Atlas › Modular forms › Weight 4 › Level 172 › 172.4.a › 172.4.a.a

Newform orbit 172.4.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 11q^{3} - q^{5} - 8q^{7} + 36q^{9} - 33q^{11} - 83q^{13} - 216q^{15} - 210q^{17} - 245q^{19} - 62q^{21} - 318q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} + 12x^{4} - 63x^{3} - 794x^{2} + 804x + 7896$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 172$ )
43	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	−11	−1	−8	−33	−83	−210	−245	−318	−79	−388	−131	6	−215	−657	627	−380	1542	981	−494	−178	143	425	−1002	1546

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.118 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, 4, 43, 86, 172]: [1, 2, 3, 12, 35, 69] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(172, 4)[0]   # 172.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 172:4 1000