Atlas › Modular forms › Weight 4 › Level 119 › 119.4.a › 119.4.a.c

Newform orbit 119.4.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 2q^{2} - 7q^{3} - 2q^{4} - 9q^{5} + 16q^{6} + 28q^{7} - 24q^{8} - 29q^{9} - 26q^{10} - 26q^{11} - 46q^{12} - 88q^{13} - 14q^{14} - 86q^{15} - 126q^{16} - 68q^{17} - 110q^{18} - 258q^{19} - 22q^{20} - 49q^{21} - 116q^{22} + 14q^{23} + 162q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 9x^{3} - 30x^{2} - 103x + 127$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
17	$+1$

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
$\operatorname{tr} a_p$	−2	−7	−9	28	−26	−88	−68	−258	14	198	−299	−360	−219	121	−584	317	546	−1359	159	352	−1189	−496	546	−244	−4807

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 7, 17, 119]: [1, 3, 6, 38] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(119, 4)[2]   # 119.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 119:4 1000