorname{tr} a_p$	−4	−5	−7	14	−22	46	−88	−46	−115	−372	−137	−19	−440	192	728	−808	−35	312	−301	−137	788	−366	−2324	1235	709

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.213 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 engine's checks

dim M^+ at levels [1, 2, 7, 11, 14, 22, 77, 154]: [1, 2, 3, 4, 8, 11, 26, 76] (certified at ell = 2147482951)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 6 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(154, 4)[5]   # 154.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 154:4 1000