Atlas › Modular forms › Weight 4 › Level 239 › 239.4.a › 239.4.a.a

Newform orbit 239.4.a.a

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q - 4q^{2} - 13q^{3} + 50q^{4} - 37q^{5} - 42q^{6} - 52q^{7} - 69q^{8} + 69q^{9} - 93q^{10} - 77q^{11} - 152q^{12} - 218q^{13} - 111q^{14} - 142q^{15} - 42q^{16} - 219q^{17} - 291q^{18} - 476q^{19} - 314q^{20} - 474q^{21} - 390q^{22} - 202q^{23} - 497q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} + 17x^{21} - 258x^{20} - 5009x^{19} + 28888x^{18} + 617005x^{17} - 2034190x^{16} - 41390734x^{15} + 107841588x^{14} + 1632029584x^{13} - 4275106893x^{12} - 37744643110x^{11} + 112010842410x^{10} + 478614868545x^{9} - 1712417124428x^{8} - 2738309177111x^{7} + 13487828332902x^{6} + 2049227391934x^{5} - 45589459911950x^{4} + 25835353515257x^{3} + 46139351873401x^{2} - 42755616102699x + 4702888740782$

Atkin–Lehner signs

$p$	$w_p$
239	$-1$

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	−13	−37	−52	−77	−218	−219	−476	−202	−307	−1001	−922	−1188	−192	−102	−580	−1528	−1631	−689	341	−2260	396	1065	−1984	−4946

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 239]: [1, 61] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 59 = 1 x 59 (phi(ord eps) x dim S_k^new); 8 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(239, 4)[0]   # 239.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 239:4 1000