Atlas › Modular forms › Weight 4 › Level 157 › 157.4.a › 157.4.a.a

Newform orbit 157.4.a.a

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 15q^{2} - 23q^{3} + 59q^{4} - 25q^{5} - 27q^{6} - 55q^{7} - 156q^{8} + 107q^{9} - 8q^{10} - 173q^{11} - 141q^{12} - 108q^{13} - 99q^{14} - 210q^{15} + 99q^{16} - 231q^{17} - 284q^{18} - 247q^{19} - 375q^{20} - 316q^{21} - 387q^{22} - 683q^{23} - 448q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} + 38x^{17} + 351x^{16} - 3064x^{15} - 60247x^{14} - 38558x^{13} + 3197723x^{12} + 9207351x^{11} - 70946691x^{10} - 262573383x^{9} + 576754029x^{8} + 2306434808x^{7} + 5702758x^{6} - 5299288274x^{5} - 4693966195x^{4} + 258235703x^{3} + 1912138744x^{2} + 815853509x + 106339027$

Atkin–Lehner signs

$p$	$w_p$
157	$-1$

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
$\operatorname{tr} a_p$	−15	−23	−25	−55	−173	−108	−231	−247	−683	−793	22	−374	−358	−422	−1064	−815	−1296	463	−2331	376	510	−801	−4254	−1395	−1990

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 157]: [1, 41] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 39 = 1 x 39 (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(157, 4)[0]   # 157.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 157:4 1000