Atlas › Modular forms › Weight 4 › Level 179 › 179.4.a › 179.4.a.a

Newform orbit 179.4.a.a

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q - 2q^{2} - 9q^{3} + 30q^{4} - 45q^{5} - 46q^{6} - 44q^{7} - 45q^{8} + 68q^{9} - 141q^{10} - 51q^{11} - 163q^{12} - 313q^{13} - 108q^{14} - 218q^{15} - 58q^{16} - 262q^{17} - 119q^{18} - 427q^{19} - 305q^{20} - 588q^{21} - 276q^{22} - 166q^{23} - 611q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} + 11x^{16} - 240x^{15} - 2374x^{14} + 22642x^{13} + 182879x^{12} - 1127218x^{11} - 6263634x^{10} + 31299768x^{9} + 93255228x^{8} - 433371096x^{7} - 477100726x^{6} + 2245936345x^{5} + 665713679x^{4} - 3314178560x^{3} + 1385196050x^{2} - 15251250x - 6890625$

Atkin–Lehner signs

$p$	$w_p$
179	$-1$

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	−9	−45	−44	−51	−313	−262	−427	−166	−521	−498	−773	−244	−1217	359	235	−545	−3811	−7	556	−842	−1134	1747	−2770	−1054

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.128 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, 179]: [1, 46] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 44 = 1 x 44 (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(179, 4)[0]   # 179.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 179:4 1000