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

Newform orbit 179.4.a.b

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q + 4q^{2} + 9q^{3} + 130q^{4} + 35q^{5} + 50q^{6} + 68q^{7} + 27q^{8} + 338q^{9} + 119q^{10} + 59q^{11} + 53q^{12} + 311q^{13} + 116q^{14} - 38q^{15} + 822q^{16} + 146q^{17} + 151q^{18} + 447q^{19} + 335q^{20} + 504q^{21} + 252q^{22} + 110q^{23} + 541q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} - 13x^{26} - 672x^{25} + 8790x^{24} + 188321x^{23} - 2435314x^{22} - 29794744x^{21} + 369207014x^{20} + 3030845625x^{19} - 34200815454x^{18} - 213154547348x^{17} + 2034521739876x^{16} + 10741612837805x^{15} - 79011158835486x^{14} - 388258532942824x^{13} + 1979172839514176x^{12} + 9776225098378369x^{11} - 30514917055488630x^{10} - 162546476683702914x^{9} + 259764918589708682x^{8} + 1649072399583355001x^{7} - 898413000353759370x^{6} - 9050320671917762982x^{5} - 795358445455265892x^{4} + 21472782517130062356x^{3} + 6603278691636885427x^{2} - 13646401920029642786x - 2791607129110945302$

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 even 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	9	35	68	59	311	146	447	110	639	680	781	1232	417	−587	233	−459	3817	−149	−332	2054	2754	−1191	−658	1934

$\operatorname{tr} a_n$ 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)[1]   # 179.4.a.b

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

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